- Email: [email protected]
Optimal investment for insurer with jump-diffusion risk process
Insurance: Mathematics and Economics 37 (2005) 615–634
Optimal investment for insurer with jump-diffusion risk process Hailiang Yanga,∗ , Lihong Zhangb a
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China b School of Economics and Management, Tsinghua University, Beijing, PR China Received September 2004; received in revised form June 2005; accepted 9 June 2005
Abstract In this paper, we study optimal investment policies of an insurer with jump-diffusion risk process. Under the assumptions that the risk process is compound Poisson process perturbed by a standard Brownian motion and the insurer can invest in the money market and in a risky asset, we obtain the close form expression of the optimal policy when the utility function is exponential. We also study the insurer’s optimal policy for general objective function, a verification theorem is proved by using martingale optimality principle and Ito’s formula for jump-diffusion process. In the case of minimizing ruin probability, numerical methods and numerical results are presented for various claim-size distributions. © 2005 Elsevier B.V. All rights reserved. MSC: IM12; IM50 Keywords: Hamilton–Jacobi–Bellman equations; Martingale; Utility; Jump-diffusion; Ito’s formula; Stochastic control
1. Introduction In recent years there has been an increasing attention in the utilization of stochastic control theory to insurance related problems. This is due to the facts that insurance companies can invest in the stock market, and can pay dividend to maximize (or minimize) a certain objective function under different constraints. The classical works on this subject are those by Gerber (1969), Buhlmann (1970), Dayananda (1970), Martin-lof (1973, 1983, 1994) and Jeanblanc-Picqu´e and Shiryaev (1995). Browne (1995) considers a model in which the aggregate claims are modelled by a Brownian motion with drift, and the risky asset is modelled by a geometric Brownian motion (see also Browne, 1997, 1999). Asmussen and Taksar (1997) consider maximizing expected value of discounted dividend paid until time of ruin under a Brownian motion with drift model. Paulsen and Gjessing (1997) consider the similar problem, but allow stochastic returns on investment in their model, and in Paulsen (2003) the same problem is studied ∗
Corresponding author. Tel.: +852 2857 8322; fax: +852 2858 9041. E-mail addresses: [email protected] (H. Yang), [email protected] (L. Zhang).
0167-6687/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2005.06.009
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
616
when dividends paid are subject to solvency constraints. It is well known that the compound Poisson process is the most popular and a simple model in risk theory, Hipp and Taksar (2000) use the compound Poisson process to model the insurance business and consider the problem of optimal choice of new business to minimize the ultimate ruin probability. Hipp and Plum (2000) use the classical Cram´er–Lundberg model to describe the surplus of the insurance company and assume that the surplus of the insurance company can be invested in a risky asset (market index) which follows a geometric Brownian motion. In the case of exponential distributed claim-size, explicit solutions can be obtained. However, they do not incorporate a risk-free asset in their model. Liu and Yang (2004) reconsider the model in Hipp and Plum (2000) incorporate a risk-free interest rate. In this case, closed-form solution cannot be obtained, they provide numerical results for optimal policy for maximizing survival probability under different claim-size distribution assumptions. Taksar (2000) presents a survey of stochastic models of risk control and dividend optimization techniques for a financial corporation, the objective in his paper is to maximize the dividend pay-out. For more recent related papers see, for example, Gerber and Shiu (2004), Højgaard and Taksar (1998a,b, 2000), Taksar and Markussen (2003), Schmidli (2002), Hipp and Plum (2003), Irgend and Paulsen (2004), Moore and Young (2004) and Milevsky et al. (2004). In this paper, we use a jump-diffusion risk process, where the diffusion term denotes the uncertainty associated with the surplus of the insurance company, to model the surplus of the insurance company and study the portfolio selection problems for an insurer. The three optimization criterions are: (1) maximizing exponential utility function for a given terminal time; (2) maximizing the survival probability (equivalently, minimizing the ruin probability) of insurance company; (3) a general objective function. Controlled jump-diffusion models of stochastic control theory are used. For the first criterion, similar to Browne (1995), a closed-form solution is obtained, but for the second criterion, similar to Liu and Yang (2004), we are unable to obtain any closed-form solution. By performing some helpful transformations similar to those in Hipp and Plum (2003), we present some numerical results. For the general objective function, a verification theorem is proved by using martingale optimality principle and Ito’s formula. Here is a brief outline of this paper. In Section 2 the model assumptions are formulated. In Section 3, we study the problem of maximizing exponential utility function, a closed-form expression for the optimal investment amount is obtained. In Section 4, we discuss a general optimal control problem. As a special case of the general optimal control, the problem of maximizing survival probability is studied in Section 5.
2. The model In this paper, we assume that the standard assumptions of continuous-time financial models hold, that is 1. continuous trading is allowed; 2. no transaction cost or tax is involved in trading; and 3. all assets are infinitely divisible. For simplicity, we assume that there are only two assets available for investors in the financial market: a risk-free asset, whose price at time t is denoted by B(t), and a risky asset, whose price at time t is denoted by P(t). B(t) is assumed to follow dB(t) = r0 B(t) dt, where r0 , the risk-free interest rate, is assumed to be a positive constant. The price of the risky asset is assumed to follow the stochastic differential equation (1)
dP(t) = µP(t) dt + σP(t) dWt ,
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
617
where µ and σ are positive constants represent the expected instantaneous rate of return of the risky asset and the (1) volatility of the risky asset price, respectively, and {Wt : t ≥ 0} is a standard Brownian motion defined on the complete probability space (Ω, F, P). As usual, we assume that µ > r0 . We are concerned with investment behavior in the presence of a stochastic cash flow, described by a jump-diffusion process, which we denoted by {R(t) : t ≥ 0}, satisfies (2)
dR(t) = α dt + β dWt
− dS(t), (2)
where α and β are constants (with β > 0), and {Wt : t ≥ 0} is another standard Brownian motion defined on the same complete probability space (Ω, F, P). The assumption β > 0 is essential in this paper (a boundary condition for the value function needs this assumption). However, this assumption excludes the classical compound Poisson model. We assume that the joint distribution of these two Brownian motions is bivariate normal, and we denote (1) (2) their correlation coefficient by ρ, i.e., E[Wt Wt ] = ρt. We will not consider the uninteresting case of ρ2 = 1, N(t) in such case there would only be one source of randomness in the model; S(t) = i=1 Yi is a compound Poisson process, denotes cumulated claims in time interval [0, t], where N(t), the number of claims occurring in the time interval [0, t], is a homogeneous Poisson process with intensity λ, and Yi , the size of the ith claim, is assumed to (1) (2) be independent of the claim number process, independent of the two Brownian motions Wt and Wt above, and independent identically distributed with common distribution function F, where F satisfies F (0) = 0. Moreover, ∞ (2) we assume that F has a mean value of µ1 and µ2 = 0 x2 dF (x) < +∞. The diffusion term βWt stands for the uncertainty associated with the surplus of the insurance company at time t. The company is allowed to invest in the risky asset and risk-free asset, and let At be the total amount of money invested in the stock market at time t when the company’s wealth is Xt , where {At : t ≥ 0} is a suitable, admissible adapted process, i.e., At is a nonanticipative function that satisfies T [At ]2 dt < ∞ a.e. for all T < ∞. 0
Let Xt denote the wealth of the company at time t, if it follows investment policy A(t), with initial wealth X0 = x. This process then evolves as dXt = At
dP(t) dB(t) + (Xt − At ) + dR(t), P(t) B(t)
or, more explicitly, (1)
dXt = [(µ − r0 )At + r0 Xt + α] dt + σAt dWt X0 = x. (1)
(2)
+ β dWt
− dS(t) (2.1)
(2)
Since Wt and Wt are correlated standard Brownian motions with correlation coefficient ρ, applying It oˆ formula for jump-diffusion process (see Cont and Tankov, 2003) to function g(t, x), we obtain that, the generator is AA g(t, x) = gt + [A(µ − r0 ) + α + r0 x]gx + [g(t, xt− + xt ) − g(t, xt− )] + 21 [A2 σ 2 + β2 + 2ρσβA]gxx , (2.2) where gx , gt and gxx denote the first order partial derivative with respect to x, the first order partial derivative with respect to t, and the second order partial derivative with respect to x respectively. Note that so long as ρ2 = 1, this model is incomplete in a very strong sense that the random cash flow, R(t), cannot be traded on the security market, and hence the risk to the investor (insurer) cannot be eliminated under any circumstance.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
618
3. Maximizing expected exponential utility of terminal wealth Suppose that the investor is interested in maximizing the utility function for his terminal wealth, say at time T. The utility function is u(x), here we assume that u > 0 and u < 0. Let V (t, x) denote the maximal utility attainable by the investor from the state x at time t, i.e., V (t, x) =
sup
{A(s):s≥t}
E[u(XTA )|XtA = x].
(3.1)
where {XtA : t ≥ 0} is the wealth process under the policy A. Since u is a concave function, there exists a unique optimal investment policy A∗ such that the expected utility reaches to its maximum. Suppose that the investor has an exponential utility function u(x) = c −
δ −γx e , γ
(3.2)
where δ > 0 and γ > 0. This utility has constant absolute risk aversion parameter γ, such utility functions play a prominent role in insurance mathematics and actuarial practice, since they are the only utility functions under which the principle of “zero utility” gives a fair premium that is independent of the level of reserves of an insurance company (see Gerber, 1979, p. 68 or Goovaerts et al., 1990, Section II.6). For the problem of maximizing utility of terminal wealth at a fixed terminal time T, the Hamilton–Jacobi–Bellman (HJB) equation becomes, for t < T (see Fleming and Rishel, 1975 or Fleming and Soner, 1993): sup{AA V (t, x)} = 0,
V (T, x) = u(x).
(3.3)
A
In other words, for each (t, x), we must solve the nonlinear partial differential equation (3.3), and find the value A which maximizes the function Vt + [α + r0 x + A(µ − r0 )]Vx + 21 [A2 σ 2 + β2 + 2ρσβA]Vxx + λE[V (t, x − Y ) − V (t, x)].
(3.4)
It is well known in stochastic control theory that, in some cases, a value function is not smooth enough to satisfy the HJB equation in the classical sense. A commonly used weak formulation of solution to the HJB equation is called the viscosity solution proposed by Crandall and Lions (1983). For more detailed discussions on viscosity solutions, see Crandall et al. (1992) and Fleming and Soner (1993). However, in many cases, we can prove that the value function is smooth enough. Similar method as that in Hipp and Plum (2003) can be used to prove that the HJB equation has a classical solution in our case. Here we will not discuss this problem in detail. Let us assume that the HJB equation (3.3) has a classical solution V, which satisfies Vx > 0 and Vxx < 0. Then differentiating with respect to A in (3.4) gives the optimizer A∗ = −
µ − r 0 Vx ρβ − . σ 2 Vxx σ
(3.5)
Plugging (3.5) into HJB equation (3.4), after simplification, we have ρβ(µ − r0 ) 1 µ − r0 2 Vx2 1 Vt + α + r0 x − Vx − + β2 (1 − ρ2 )Vxx + λE[V (t, x − Y ) − V (t, x)] = 0 σ 2 σ Vxx 2 for t < T. To solve Eq. (3.6), inspired by Browne (1995), we try to fit a solution of the form 2 δ 1 µ − r 0 V (t, x) = c − exp −γx er0 (T −t) − (T − t) + h(T − t) , γ 2 σ
(3.6)
(3.7)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
619
where h(·) is a suitable function such that (3.7) is a solution of (3.6), and the boundary condition V (T, x) = u(x) = c −
δ −γx e γ
implies that h(0) = 0. Note that for this solution, we have Vx = [V (t, x) − c][−γ er0 (T −t) ],
Vxx = [V (t, x) − c][γ 2 e2r0 (T −t) ], 2 1 µ − r 0 Vt = [V (t, x) − c] xr0 γ er0 (T −t) + − h (T − t) . 2 σ
Substitute the values for Vx and Vxx into (3.5), we find the optimal investment policy is given by A∗ =
µ − r0 −r0 (T −t) ρβ e − . γσ 2 σ
(3.8)
Note that, in this situation, the amount invested in the risky asset is independent of the wealth level. This is due to the fact that the exponential utility function has constant absolute risk aversion. When the utility function is exponential, we conclude that even if there are claims that should be paid by the insurer, the investment amount on risky asset is independent of insurer’s wealth when there is only one risky asset and one risk-free asset in the market. This conclusion is the same as that in the case of individual’s simple investment behavior, see Huang and Litzenberger (1987). It is interesting to observe that this policy invests more as the deadline gets closer, this is consistent with intuition. In order to obtain the value function, we need to calculate λE[V (t, x − Y ) − V (t, x)] = λ[V (t, x) − c]
∞
r0 (T −t)
exp{γy e
} dF (y) − 1
0
= λ[V (t, x) − c]E[exp{γY er0 (T −t) } − 1]. Plugging the values of Vx , Vxx and Vt into (3.6), together with the result above, this leads to that we require ρβ(µ − r0 ) r0 (T −t) 1 2 2 e + γ β (1 − ρ2 ) e2r0 (T −t) + λE[exp{γY exp{r0 (T − t)}} − 1] h (T − t) = −γ α − σ 2
(3.9) together with the initial condition h(0) = 0. If we know the distribution of Y, we can obtain the close form expression of h(·). When the investment is optimal, the optimal wealth process follows (µ − r0 )2 −r0 (T −t) (µ − r0 )ρβ ∗ dXt∗ = e + α + r − X 0 t dt γσ 2 σ µ − r0 −r0 (T −t) (1) (2) + − ρβ dWt + β dWt − dS(t) e γσ
(3.10)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
620
(µ − r0 )2 −r0 (T −t) (µ − r0 )ρβ ∗ = e − + α + r0 X (t) dt γσ 2 σ µ − r0 −r (T −t) 2 (3) 0 e + + β2 (1 − ρ2 ) dWt − dS(t), γρ
(3.11)
where
(3)
Wt
(1) (2) e−r0 (T −t) − ρβ Wt + βWt = 2 µ−r0 −r0 (T −t) e + β2 (1 − ρ2 ) γρ µ−r0 γσ
(1)
(2)
is another standard Brownian motion on the same probability space (Ω, F, P) by the assumptions on Wt and Wt . Since this is a linear stochastic differential equation plus a jump process (compound Poisson process), standard results (e.g., Krylov, 1980 or Karatzas and Shreve, 1988) tell us it admits the unique strong solution. This, together with the fact that the value function is twice-continuously differentiable, gives the following theorem. Theorem 1. Under our assumptions, and assume that the objective is to maximize utility of terminal wealth, at the fixed terminal time T, and the utility function is given by u(x) = c −
δ −γx e , γ
then the optimal value function is given by δ 1 µ − r0 2 r0 (T −t) V (t, x) = c − exp −γx e − (T − t) + h(T − t) , γ 2 σ
(3.12)
where the function h(·) is given by (3.9) with initial condition h(0) = 0, and the optimal policy is to invest A∗ =
µ − r0 −r0 (T −t) ρβ e − γσ 2 σ
(3.13)
in the risky asset at time t. Proof. It is not difficult to check that all the conditions of the verification theorem are satisfied (see Theorem VI 4.2 in Fleming and Rishel (1975)). Therefore the investment strategy given by (3.13) is optimal. It is instructive to verify the optimality by using the martingale optimality principle, which entails finding an appropriate functional which is a uniformly integrable martingale under the optimal policy, but a super-martingale under any other admissible policy. By applying Ito’s formula, we have that {V (t, XtA ) − c : 0 < t < T } is a super-martingale under any admissible policy A. Note that we have a boundary condition V (T, XtA ) = u(XtA ), the proof of the theorem will be completed if we have the following lemma. Lemma 1. Under the optimal policy, the process {V (t, Xt∗ ) − c : t ≤ T } is a martingale. Proof. Apply Ito’s formula for jump-diffusion process (see Cont and Tankov, 2004) to the function V (t, Xt∗ ) − c, we obtain
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
d[V (t, Xt∗ ) − c] =
621
1 Vt + [r0 Xt∗ + γ1 + γγ2 e−r0 (T −t) ]Vx + [γ3 + γ2 e−2r0 (T −t) ]Vxx dt 2 (3) + γ3 + γ2 e−2r0 (T −t) Vx dWt + λE[V (t, Xt∗ − Y ) − V (t, Xt∗ )],
(3.14)
2 µ−r0 where γ1 = α − ρβ , and γ3 = β2 (1 − ρ2 ). Note that the last expectation is taken only with σ (µ − r0 ), γ2 = γσ respect to random variable Y. Plugging all the expressions of Vx , Vxx , Vt and h(·) into (3.14) and simplify to obtain (3) d[V (t, Xt∗ ) − c] = [V (t, Xt∗ ) − c][−γ γ2 + γ3 e2r0 (T −t) dWt ]. (3.15) This is a linear stochastic differential equation with solution 2 t T V (t, Xt∗ ) − c γ 2r0 (T −u) (3) 2r (T −u) 0 . = exp − (γ2 + γ3 e ) du − γ γ2 + γ 3 e dWu V (0, X0∗ ) − c 2 0 0
(3.16)
This is the exponential martingale. Hence, for all t ≤ T , the process {V (t, Xt∗ ) − c} is a martingale. By the martingale optimality principle, the theorem is proved.
4. A general optimal control problem Many problems considered in the literature so far, such as maximizing the survival probability, minimizing the expected (discounted) penalty of ruin, etc., are special cases of the optimal control problem below: for each admissible control process {At : t ≥ 0}, let τzA = inf{t > 0 : XtA = z} denote the first hitting time to the point z of the associated wealth process {XtA } of (2.1). For given numbers (l, u) with l < x < u, let τ A = min{τlA , τuA } denote the first escape time from the interval (l, u). For a given nonnegative continuous function θ(x) ≥ 0, given a real bounded continuous function g(x) and a function h(x), given l and u, let V A (x) be defined by
A A τ
V A (x) = Ex
0
−
g(XtA ) e
t
0
θ(XtA ) ds
−
dt + h(XτAA ) e
τ
0
θ(XsA ) ds
(4.1)
with V (x) = supA∈G V A (x). Here we assume that the optimal strategy exists and let A∗ = arg supA∈G V A (x), where G denotes the set of admissible policies, Ex [·] denotes conditional expectation given X0 = x. We note that we are only interested in controls with V A (x) < ∞. Theorem 2. Suppose that w(x) : (l, u) → (−∞, +∞) is a C2 function and is the concave increasing (i.e., wx > 0, wxx < 0) solution to the equation ρβ(µ − r0 ) 1 µ − r0 2 w2x (x) 1 α + r0 x − wx (x) − + β2 (1 − ρ2 )wxx (x) + λE[w(x − Y ) − w(x)] σ 2 σ wxx (x) 2 + g(x) − θ(x)w(x) = 0
(4.2)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
622
with w(l) = h(l) and
w(u) = h(u)
(4.3)
and satisfies the following conditions: w2 (x)
1. wxxx (x) is bounded for all x ∈ (l, u), 2. there exists an integrable random variable Y such that for all t ≥ 0, w(XtA ) ≥ Y , 3. wwxxx (x) (x) is locally Lipschitz continuous, then w(x) is the optimal value function, i.e., w(x) = V (x), and the optimal investment amount A∗ can be written as A∗ = −
µ − r0 wx (x) ρβ − 2 σ wxx (x) σ
for x ∈ (l, u).
(4.4)
Proof. The HJB equation is given by sup [A(µ − r0 ) + α + r0 x]Vx + 21 [A2 σ 2 + β2 + 2ρσβA]Vxx + λE[V (x − Y ) − V (x)] + g − θV = 0,
A∈G
(4.5) subject to the boundary condition V (l) = h(l), V (u) = h(u). Assuming that (4.5) has a classical solution with Vx > 0, Vxx < 0. We can obtain the maximizer A∗ = −
µ − r0 wx (x) ρβ − σ 2 wxx (x) σ
for x ∈ (l, u).
Plugging this expression into (4.5) and after simplification, we obtain the nonlinear equation (4.2) (with V = w). Next, we need that the policy A∗ is indeed optimal. Here martingale optimal principle is used again. t to verify A A Let Θ (s, t) = s θ(xv ) dv, and define process t A A M(t, xtA ) = e−Θ (0,t) w(xtA ) + e−Θ (0,s) g(xsA ) ds (4.6) 0
for t ∈ [0, τ A ], where w is the concave and increasing solution of (4.2). Optimality of A∗ is then a direct consequence of the following lemma.
Lemma 2. For any admissible policy A, we have A E[M(t ∧ τ A , Xt∧τ A )] ≤ M(0, X0 ) = w(x)
(4.7)
with equality holding if and only if A = A∗ , where A∗ is the policy given by (4.4). Moreover, under the optimal ∗ policy A∗ , the process {M(t ∧ τ A , Xt∧τ A )} is a uniform integrable martingale. Proof of Lemma 2. First, we rewrite (2.1) into the following form: (4) dXtA = [(µ − r0 )A + r0 XtA + α] dt + σ 2 A2 + β2 + 2ρβσA dWt − dS(t), where (1)
(4)
Wt
(2)
σAWt + βWt = σ 2 A2 + β2 + 2ρβσA
(4.8)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
623 (1)
is a standard Brownian motion on the same probability space (Ω, F, P) due to the assumptions on Wt (2) and Wt . Applying Ito’s formula for jump-diffusion process (2.1) to M(t, xtA ), we have that, for 0 ≤ s ≤ t ≤ τ A , M(t, XtA ) = M(s, XsA ) +
+
s≤Ti ≤t
t
e−Θ
A (s,v)
s
Q(A, XvA ) dv +
t
e−Θ
A (s,v)
s
σ 2 A2 + 2ρσβA + β2 wx (XvA ) dWv(4)
E[M(Ti , XTAi − − Y ) − M(Ti −, XTAi − )].
(4.9)
Note that the last expectation is taken only with respect to Y, and Ti is the ith claim time, where Q(z; y) denotes the quadratic (in z) term defined by Q(z; y) = z2 [ 21 σ 2 wxx (y)] + z[(µ − r0 )wx (y) + ρσβwxx (y)] + [α + r0 x]wx (y) + 21 β2 wxx (y) + g(y) − θ(y)w(y).
(4.10)
Note that since Qzz (z; y) = σ 2 wxx (y) < 0, and the maximum of Qzz (z; y) is achieved at the value z∗ (y) = −
µ − r0 wx (y) ρβ − σ 2 wxx (y) σ
with corresponding maximal value ρβ(µ − r0 ) 1 µ − r0 2 w2x (x) 1 wx (x)− Q(z∗ ; y) = α + r0 x− + β2 (1 − ρ2 )wxx (x) + g(x) − θ(x)w(x) σ 2 σ wxx (x) 2 = − λE[w(x − Y ) − w(x)]. The last equality follows from (4.2). Therefore the sum of the second term and the last term on the right-hand side of (4.9) is always less than or equals to 0. Moreover,
t∧τ A
e−Θ
A (s,v)
0
(σ 2 A2 + 2ρσβA + β2 )wx (XvA ) dWv(4)
= M(t ∧ τ −
A
A , Xt∧τ A ) − w(x) −
t∧τ A
e−Θ
A (s,v)
0
Q(A, XvA ) dv
A E[M(Ti , XTAi − − Y ) − M(Ti −, XTAi − )] ≥ M(t ∧ τ A , Xt∧τ A ) − w(x).
0≤Ti ≤t∧τ A
Thus by assumption (2) we see that the stochastic integral term in (4.9) is a local martingale, and in fact it is a super-martingale. Hence, taking expectations on both sides of (4.9), and let s = 0, we have
A A E[M(t ∧ τ A , Xt∧τ A )] ≤ w(x) + E
+
0≤Ti ≤t∧τ A
t∧τ
0
e−Θ
A (s,v)
Q(A, XvA ) dv
E[M(Ti , XTAi − − Y ) − M(Ti −, XTAi − )]
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
624
≤ w(x) + E
t∧τ A
e
−ΘA (s,v)
0
+
0≤Ti
sup[Q(A, XvA )] dv A
E[M(Ti , XTAi − − Y ) − M(Ti −, XTAi − )] = w(x).
(4.11)
≤t∧τ A
The above equality holds if and only if the policy is A∗ . Note that under the optimal policy A∗ , the wealth process X∗ satisfies the following stochastic differential equation: (µ − r0 )2 wx (Xt∗ ) βρ(µ − r0 ) ∗ dXt = α + r0 x − dt − dS(t) − σ2 wxx (Xt∗ ) σ
2 ∗) µ − r0 wx (Xt∗ ) µ − r ρβ w βρ (X 0 x (4) t + − 2βρσ + + + β2 dWt σ 2 wxx (Xt∗ ) σ σ 2 wxx (Xt∗ ) σ for 0 < t < τ A , by assumption (3), this equation admits a unique strong solution. Moreover, under the optimal ∗ policy A∗ , we have, for all 0 ≤ s ≤ t ≤ τ ∗ (where τ ∗ = τ A ), M(t, Xt∗ )
=
M(s, Xs∗ ) −
t
e
−
v
θ(Xu∗ ) du s
s
µ − r0 σ
2
wx (Xv∗ ) wxx (Xv∗ )
2
+ β2 (1 − ρ2 )wx (Xv∗ ) dWv(4) .
By assumption (1) it is easy to see that expression above is a uniform integrable martingale. Thus the lemma is proved. This also completes the proof of the theorem. 5. Maximizing the survival probability In this section, we consider the problem of minimizing the probability of ruin, i.e., maximizing the survival probability. Note that this is simply a special case of the control problem solved in Theorem 2 with θ(x) = 0, g(x) = 0, u = ∞ and l = 0. Let V (x) denote the survival probability of risk process Xt with initial surplus x, then the nonlinear equation for the optimal value function V becomes, in this case, ρβ(µ − r0 ) 1 µ − r0 2 Vx2 (x) 1 2 α + r0 x − Vx (x)− + β (1 − ρ2 )Vxx (x)+λE[V (x − Y ) − V (x)] = 0. σ 2 σ Vxx (x) 2 (5.1) The same method as that in Hipp and Plum (2003) can be used to prove that the Bellman equation has a classical solution in this case. Using the results in Hipp and Plum (2003), Cai and Yang (2005) pointed out that the survival probability is twice continuously differentiable in some insurance risk models with stochastic interest rate (see also Wang and Wu, 2001). Under proper conditions on the claim-size distribution, the conditions 1 and 3 in Theorem 2 hold. Condition 2 holds automatically by taking Y = 0. So the solution of (5.1) is the optimal value function, i.e., the survival probability under optimal policy. In order to obtain the initial value easily, we assume that the coefficient correlation ρ ≤ 0 in this section. From (4.4) we can easily know that the optimal investment amount A ≥ 0. It is obvious that A∗ (0) = 0 and V (0) = 0 due to the fluctuation property of the Brownian motion. Henceforth for maximizing the survival probability, we need to solve the nonlinear equation (5.1) with initial condition V (0) = 0. In the following, we present numerical solutions of (5.1) under different assumptions on the claim-size distributions.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
625
We rewrite HJB equation (5.1) in a form that is more readily solvable. 2 λ α 0 2 ¯ r0 ¯ and Rβ 0 by β¯ 2 , (5.1) becomes Denoting ( µ−r σ ) by R0 , R0 by λ, R0 by r¯0 , R0 by α 2 1 ¯ x (x) = 1 Vx (x) . ¯ λE[V (x − Y ) − V (x)] + β¯ 2 (1 − ρ2 )Vxx (x) + [α¯ + r¯0 x − ρβ]V 2 2 Vxx (x)
(5.2)
Let F (y) be the distribution function of the claim-size, H(y) = 1 − F (y), then (5.2) can be rewritten as x 2 1 ¯2 2 ¯ ¯ ¯ (0)H(x) = 1 Vx (x) . (α¯ + r¯0 x − ρβ)Vx (x) + β (1 − ρ )Vxx (x) − λ Vx (x − y)H(y) dy − λV 2 2 Vxx (x) 0 Let u(x) = Vx , then 1 ¯ (α¯ + r¯0 x − ρβ)u(x) + β¯ 2 (1 − ρ2 )u (x) − λ¯ 2
x
¯ (0)H(x) = u(x − y)H(y) dy − λV
0
1 u2 (x) . 2 u (x)
(5.3)
(5.4)
The following procedures are essentially the same as those proposed by Hipp and Plum (2000) (see also Liu and ¯ are incorporated. The transforms to be used in Yang, 2004) except that extra terms 21 β¯ 2 (1 − ρ2 )u (x) and −ρβu(x) the following subsections are the same as those in Hipp and Plum (2003). 5.1. Exponential claim-size distribution Assume that the probability density function of the claim size is f (y) = k e−ky , then F (y) = 1 − e−ky , H(y) = Let v(y) = u(y) eky , then
e−ky .
v (y) = u (y) eky + kv(y),
v (y) − kv(y) u (y) = . v(y) u(y)
Substituting them into Eq. (5.4) yields s 2 ¯ ¯ (0) + 1 β¯ 2 (1 − ρ2 )(v (s) − kv(s)) = 1 (v(s) ) . − λ¯ v(y) dy + (¯r0 s + α¯ − ρβ)v(s) − λV 2 2 v (s) − kv(s) 0 2 v(s) , since u = Vxx < 0 and u = Vx > 0, we have Let w(s) = v (s)−kv(s)
(5.5)
(5.6)
v (y) = u (y) eky + kv(y) < kv(y). Thus √
1 kv(s) − v (s) v (s) = =k− . v(s) v(s) w(s)
From Eq. (5.6), we have s 1 −v(s) ¯ ¯ (0) + 1 β¯ 2 (1 − ρ2 ) √ λ¯ v(y) dy + (¯r0 s + α¯ − ρβ)v(s) − λV = − v(s) w(s). 2 2 w(s) 0
(5.7)
(5.8)
Differentiate both sides with respect to s and then divide both sides by v(s), we obtain v (s) √ v(s) 1 √ w (s) (s) 1 v 1 w (s) 1 v (s) v(s) w(s) − v(s) 2 w(s) 2 2 ¯ . − β¯ (1 − ρ ) w(s) − √ =− (−λ¯ + r¯0 ) + (¯r0 s + α¯ − ρβ) v(s) 2 2 v(s) 4 w(s) w(s) (5.9)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
626
Plugging (5.7) into (5.9), we have ¯ − 1] w (s) = [β¯ 2 (1 − ρ2 ) + w(s)]−1 {−4(w(s))3/2 [−λ¯ + r¯0 + k(¯r0 s + α¯ − ρβ) 2 √ 1 ¯2 2 2 2 2 ¯ ¯ + 4[¯r0 s + α¯ − ρβ + 2 kβ (1 − ρ )]w(s) − 2k(w(s)) − 2β (1 − ρ ) w(s)}
(5.10)
which is a nonlinear ordinary differential equation with respect to function w(s). Now we need to provide the initial condition for w(s), because w(s) =
v(s) u(s) Vx σ2 =− =− = kv(s) − v (s) u (s) Vxx µ − r0
A∗ (s) +
ρβ σ
,
so w(0) =
σ2 µ − r0
2
A∗ (0) +
ρβ σ
2 =
σ 2 β2 ρ2 . (µ − r0 )2
We solve the ordinary differential equation by the finite-difference method. Let h be the length of the interval used in the numerical scheme. Denote w(nh) by wn , discretize the equation as ¯ − 1] wn+1 = wn + h[β¯ 2 (1 − ρ2 ) + wn ]−1 {−4(wn )3/2 [−λ¯ + r¯0 + k(¯r0 nh + α¯ − ρβ) 2 √ 1 ¯2 2 2 2 2 ¯ ¯ + 4[¯r0 nh + α¯ − ρβ + 2 kβ (1 − ρ )]wn − 2k(wn ) − 2β (1 − ρ ) wn }. By this recursive formula and the initial condition, the numerical solution of w(s) can be obtained. The optimal investment policy A∗ (s) can be obtained by A∗ (s) =
µ − r0 ρβ w(s) − . σ2 σ
After obtaining A∗ , survival probability V (s) can also be found from the function relationship between V (s) and w(s). Example 1. Let µ = 0.1, r0 = 0.04, σ = 0.3, k = 1, λ = 3, safety loading θ = 0.2, and h = 0.01, then α = (1 + θ)λE(Y ) = 3.6. The results are shown in Figs. 1 and 2. In Fig. 1, we let β = 0.1, and let ρ take values −0.1, −0.5 and −0.9. From the figure we conclude that the higher coefficient correlation between two Brownian motions, the larger optimal investment amount in risky asset. In Fig. 2, we let ρ = −0.1, and let β take values of 0.1, 0.5 and 0.9. From the figure we conclude that if the volatility of risk process is big, i.e., β increases, then we should invest more in the risky asset (the volatility of risky asset does not change) when the surplus is relatively large. Similar to Liu and Yang’s (2004) interpretation, we can see that for the same curve, when the surplus is small the insurer will invest more in the risky asset. The insurer takes a leverage. (Our setup assumes that there is no difference between the borrowing rate and lending rate.) The insurer is willing to achieve a higher return to prevent bankruptcy despite the higher risk involved. However, as the insurer’s surplus increases, the proportion and even the amount invested in risky asset decrease. 5.2. Gamma claim-size distribution In this subsection, we consider the case when claim-size is gamma distributed. The technique of transforming the integro differential equation into an ordinary differential equation in the previous subsection can be applied to this case, because gamma distribution also contains an exponential factor.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
627
Fig. 1. Exponential claim-size with β = 0.1. n n−1
y Suppose that the density function is f (y) = k(n−1)! e−ky , then the corresponding distribution function F (y) = y kn t n−1 −kt n (ky)i−1 −ky n t i−1 i−1 −ky . Let S (t) = dt = 1 − i=1 (i−1)! e , and H(y) = 1 − F (y) = ni=1 (ky) n i=1 (i−1)! , 0 (n−1)! e (i−1)! e n−1 t i−1 −ky then H(y) = Sn (ky) e . Note that Sn (t) = i=1 (i−1)! = Sn−1 (t), so the HJB equation (5.4) becomes
s
−λ¯ 0
2 ¯ ¯ (0)Sn (ks) e−ks + 1 β¯ 2 (1 − ρ2 )u (s) = 1 (u(s)) . u(s − y)Sn (ky) e−ky dy + (¯r0 s + α¯ − βρ)u(s) − λV 2 2 u (s)
Again let v(y) = u(y) eky , by transforming the equation in the same way as in the exponential case, we have s ¯ ¯ (0)Sn (ks) + 1 β¯ 2 (1 − ρ2 )(v (s) − kv(s)) v(y)Sn (k(s − y)) dy + (¯r0 s + α¯ − βρ)v(s) − λV −λ¯ 2 0 =
1 (v(s))2 . 2 v (s) − kv(s)
When n = 2, w(s) =
v(s) kv(s)−v (s)
(5.11) 2 , the HJB equation becomes
−v(s) ¯ ¯ (0)(1 + ks) + 1 β¯ 2 (1 − ρ2 ) √ v(y)[1 + k(s − y)] dy + (¯r0 s + α¯ − βρ)v(s) − λV 2 w(s) 0 1 = − v(s) w(s). 2
−λ¯
s
(5.12)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
628
Fig. 2. Exponential claim-size with ρ = −0.1.
Similar to the exponential case, let z(s) =
s 0
¯ v(y) dy and m1 (s) = λz(s) +
¯ (0) λV z (s) ,
after some calculation, we obtain
1 2 2 −1 3/2 ¯ ¯ ¯ w (s) = [β (1 − ρ ) + w(s)] −4(w(s)) −λ + r¯0 + k(¯r0 s + α¯ − ρβ) − − km1 (s) 2 1 + 4[¯r0 s + α¯ − ρβ¯ + kβ¯ 2 (1 − ρ2 )]w(s) − 2k(w(s))2 − 2β¯ 2 (1 − ρ2 ) w(s) . 2
Let m1n = m1 (nh), we discretize the system of differential equations to obtain 1 2 2 −1 3/2 ¯ ¯ ¯ −4(wn ) −λ + r¯0 + k(¯r0 nh + α¯ − ρβ) − − km1n wn+1 = wn + h [β (1 − ρ ) + wn ] 2 √ 1 + 4 r¯0 nh + α¯ − ρβ¯ + kβ¯ 2 (1 − ρ2 ) wn − 2k(wn )2 − 2β¯ 2 (1 − ρ2 ) wn , 2 1 m1n+1 = m1n + h λ¯ − m1n+1 k − √ wn+1
σ2 µ − r0
2
A∗ (0) +
ρβ σ
2 =
ρ2 σ 2 β2 , (µ − r0 )2
(5.14)
(5.15)
and w0 = w(0) =
(5.13)
m10 = 0.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
629
Then the optimal policy can be obtained by A∗ (s) =
µ − r0 βρ w(s) − 2 σ σ
and the value function V (s) is obtained from its relationship with w(s). Example 2. Let µ = 0.1, r0 = 0.04, σ = 0.3, n = 2, k = 1, λ = 3, safety loading θ = 0.2, and h = 0.01, then α = (1 + θ)λE(Y ) = 7.2. The results are shown in Figs. 3 and 4. In Fig. 3, we let β = 0.1, and let ρ take values −0.1, −0.5 and −0.9. It can be seen that as the coefficient correlation between the two standard Brownian motions becomes large, the optimal investment amount in risky asset will be large. In Fig. 4, we let ρ = −0.1, and let β take values of 0.1, 0.5 and 0.9. From the figure we conclude that if the volatility of risk process is large, i.e., β increases, the optimal investment amount in the risky asset will be larger. This is because the negative coefficient correlation between the two Brownian motions. In the gamma distribution case, we also have that the optimal investment decreases as the surplus increases. The interpretation is the same as that in Liu and Yang (2004). 5.3. Pareto claim-size distribution As shown in Embrechts et al. (1997), a heavy-tailed distribution should be used to model the insurance claim distribution. In this subsection, we discuss the case when the claim-size has a Pareto distribution. The procedures
Fig. 3. Gamma claim-size with β = 0.1.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
630
Fig. 4. Gamma claim-size with ρ = −0.1.
employed in the case of exponential claim distribution cannot be directly applied to transform the integro differential equation to a system of ordinary differential equations in this case. However, the procedures above do provide a way of stabilizing the integro differential equation. aba The probability density of the Pareto distribution is f (x) = (x+b) a+1 , where a, b are strictly positive constants, a a b b and the distribution function is F (x) = 1 − x+b , hence H(x) = x+b . Substitute them into Eq. (5.4) we have −λ¯
s
u(s − y)
0
b y+b
a
¯ ¯ (0) dy + (¯r0 s + α¯ − ρβ)u(s) − λV
b s+b
a
1 1 (u(s))2 . + β¯ 2 (1 − ρ2 )u (s) = 2 2 u (s)
Differentiate both sides with respect to s, then divide both sides of the equation by u (s), we have −
s λ¯ u (s − y) u (s) 0 ¯ (0) + λV =
b y+b
a
dy − λ¯
b s+b
a
1 u (s)
+ r¯0 s + α¯ − ρβ¯
1 u(s) 1 1 ¯2 1 ¯2 w (s) aba 2 2 u(s) √ √ (1 − ρ ) (1 − ρ ) + + r ¯ − β β 0 (s + b)a+1 u (s) u (s) 2 u (s) ( w(s))3 w(s) 4
1 u(s) 1 u(s) w (s) √ , − 2 u (s) 4 u (s) w(s)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
where
√
631
w(s) = − uu(s) (s) , plugging this into the expression above, after some calculations, we obtain
a b 1 ¯ w(s) −λm(s) + λ¯ s+b u(s) a ab 1 1 1 1 ¯2 2 ¯ ¯ − λV (0) − r¯0 + + (¯r0 s + α¯ − ρβ) , − β (1 − ρ ) √ (s + b)a+1 u(s) 2 2 w(s)
w (s) = 4w(s)[β¯ 2 (1 − ρ2 ) + w(s)]−1
(5.16)
where m(s) = 0
s
√
1 u(s − y) w(s − y) u(s)
b y+b
a dy.
√ Together with w(s) = − uu(s) (s) , we have a system of integro differential equation for u(s), w(s) and m(s). Now we discretize the system of equations and solve them numerically. Define wn = w(nh), un = u(nh), we have a √ 1 b ¯ n + λ¯ wn+1 = wn + 4h wn [β¯ 2 (1 − ρ2 ) + wn ]−1 wn −λm nh + b un aba 1 1 1 1 ¯2 2 ¯ ¯ (0) ¯ − λV β − r ¯ + (1 − ρ ) + (¯ r nh + α − ρ β) , (5.17) − √ 0 0 (nh + b)a+1 un 2 2 wn
Fig. 5. Pareto claim-size with ρ = −0.1.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
632
nh
mn = 0
un+1 − un 1 =− , √ wn hun
un+1
a a n b b 1 un−i dy ≈ h, √ y+b wn−i un ih + b i=1 h . = un 1 − √ wn
1 u(nh − y) √ un w(nh − y)
(5.18)
(5.19)
The optimal investment policy can be found by using A∗n
=
µ − r0 σ
2
√
wn −
βρ . σ
The survival probability can be obtained by V (s) =
s 0
u(y) dy or Vn ≈
n
i=1 ui h.
Example 3. For Pareto claim-size distribution, we choose µ = 0.1, r0 = 0.04, σ = 0.3, n = 2, k = 1, λ = 3, a = 3, b = 2, safety loading θ = 0.2, and h = 0.01, then α = (1 + θ)λE(Y ) = 3.6. The results are shown in Figs. 5 and 6. In Fig. 5, we let ρ = −0.1, and let β take values of 0.1, 0.5 and 0.9. It can be seen from the figure that, unlike exponential and gamma cases, the larger the value of β, the less optimal investment amount in risky asset. This is due to the fact that Pareto is heavy-tailed. For the same β, the optimal investment amount in the risky asset increases as the surplus increases. This is very different from cases where the claim-size is exponential or gamma. In Fig. 6, we let β = 0.1, and let ρ = −0.1, −0.5, −0.9. It can be seen that the larger the coefficient between the two Brownian motions, the larger the optimal investment amount in risky asset. For the same ρ, the optimal investment amount in risky asset increases as the surplus increases.
Fig. 6. Pareto claim-size with β = 0.1.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
633
Comparing Figs. 5 and 6, we can see that the optimal investment amount in risky asset is more sensitive to β than it is to ρ. From the figures one can see that the optimal strategy is asymptotically linear. For the compound Poisson model, this can be shown mathematically, see Gaier and Grandits (2002, 2004) and Schmidli (2005).
Acknowledgments The authors would like to thank the referee for careful reading of the paper and helpful suggestions. The work described in this paper was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7239/04H) and the Natural Science Foundation of China (Grant No. 70371002).
References Asmussen, S., Taksar, M., 1997. Controlled diffusion models for optimal dividend pay-out. Insurance: Mathematics and Economics 20, 1–15. Browne, Sid, 1995. Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Mathematics of Operations Research 20, 937–958. Browne, Sid, 1997. Survival and growth with a liability: optimal portfolio strategies in continuous time. Mathematics of Operations Research 22, 468–493. Browne, Sid, 1999. Beating a moving target: optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastics 3, 275–294. B¨uhlmann, H., 1970. Mathematical Methods in Risk Theory. Springer-Verlag, Berlin. Cai, J., Yang, H., 2005. Ruin in the perturbed compound Poisson risk process under interest force. Advances in Applied Probability, 37, in press. Cont, R., Tankov, P., 2003. Financial Modelling With Jump Processes. Chapman & Hall/CRC Financial Mathematics Series. Crandall, M.G., Lions, P.L., 1983. Viscosity solutions of Hamilton–Jacobi equations. Transactions of the American Mathematical Society 277, 1–42. Crandall, M.G., Ishii, H., Lions, P.L., 1992. User’s guide to viscosity solutions of second order partial differential equations. Bulletin (new series) of the American Mathematical Society 27, 1–67. Dayananda, P.W.A., 1970. Optimal reinsurance. Journal of Applied Probability 7, 134–156. Embrechts, P., Kl¨uppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer-Verlag, NY. Fleming, W.H., Rishel, R.W., 1975. Deterministic and Stochastic Optimal Control. Springer-Verlag, NY. Fleming, W.H., Soner, H.M., 1993. Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, NY. Gaier, J., Grandits, P., 2002. Ruin probabilities in the presence of regularly varying tails and optimal investment. Insurance: Mathematics and Economics 30, 211–217. Gaier, J., Grandits, P., 2004. Ruin probabilities and investment under interest force in the presence of regularly varying tails. Scandinavian Actuarial Journal, 256–278. Gerber, H., 1969. Entscheidigungskriterien f¨urden zusammengesetzten Poisson Prozess. Mitteilungen der Vereinigung Schweizer Versicherungsmathematiker 19, 185–228. Gerber, H., 1979. An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph No. 8. Gerber, H., Shiu, E.S.W., 2004. Optimal dividends: analysis with Brownian motion. North American Actuarial Journal 8 (1), 1–20. Goovaerts, M.J., Kass, R., van Heerwarrden, A.E., Bauwelinckx, T., 1990. Effective Actuarial Methods. North-Holland, Amsterdam. Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215–228. Hipp, C., Plum, M., 2003. Optimal investment for investors with state dependent income, and for insurers. Finance and Stochastics 7, 299–321. Hipp, C., Taksar, M., 2000. Stochastic control for optimal new business. Insurance: Mathematics and Economics 26, 185–192. Huang, C.H., Litzenberger, R.H., 1987. Foundations for Financial Economics. Prentice-Hall, NJ. Højgaard, B., Taksar, M., 1998. Optimal proportional reinsurance policies for diffusion models. Scandinavian Actuarial Journal 2, 166–180. Højgaard, B., Taksar, M., 1998. Optimal proportional reinsurance policies for diffusion models with transaction costs. Insurance: Mathematics and Economics 22, 41–51. Højgaard, B., Taksar, M., 2000. Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Preprint. Irgend, C., Paulsen, J., 2004. Optimal control of risk exposure, reinsurance and investments for insurance portfolios. Insurance: Mathematics and Economics 35, 21–51.
634
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
Jeanblanc-Picqu´e, M., Shiryaev, A.N., 1995. Optimization of the flow of dividends. Russian Mathematical Surveys 50, 257–277. Karatzas, I., Shreve, S., 1988. Brownian Motion and Stochastic Calculus. Springer-Verlag, NY. Krylov, N.V., 1980. Controlled Diffusion Processes. Springer-Verlag, NY. Liu, C.S., Yang, H., 2004. Optimal investment for a insurer to minimize its probability of ruin. North American Actuarial Journal 8 (2), 11–31. Martin-l¨of, A., 1973. A method for finding the optimal decision rule for a policy holder of an insurance with a bonus system. Scandinavian Actuarial Journal 23–39. Martin-l¨of, A., 1983. Premium control in an insurance system, an approach using linear control theory. Scandinavian Actuarial Journal, 1–27. Martin-l¨of, A., 1994. Lectures on the use of control theory in insurance. Scandinavian Actuarial Journal, 1–25. Moore, K.S., Young, V.R., 2004. Optimal insurance in a continuous-time model. Preprint. Milevsky, M.A., Moore, K.S., Young, V.R., 2004. Optimal asset allocation and ruin minimization annuitization strategies. Preprint. Paulsen, J., 2003. Optimal dividend payouts for diffusions with solvency constraints. Finance and Stochastics 4, 457–474. Paulsen, J., Gjessing, H.K., 1997. Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance: Mathematics and Economics 20, 215–223. Schmidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Annals of Applied Probability 12, 890–907. Schmidli, H., 2005. On optimal investment and subexponential claims. Insurance: Mathematics and Economics 36, 25–35. Taksar, M., 2000. Optimal risk and dividend distribution control for an insurance company. Mathematical Methods of Operations Research 1, 1–42. Taksar, M., Markussen, C., 2003. Optimal dynamic reinsurance policies for large insurance portfolios. Finance and Stochastics 7, 97–121. Wang, G., Wu, R., 2001. Distributions for risk process with a stochastic return on investment. Stochastic Processes and their Application 95, 329–341.
Optimal investment for insurer with jump-diffusion risk process Hailiang Yanga,∗ , Lihong Zhangb a
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China b School of Economics and Management, Tsinghua University, Beijing, PR China Received September 2004; received in revised form June 2005; accepted 9 June 2005
Abstract In this paper, we study optimal investment policies of an insurer with jump-diffusion risk process. Under the assumptions that the risk process is compound Poisson process perturbed by a standard Brownian motion and the insurer can invest in the money market and in a risky asset, we obtain the close form expression of the optimal policy when the utility function is exponential. We also study the insurer’s optimal policy for general objective function, a verification theorem is proved by using martingale optimality principle and Ito’s formula for jump-diffusion process. In the case of minimizing ruin probability, numerical methods and numerical results are presented for various claim-size distributions. © 2005 Elsevier B.V. All rights reserved. MSC: IM12; IM50 Keywords: Hamilton–Jacobi–Bellman equations; Martingale; Utility; Jump-diffusion; Ito’s formula; Stochastic control
1. Introduction In recent years there has been an increasing attention in the utilization of stochastic control theory to insurance related problems. This is due to the facts that insurance companies can invest in the stock market, and can pay dividend to maximize (or minimize) a certain objective function under different constraints. The classical works on this subject are those by Gerber (1969), Buhlmann (1970), Dayananda (1970), Martin-lof (1973, 1983, 1994) and Jeanblanc-Picqu´e and Shiryaev (1995). Browne (1995) considers a model in which the aggregate claims are modelled by a Brownian motion with drift, and the risky asset is modelled by a geometric Brownian motion (see also Browne, 1997, 1999). Asmussen and Taksar (1997) consider maximizing expected value of discounted dividend paid until time of ruin under a Brownian motion with drift model. Paulsen and Gjessing (1997) consider the similar problem, but allow stochastic returns on investment in their model, and in Paulsen (2003) the same problem is studied ∗
Corresponding author. Tel.: +852 2857 8322; fax: +852 2858 9041. E-mail addresses: [email protected] (H. Yang), [email protected] (L. Zhang).
0167-6687/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2005.06.009
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
616
when dividends paid are subject to solvency constraints. It is well known that the compound Poisson process is the most popular and a simple model in risk theory, Hipp and Taksar (2000) use the compound Poisson process to model the insurance business and consider the problem of optimal choice of new business to minimize the ultimate ruin probability. Hipp and Plum (2000) use the classical Cram´er–Lundberg model to describe the surplus of the insurance company and assume that the surplus of the insurance company can be invested in a risky asset (market index) which follows a geometric Brownian motion. In the case of exponential distributed claim-size, explicit solutions can be obtained. However, they do not incorporate a risk-free asset in their model. Liu and Yang (2004) reconsider the model in Hipp and Plum (2000) incorporate a risk-free interest rate. In this case, closed-form solution cannot be obtained, they provide numerical results for optimal policy for maximizing survival probability under different claim-size distribution assumptions. Taksar (2000) presents a survey of stochastic models of risk control and dividend optimization techniques for a financial corporation, the objective in his paper is to maximize the dividend pay-out. For more recent related papers see, for example, Gerber and Shiu (2004), Højgaard and Taksar (1998a,b, 2000), Taksar and Markussen (2003), Schmidli (2002), Hipp and Plum (2003), Irgend and Paulsen (2004), Moore and Young (2004) and Milevsky et al. (2004). In this paper, we use a jump-diffusion risk process, where the diffusion term denotes the uncertainty associated with the surplus of the insurance company, to model the surplus of the insurance company and study the portfolio selection problems for an insurer. The three optimization criterions are: (1) maximizing exponential utility function for a given terminal time; (2) maximizing the survival probability (equivalently, minimizing the ruin probability) of insurance company; (3) a general objective function. Controlled jump-diffusion models of stochastic control theory are used. For the first criterion, similar to Browne (1995), a closed-form solution is obtained, but for the second criterion, similar to Liu and Yang (2004), we are unable to obtain any closed-form solution. By performing some helpful transformations similar to those in Hipp and Plum (2003), we present some numerical results. For the general objective function, a verification theorem is proved by using martingale optimality principle and Ito’s formula. Here is a brief outline of this paper. In Section 2 the model assumptions are formulated. In Section 3, we study the problem of maximizing exponential utility function, a closed-form expression for the optimal investment amount is obtained. In Section 4, we discuss a general optimal control problem. As a special case of the general optimal control, the problem of maximizing survival probability is studied in Section 5.
2. The model In this paper, we assume that the standard assumptions of continuous-time financial models hold, that is 1. continuous trading is allowed; 2. no transaction cost or tax is involved in trading; and 3. all assets are infinitely divisible. For simplicity, we assume that there are only two assets available for investors in the financial market: a risk-free asset, whose price at time t is denoted by B(t), and a risky asset, whose price at time t is denoted by P(t). B(t) is assumed to follow dB(t) = r0 B(t) dt, where r0 , the risk-free interest rate, is assumed to be a positive constant. The price of the risky asset is assumed to follow the stochastic differential equation (1)
dP(t) = µP(t) dt + σP(t) dWt ,
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
617
where µ and σ are positive constants represent the expected instantaneous rate of return of the risky asset and the (1) volatility of the risky asset price, respectively, and {Wt : t ≥ 0} is a standard Brownian motion defined on the complete probability space (Ω, F, P). As usual, we assume that µ > r0 . We are concerned with investment behavior in the presence of a stochastic cash flow, described by a jump-diffusion process, which we denoted by {R(t) : t ≥ 0}, satisfies (2)
dR(t) = α dt + β dWt
− dS(t), (2)
where α and β are constants (with β > 0), and {Wt : t ≥ 0} is another standard Brownian motion defined on the same complete probability space (Ω, F, P). The assumption β > 0 is essential in this paper (a boundary condition for the value function needs this assumption). However, this assumption excludes the classical compound Poisson model. We assume that the joint distribution of these two Brownian motions is bivariate normal, and we denote (1) (2) their correlation coefficient by ρ, i.e., E[Wt Wt ] = ρt. We will not consider the uninteresting case of ρ2 = 1, N(t) in such case there would only be one source of randomness in the model; S(t) = i=1 Yi is a compound Poisson process, denotes cumulated claims in time interval [0, t], where N(t), the number of claims occurring in the time interval [0, t], is a homogeneous Poisson process with intensity λ, and Yi , the size of the ith claim, is assumed to (1) (2) be independent of the claim number process, independent of the two Brownian motions Wt and Wt above, and independent identically distributed with common distribution function F, where F satisfies F (0) = 0. Moreover, ∞ (2) we assume that F has a mean value of µ1 and µ2 = 0 x2 dF (x) < +∞. The diffusion term βWt stands for the uncertainty associated with the surplus of the insurance company at time t. The company is allowed to invest in the risky asset and risk-free asset, and let At be the total amount of money invested in the stock market at time t when the company’s wealth is Xt , where {At : t ≥ 0} is a suitable, admissible adapted process, i.e., At is a nonanticipative function that satisfies T [At ]2 dt < ∞ a.e. for all T < ∞. 0
Let Xt denote the wealth of the company at time t, if it follows investment policy A(t), with initial wealth X0 = x. This process then evolves as dXt = At
dP(t) dB(t) + (Xt − At ) + dR(t), P(t) B(t)
or, more explicitly, (1)
dXt = [(µ − r0 )At + r0 Xt + α] dt + σAt dWt X0 = x. (1)
(2)
+ β dWt
− dS(t) (2.1)
(2)
Since Wt and Wt are correlated standard Brownian motions with correlation coefficient ρ, applying It oˆ formula for jump-diffusion process (see Cont and Tankov, 2003) to function g(t, x), we obtain that, the generator is AA g(t, x) = gt + [A(µ − r0 ) + α + r0 x]gx + [g(t, xt− + xt ) − g(t, xt− )] + 21 [A2 σ 2 + β2 + 2ρσβA]gxx , (2.2) where gx , gt and gxx denote the first order partial derivative with respect to x, the first order partial derivative with respect to t, and the second order partial derivative with respect to x respectively. Note that so long as ρ2 = 1, this model is incomplete in a very strong sense that the random cash flow, R(t), cannot be traded on the security market, and hence the risk to the investor (insurer) cannot be eliminated under any circumstance.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
618
3. Maximizing expected exponential utility of terminal wealth Suppose that the investor is interested in maximizing the utility function for his terminal wealth, say at time T. The utility function is u(x), here we assume that u > 0 and u < 0. Let V (t, x) denote the maximal utility attainable by the investor from the state x at time t, i.e., V (t, x) =
sup
{A(s):s≥t}
E[u(XTA )|XtA = x].
(3.1)
where {XtA : t ≥ 0} is the wealth process under the policy A. Since u is a concave function, there exists a unique optimal investment policy A∗ such that the expected utility reaches to its maximum. Suppose that the investor has an exponential utility function u(x) = c −
δ −γx e , γ
(3.2)
where δ > 0 and γ > 0. This utility has constant absolute risk aversion parameter γ, such utility functions play a prominent role in insurance mathematics and actuarial practice, since they are the only utility functions under which the principle of “zero utility” gives a fair premium that is independent of the level of reserves of an insurance company (see Gerber, 1979, p. 68 or Goovaerts et al., 1990, Section II.6). For the problem of maximizing utility of terminal wealth at a fixed terminal time T, the Hamilton–Jacobi–Bellman (HJB) equation becomes, for t < T (see Fleming and Rishel, 1975 or Fleming and Soner, 1993): sup{AA V (t, x)} = 0,
V (T, x) = u(x).
(3.3)
A
In other words, for each (t, x), we must solve the nonlinear partial differential equation (3.3), and find the value A which maximizes the function Vt + [α + r0 x + A(µ − r0 )]Vx + 21 [A2 σ 2 + β2 + 2ρσβA]Vxx + λE[V (t, x − Y ) − V (t, x)].
(3.4)
It is well known in stochastic control theory that, in some cases, a value function is not smooth enough to satisfy the HJB equation in the classical sense. A commonly used weak formulation of solution to the HJB equation is called the viscosity solution proposed by Crandall and Lions (1983). For more detailed discussions on viscosity solutions, see Crandall et al. (1992) and Fleming and Soner (1993). However, in many cases, we can prove that the value function is smooth enough. Similar method as that in Hipp and Plum (2003) can be used to prove that the HJB equation has a classical solution in our case. Here we will not discuss this problem in detail. Let us assume that the HJB equation (3.3) has a classical solution V, which satisfies Vx > 0 and Vxx < 0. Then differentiating with respect to A in (3.4) gives the optimizer A∗ = −
µ − r 0 Vx ρβ − . σ 2 Vxx σ
(3.5)
Plugging (3.5) into HJB equation (3.4), after simplification, we have ρβ(µ − r0 ) 1 µ − r0 2 Vx2 1 Vt + α + r0 x − Vx − + β2 (1 − ρ2 )Vxx + λE[V (t, x − Y ) − V (t, x)] = 0 σ 2 σ Vxx 2 for t < T. To solve Eq. (3.6), inspired by Browne (1995), we try to fit a solution of the form 2 δ 1 µ − r 0 V (t, x) = c − exp −γx er0 (T −t) − (T − t) + h(T − t) , γ 2 σ
(3.6)
(3.7)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
619
where h(·) is a suitable function such that (3.7) is a solution of (3.6), and the boundary condition V (T, x) = u(x) = c −
δ −γx e γ
implies that h(0) = 0. Note that for this solution, we have Vx = [V (t, x) − c][−γ er0 (T −t) ],
Vxx = [V (t, x) − c][γ 2 e2r0 (T −t) ], 2 1 µ − r 0 Vt = [V (t, x) − c] xr0 γ er0 (T −t) + − h (T − t) . 2 σ
Substitute the values for Vx and Vxx into (3.5), we find the optimal investment policy is given by A∗ =
µ − r0 −r0 (T −t) ρβ e − . γσ 2 σ
(3.8)
Note that, in this situation, the amount invested in the risky asset is independent of the wealth level. This is due to the fact that the exponential utility function has constant absolute risk aversion. When the utility function is exponential, we conclude that even if there are claims that should be paid by the insurer, the investment amount on risky asset is independent of insurer’s wealth when there is only one risky asset and one risk-free asset in the market. This conclusion is the same as that in the case of individual’s simple investment behavior, see Huang and Litzenberger (1987). It is interesting to observe that this policy invests more as the deadline gets closer, this is consistent with intuition. In order to obtain the value function, we need to calculate λE[V (t, x − Y ) − V (t, x)] = λ[V (t, x) − c]
∞
r0 (T −t)
exp{γy e
} dF (y) − 1
0
= λ[V (t, x) − c]E[exp{γY er0 (T −t) } − 1]. Plugging the values of Vx , Vxx and Vt into (3.6), together with the result above, this leads to that we require ρβ(µ − r0 ) r0 (T −t) 1 2 2 e + γ β (1 − ρ2 ) e2r0 (T −t) + λE[exp{γY exp{r0 (T − t)}} − 1] h (T − t) = −γ α − σ 2
(3.9) together with the initial condition h(0) = 0. If we know the distribution of Y, we can obtain the close form expression of h(·). When the investment is optimal, the optimal wealth process follows (µ − r0 )2 −r0 (T −t) (µ − r0 )ρβ ∗ dXt∗ = e + α + r − X 0 t dt γσ 2 σ µ − r0 −r0 (T −t) (1) (2) + − ρβ dWt + β dWt − dS(t) e γσ
(3.10)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
620
(µ − r0 )2 −r0 (T −t) (µ − r0 )ρβ ∗ = e − + α + r0 X (t) dt γσ 2 σ µ − r0 −r (T −t) 2 (3) 0 e + + β2 (1 − ρ2 ) dWt − dS(t), γρ
(3.11)
where
(3)
Wt
(1) (2) e−r0 (T −t) − ρβ Wt + βWt = 2 µ−r0 −r0 (T −t) e + β2 (1 − ρ2 ) γρ µ−r0 γσ
(1)
(2)
is another standard Brownian motion on the same probability space (Ω, F, P) by the assumptions on Wt and Wt . Since this is a linear stochastic differential equation plus a jump process (compound Poisson process), standard results (e.g., Krylov, 1980 or Karatzas and Shreve, 1988) tell us it admits the unique strong solution. This, together with the fact that the value function is twice-continuously differentiable, gives the following theorem. Theorem 1. Under our assumptions, and assume that the objective is to maximize utility of terminal wealth, at the fixed terminal time T, and the utility function is given by u(x) = c −
δ −γx e , γ
then the optimal value function is given by δ 1 µ − r0 2 r0 (T −t) V (t, x) = c − exp −γx e − (T − t) + h(T − t) , γ 2 σ
(3.12)
where the function h(·) is given by (3.9) with initial condition h(0) = 0, and the optimal policy is to invest A∗ =
µ − r0 −r0 (T −t) ρβ e − γσ 2 σ
(3.13)
in the risky asset at time t. Proof. It is not difficult to check that all the conditions of the verification theorem are satisfied (see Theorem VI 4.2 in Fleming and Rishel (1975)). Therefore the investment strategy given by (3.13) is optimal. It is instructive to verify the optimality by using the martingale optimality principle, which entails finding an appropriate functional which is a uniformly integrable martingale under the optimal policy, but a super-martingale under any other admissible policy. By applying Ito’s formula, we have that {V (t, XtA ) − c : 0 < t < T } is a super-martingale under any admissible policy A. Note that we have a boundary condition V (T, XtA ) = u(XtA ), the proof of the theorem will be completed if we have the following lemma. Lemma 1. Under the optimal policy, the process {V (t, Xt∗ ) − c : t ≤ T } is a martingale. Proof. Apply Ito’s formula for jump-diffusion process (see Cont and Tankov, 2004) to the function V (t, Xt∗ ) − c, we obtain
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
d[V (t, Xt∗ ) − c] =
621
1 Vt + [r0 Xt∗ + γ1 + γγ2 e−r0 (T −t) ]Vx + [γ3 + γ2 e−2r0 (T −t) ]Vxx dt 2 (3) + γ3 + γ2 e−2r0 (T −t) Vx dWt + λE[V (t, Xt∗ − Y ) − V (t, Xt∗ )],
(3.14)
2 µ−r0 where γ1 = α − ρβ , and γ3 = β2 (1 − ρ2 ). Note that the last expectation is taken only with σ (µ − r0 ), γ2 = γσ respect to random variable Y. Plugging all the expressions of Vx , Vxx , Vt and h(·) into (3.14) and simplify to obtain (3) d[V (t, Xt∗ ) − c] = [V (t, Xt∗ ) − c][−γ γ2 + γ3 e2r0 (T −t) dWt ]. (3.15) This is a linear stochastic differential equation with solution 2 t T V (t, Xt∗ ) − c γ 2r0 (T −u) (3) 2r (T −u) 0 . = exp − (γ2 + γ3 e ) du − γ γ2 + γ 3 e dWu V (0, X0∗ ) − c 2 0 0
(3.16)
This is the exponential martingale. Hence, for all t ≤ T , the process {V (t, Xt∗ ) − c} is a martingale. By the martingale optimality principle, the theorem is proved.
4. A general optimal control problem Many problems considered in the literature so far, such as maximizing the survival probability, minimizing the expected (discounted) penalty of ruin, etc., are special cases of the optimal control problem below: for each admissible control process {At : t ≥ 0}, let τzA = inf{t > 0 : XtA = z} denote the first hitting time to the point z of the associated wealth process {XtA } of (2.1). For given numbers (l, u) with l < x < u, let τ A = min{τlA , τuA } denote the first escape time from the interval (l, u). For a given nonnegative continuous function θ(x) ≥ 0, given a real bounded continuous function g(x) and a function h(x), given l and u, let V A (x) be defined by
A A τ
V A (x) = Ex
0
−
g(XtA ) e
t
0
θ(XtA ) ds
−
dt + h(XτAA ) e
τ
0
θ(XsA ) ds
(4.1)
with V (x) = supA∈G V A (x). Here we assume that the optimal strategy exists and let A∗ = arg supA∈G V A (x), where G denotes the set of admissible policies, Ex [·] denotes conditional expectation given X0 = x. We note that we are only interested in controls with V A (x) < ∞. Theorem 2. Suppose that w(x) : (l, u) → (−∞, +∞) is a C2 function and is the concave increasing (i.e., wx > 0, wxx < 0) solution to the equation ρβ(µ − r0 ) 1 µ − r0 2 w2x (x) 1 α + r0 x − wx (x) − + β2 (1 − ρ2 )wxx (x) + λE[w(x − Y ) − w(x)] σ 2 σ wxx (x) 2 + g(x) − θ(x)w(x) = 0
(4.2)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
622
with w(l) = h(l) and
w(u) = h(u)
(4.3)
and satisfies the following conditions: w2 (x)
1. wxxx (x) is bounded for all x ∈ (l, u), 2. there exists an integrable random variable Y such that for all t ≥ 0, w(XtA ) ≥ Y , 3. wwxxx (x) (x) is locally Lipschitz continuous, then w(x) is the optimal value function, i.e., w(x) = V (x), and the optimal investment amount A∗ can be written as A∗ = −
µ − r0 wx (x) ρβ − 2 σ wxx (x) σ
for x ∈ (l, u).
(4.4)
Proof. The HJB equation is given by sup [A(µ − r0 ) + α + r0 x]Vx + 21 [A2 σ 2 + β2 + 2ρσβA]Vxx + λE[V (x − Y ) − V (x)] + g − θV = 0,
A∈G
(4.5) subject to the boundary condition V (l) = h(l), V (u) = h(u). Assuming that (4.5) has a classical solution with Vx > 0, Vxx < 0. We can obtain the maximizer A∗ = −
µ − r0 wx (x) ρβ − σ 2 wxx (x) σ
for x ∈ (l, u).
Plugging this expression into (4.5) and after simplification, we obtain the nonlinear equation (4.2) (with V = w). Next, we need that the policy A∗ is indeed optimal. Here martingale optimal principle is used again. t to verify A A Let Θ (s, t) = s θ(xv ) dv, and define process t A A M(t, xtA ) = e−Θ (0,t) w(xtA ) + e−Θ (0,s) g(xsA ) ds (4.6) 0
for t ∈ [0, τ A ], where w is the concave and increasing solution of (4.2). Optimality of A∗ is then a direct consequence of the following lemma.
Lemma 2. For any admissible policy A, we have A E[M(t ∧ τ A , Xt∧τ A )] ≤ M(0, X0 ) = w(x)
(4.7)
with equality holding if and only if A = A∗ , where A∗ is the policy given by (4.4). Moreover, under the optimal ∗ policy A∗ , the process {M(t ∧ τ A , Xt∧τ A )} is a uniform integrable martingale. Proof of Lemma 2. First, we rewrite (2.1) into the following form: (4) dXtA = [(µ − r0 )A + r0 XtA + α] dt + σ 2 A2 + β2 + 2ρβσA dWt − dS(t), where (1)
(4)
Wt
(2)
σAWt + βWt = σ 2 A2 + β2 + 2ρβσA
(4.8)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
623 (1)
is a standard Brownian motion on the same probability space (Ω, F, P) due to the assumptions on Wt (2) and Wt . Applying Ito’s formula for jump-diffusion process (2.1) to M(t, xtA ), we have that, for 0 ≤ s ≤ t ≤ τ A , M(t, XtA ) = M(s, XsA ) +
+
s≤Ti ≤t
t
e−Θ
A (s,v)
s
Q(A, XvA ) dv +
t
e−Θ
A (s,v)
s
σ 2 A2 + 2ρσβA + β2 wx (XvA ) dWv(4)
E[M(Ti , XTAi − − Y ) − M(Ti −, XTAi − )].
(4.9)
Note that the last expectation is taken only with respect to Y, and Ti is the ith claim time, where Q(z; y) denotes the quadratic (in z) term defined by Q(z; y) = z2 [ 21 σ 2 wxx (y)] + z[(µ − r0 )wx (y) + ρσβwxx (y)] + [α + r0 x]wx (y) + 21 β2 wxx (y) + g(y) − θ(y)w(y).
(4.10)
Note that since Qzz (z; y) = σ 2 wxx (y) < 0, and the maximum of Qzz (z; y) is achieved at the value z∗ (y) = −
µ − r0 wx (y) ρβ − σ 2 wxx (y) σ
with corresponding maximal value ρβ(µ − r0 ) 1 µ − r0 2 w2x (x) 1 wx (x)− Q(z∗ ; y) = α + r0 x− + β2 (1 − ρ2 )wxx (x) + g(x) − θ(x)w(x) σ 2 σ wxx (x) 2 = − λE[w(x − Y ) − w(x)]. The last equality follows from (4.2). Therefore the sum of the second term and the last term on the right-hand side of (4.9) is always less than or equals to 0. Moreover,
t∧τ A
e−Θ
A (s,v)
0
(σ 2 A2 + 2ρσβA + β2 )wx (XvA ) dWv(4)
= M(t ∧ τ −
A
A , Xt∧τ A ) − w(x) −
t∧τ A
e−Θ
A (s,v)
0
Q(A, XvA ) dv
A E[M(Ti , XTAi − − Y ) − M(Ti −, XTAi − )] ≥ M(t ∧ τ A , Xt∧τ A ) − w(x).
0≤Ti ≤t∧τ A
Thus by assumption (2) we see that the stochastic integral term in (4.9) is a local martingale, and in fact it is a super-martingale. Hence, taking expectations on both sides of (4.9), and let s = 0, we have
A A E[M(t ∧ τ A , Xt∧τ A )] ≤ w(x) + E
+
0≤Ti ≤t∧τ A
t∧τ
0
e−Θ
A (s,v)
Q(A, XvA ) dv
E[M(Ti , XTAi − − Y ) − M(Ti −, XTAi − )]
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
624
≤ w(x) + E
t∧τ A
e
−ΘA (s,v)
0
+
0≤Ti
sup[Q(A, XvA )] dv A
E[M(Ti , XTAi − − Y ) − M(Ti −, XTAi − )] = w(x).
(4.11)
≤t∧τ A
The above equality holds if and only if the policy is A∗ . Note that under the optimal policy A∗ , the wealth process X∗ satisfies the following stochastic differential equation: (µ − r0 )2 wx (Xt∗ ) βρ(µ − r0 ) ∗ dXt = α + r0 x − dt − dS(t) − σ2 wxx (Xt∗ ) σ
2 ∗) µ − r0 wx (Xt∗ ) µ − r ρβ w βρ (X 0 x (4) t + − 2βρσ + + + β2 dWt σ 2 wxx (Xt∗ ) σ σ 2 wxx (Xt∗ ) σ for 0 < t < τ A , by assumption (3), this equation admits a unique strong solution. Moreover, under the optimal ∗ policy A∗ , we have, for all 0 ≤ s ≤ t ≤ τ ∗ (where τ ∗ = τ A ), M(t, Xt∗ )
=
M(s, Xs∗ ) −
t
e
−
v
θ(Xu∗ ) du s
s
µ − r0 σ
2
wx (Xv∗ ) wxx (Xv∗ )
2
+ β2 (1 − ρ2 )wx (Xv∗ ) dWv(4) .
By assumption (1) it is easy to see that expression above is a uniform integrable martingale. Thus the lemma is proved. This also completes the proof of the theorem. 5. Maximizing the survival probability In this section, we consider the problem of minimizing the probability of ruin, i.e., maximizing the survival probability. Note that this is simply a special case of the control problem solved in Theorem 2 with θ(x) = 0, g(x) = 0, u = ∞ and l = 0. Let V (x) denote the survival probability of risk process Xt with initial surplus x, then the nonlinear equation for the optimal value function V becomes, in this case, ρβ(µ − r0 ) 1 µ − r0 2 Vx2 (x) 1 2 α + r0 x − Vx (x)− + β (1 − ρ2 )Vxx (x)+λE[V (x − Y ) − V (x)] = 0. σ 2 σ Vxx (x) 2 (5.1) The same method as that in Hipp and Plum (2003) can be used to prove that the Bellman equation has a classical solution in this case. Using the results in Hipp and Plum (2003), Cai and Yang (2005) pointed out that the survival probability is twice continuously differentiable in some insurance risk models with stochastic interest rate (see also Wang and Wu, 2001). Under proper conditions on the claim-size distribution, the conditions 1 and 3 in Theorem 2 hold. Condition 2 holds automatically by taking Y = 0. So the solution of (5.1) is the optimal value function, i.e., the survival probability under optimal policy. In order to obtain the initial value easily, we assume that the coefficient correlation ρ ≤ 0 in this section. From (4.4) we can easily know that the optimal investment amount A ≥ 0. It is obvious that A∗ (0) = 0 and V (0) = 0 due to the fluctuation property of the Brownian motion. Henceforth for maximizing the survival probability, we need to solve the nonlinear equation (5.1) with initial condition V (0) = 0. In the following, we present numerical solutions of (5.1) under different assumptions on the claim-size distributions.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
625
We rewrite HJB equation (5.1) in a form that is more readily solvable. 2 λ α 0 2 ¯ r0 ¯ and Rβ 0 by β¯ 2 , (5.1) becomes Denoting ( µ−r σ ) by R0 , R0 by λ, R0 by r¯0 , R0 by α 2 1 ¯ x (x) = 1 Vx (x) . ¯ λE[V (x − Y ) − V (x)] + β¯ 2 (1 − ρ2 )Vxx (x) + [α¯ + r¯0 x − ρβ]V 2 2 Vxx (x)
(5.2)
Let F (y) be the distribution function of the claim-size, H(y) = 1 − F (y), then (5.2) can be rewritten as x 2 1 ¯2 2 ¯ ¯ ¯ (0)H(x) = 1 Vx (x) . (α¯ + r¯0 x − ρβ)Vx (x) + β (1 − ρ )Vxx (x) − λ Vx (x − y)H(y) dy − λV 2 2 Vxx (x) 0 Let u(x) = Vx , then 1 ¯ (α¯ + r¯0 x − ρβ)u(x) + β¯ 2 (1 − ρ2 )u (x) − λ¯ 2
x
¯ (0)H(x) = u(x − y)H(y) dy − λV
0
1 u2 (x) . 2 u (x)
(5.3)
(5.4)
The following procedures are essentially the same as those proposed by Hipp and Plum (2000) (see also Liu and ¯ are incorporated. The transforms to be used in Yang, 2004) except that extra terms 21 β¯ 2 (1 − ρ2 )u (x) and −ρβu(x) the following subsections are the same as those in Hipp and Plum (2003). 5.1. Exponential claim-size distribution Assume that the probability density function of the claim size is f (y) = k e−ky , then F (y) = 1 − e−ky , H(y) = Let v(y) = u(y) eky , then
e−ky .
v (y) = u (y) eky + kv(y),
v (y) − kv(y) u (y) = . v(y) u(y)
Substituting them into Eq. (5.4) yields s 2 ¯ ¯ (0) + 1 β¯ 2 (1 − ρ2 )(v (s) − kv(s)) = 1 (v(s) ) . − λ¯ v(y) dy + (¯r0 s + α¯ − ρβ)v(s) − λV 2 2 v (s) − kv(s) 0 2 v(s) , since u = Vxx < 0 and u = Vx > 0, we have Let w(s) = v (s)−kv(s)
(5.5)
(5.6)
v (y) = u (y) eky + kv(y) < kv(y). Thus √
1 kv(s) − v (s) v (s) = =k− . v(s) v(s) w(s)
From Eq. (5.6), we have s 1 −v(s) ¯ ¯ (0) + 1 β¯ 2 (1 − ρ2 ) √ λ¯ v(y) dy + (¯r0 s + α¯ − ρβ)v(s) − λV = − v(s) w(s). 2 2 w(s) 0
(5.7)
(5.8)
Differentiate both sides with respect to s and then divide both sides by v(s), we obtain v (s) √ v(s) 1 √ w (s) (s) 1 v 1 w (s) 1 v (s) v(s) w(s) − v(s) 2 w(s) 2 2 ¯ . − β¯ (1 − ρ ) w(s) − √ =− (−λ¯ + r¯0 ) + (¯r0 s + α¯ − ρβ) v(s) 2 2 v(s) 4 w(s) w(s) (5.9)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
626
Plugging (5.7) into (5.9), we have ¯ − 1] w (s) = [β¯ 2 (1 − ρ2 ) + w(s)]−1 {−4(w(s))3/2 [−λ¯ + r¯0 + k(¯r0 s + α¯ − ρβ) 2 √ 1 ¯2 2 2 2 2 ¯ ¯ + 4[¯r0 s + α¯ − ρβ + 2 kβ (1 − ρ )]w(s) − 2k(w(s)) − 2β (1 − ρ ) w(s)}
(5.10)
which is a nonlinear ordinary differential equation with respect to function w(s). Now we need to provide the initial condition for w(s), because w(s) =
v(s) u(s) Vx σ2 =− =− = kv(s) − v (s) u (s) Vxx µ − r0
A∗ (s) +
ρβ σ
,
so w(0) =
σ2 µ − r0
2
A∗ (0) +
ρβ σ
2 =
σ 2 β2 ρ2 . (µ − r0 )2
We solve the ordinary differential equation by the finite-difference method. Let h be the length of the interval used in the numerical scheme. Denote w(nh) by wn , discretize the equation as ¯ − 1] wn+1 = wn + h[β¯ 2 (1 − ρ2 ) + wn ]−1 {−4(wn )3/2 [−λ¯ + r¯0 + k(¯r0 nh + α¯ − ρβ) 2 √ 1 ¯2 2 2 2 2 ¯ ¯ + 4[¯r0 nh + α¯ − ρβ + 2 kβ (1 − ρ )]wn − 2k(wn ) − 2β (1 − ρ ) wn }. By this recursive formula and the initial condition, the numerical solution of w(s) can be obtained. The optimal investment policy A∗ (s) can be obtained by A∗ (s) =
µ − r0 ρβ w(s) − . σ2 σ
After obtaining A∗ , survival probability V (s) can also be found from the function relationship between V (s) and w(s). Example 1. Let µ = 0.1, r0 = 0.04, σ = 0.3, k = 1, λ = 3, safety loading θ = 0.2, and h = 0.01, then α = (1 + θ)λE(Y ) = 3.6. The results are shown in Figs. 1 and 2. In Fig. 1, we let β = 0.1, and let ρ take values −0.1, −0.5 and −0.9. From the figure we conclude that the higher coefficient correlation between two Brownian motions, the larger optimal investment amount in risky asset. In Fig. 2, we let ρ = −0.1, and let β take values of 0.1, 0.5 and 0.9. From the figure we conclude that if the volatility of risk process is big, i.e., β increases, then we should invest more in the risky asset (the volatility of risky asset does not change) when the surplus is relatively large. Similar to Liu and Yang’s (2004) interpretation, we can see that for the same curve, when the surplus is small the insurer will invest more in the risky asset. The insurer takes a leverage. (Our setup assumes that there is no difference between the borrowing rate and lending rate.) The insurer is willing to achieve a higher return to prevent bankruptcy despite the higher risk involved. However, as the insurer’s surplus increases, the proportion and even the amount invested in risky asset decrease. 5.2. Gamma claim-size distribution In this subsection, we consider the case when claim-size is gamma distributed. The technique of transforming the integro differential equation into an ordinary differential equation in the previous subsection can be applied to this case, because gamma distribution also contains an exponential factor.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
627
Fig. 1. Exponential claim-size with β = 0.1. n n−1
y Suppose that the density function is f (y) = k(n−1)! e−ky , then the corresponding distribution function F (y) = y kn t n−1 −kt n (ky)i−1 −ky n t i−1 i−1 −ky . Let S (t) = dt = 1 − i=1 (i−1)! e , and H(y) = 1 − F (y) = ni=1 (ky) n i=1 (i−1)! , 0 (n−1)! e (i−1)! e n−1 t i−1 −ky then H(y) = Sn (ky) e . Note that Sn (t) = i=1 (i−1)! = Sn−1 (t), so the HJB equation (5.4) becomes
s
−λ¯ 0
2 ¯ ¯ (0)Sn (ks) e−ks + 1 β¯ 2 (1 − ρ2 )u (s) = 1 (u(s)) . u(s − y)Sn (ky) e−ky dy + (¯r0 s + α¯ − βρ)u(s) − λV 2 2 u (s)
Again let v(y) = u(y) eky , by transforming the equation in the same way as in the exponential case, we have s ¯ ¯ (0)Sn (ks) + 1 β¯ 2 (1 − ρ2 )(v (s) − kv(s)) v(y)Sn (k(s − y)) dy + (¯r0 s + α¯ − βρ)v(s) − λV −λ¯ 2 0 =
1 (v(s))2 . 2 v (s) − kv(s)
When n = 2, w(s) =
v(s) kv(s)−v (s)
(5.11) 2 , the HJB equation becomes
−v(s) ¯ ¯ (0)(1 + ks) + 1 β¯ 2 (1 − ρ2 ) √ v(y)[1 + k(s − y)] dy + (¯r0 s + α¯ − βρ)v(s) − λV 2 w(s) 0 1 = − v(s) w(s). 2
−λ¯
s
(5.12)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
628
Fig. 2. Exponential claim-size with ρ = −0.1.
Similar to the exponential case, let z(s) =
s 0
¯ v(y) dy and m1 (s) = λz(s) +
¯ (0) λV z (s) ,
after some calculation, we obtain
1 2 2 −1 3/2 ¯ ¯ ¯ w (s) = [β (1 − ρ ) + w(s)] −4(w(s)) −λ + r¯0 + k(¯r0 s + α¯ − ρβ) − − km1 (s) 2 1 + 4[¯r0 s + α¯ − ρβ¯ + kβ¯ 2 (1 − ρ2 )]w(s) − 2k(w(s))2 − 2β¯ 2 (1 − ρ2 ) w(s) . 2
Let m1n = m1 (nh), we discretize the system of differential equations to obtain 1 2 2 −1 3/2 ¯ ¯ ¯ −4(wn ) −λ + r¯0 + k(¯r0 nh + α¯ − ρβ) − − km1n wn+1 = wn + h [β (1 − ρ ) + wn ] 2 √ 1 + 4 r¯0 nh + α¯ − ρβ¯ + kβ¯ 2 (1 − ρ2 ) wn − 2k(wn )2 − 2β¯ 2 (1 − ρ2 ) wn , 2 1 m1n+1 = m1n + h λ¯ − m1n+1 k − √ wn+1
σ2 µ − r0
2
A∗ (0) +
ρβ σ
2 =
ρ2 σ 2 β2 , (µ − r0 )2
(5.14)
(5.15)
and w0 = w(0) =
(5.13)
m10 = 0.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
629
Then the optimal policy can be obtained by A∗ (s) =
µ − r0 βρ w(s) − 2 σ σ
and the value function V (s) is obtained from its relationship with w(s). Example 2. Let µ = 0.1, r0 = 0.04, σ = 0.3, n = 2, k = 1, λ = 3, safety loading θ = 0.2, and h = 0.01, then α = (1 + θ)λE(Y ) = 7.2. The results are shown in Figs. 3 and 4. In Fig. 3, we let β = 0.1, and let ρ take values −0.1, −0.5 and −0.9. It can be seen that as the coefficient correlation between the two standard Brownian motions becomes large, the optimal investment amount in risky asset will be large. In Fig. 4, we let ρ = −0.1, and let β take values of 0.1, 0.5 and 0.9. From the figure we conclude that if the volatility of risk process is large, i.e., β increases, the optimal investment amount in the risky asset will be larger. This is because the negative coefficient correlation between the two Brownian motions. In the gamma distribution case, we also have that the optimal investment decreases as the surplus increases. The interpretation is the same as that in Liu and Yang (2004). 5.3. Pareto claim-size distribution As shown in Embrechts et al. (1997), a heavy-tailed distribution should be used to model the insurance claim distribution. In this subsection, we discuss the case when the claim-size has a Pareto distribution. The procedures
Fig. 3. Gamma claim-size with β = 0.1.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
630
Fig. 4. Gamma claim-size with ρ = −0.1.
employed in the case of exponential claim distribution cannot be directly applied to transform the integro differential equation to a system of ordinary differential equations in this case. However, the procedures above do provide a way of stabilizing the integro differential equation. aba The probability density of the Pareto distribution is f (x) = (x+b) a+1 , where a, b are strictly positive constants, a a b b and the distribution function is F (x) = 1 − x+b , hence H(x) = x+b . Substitute them into Eq. (5.4) we have −λ¯
s
u(s − y)
0
b y+b
a
¯ ¯ (0) dy + (¯r0 s + α¯ − ρβ)u(s) − λV
b s+b
a
1 1 (u(s))2 . + β¯ 2 (1 − ρ2 )u (s) = 2 2 u (s)
Differentiate both sides with respect to s, then divide both sides of the equation by u (s), we have −
s λ¯ u (s − y) u (s) 0 ¯ (0) + λV =
b y+b
a
dy − λ¯
b s+b
a
1 u (s)
+ r¯0 s + α¯ − ρβ¯
1 u(s) 1 1 ¯2 1 ¯2 w (s) aba 2 2 u(s) √ √ (1 − ρ ) (1 − ρ ) + + r ¯ − β β 0 (s + b)a+1 u (s) u (s) 2 u (s) ( w(s))3 w(s) 4
1 u(s) 1 u(s) w (s) √ , − 2 u (s) 4 u (s) w(s)
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
where
√
631
w(s) = − uu(s) (s) , plugging this into the expression above, after some calculations, we obtain
a b 1 ¯ w(s) −λm(s) + λ¯ s+b u(s) a ab 1 1 1 1 ¯2 2 ¯ ¯ − λV (0) − r¯0 + + (¯r0 s + α¯ − ρβ) , − β (1 − ρ ) √ (s + b)a+1 u(s) 2 2 w(s)
w (s) = 4w(s)[β¯ 2 (1 − ρ2 ) + w(s)]−1
(5.16)
where m(s) = 0
s
√
1 u(s − y) w(s − y) u(s)
b y+b
a dy.
√ Together with w(s) = − uu(s) (s) , we have a system of integro differential equation for u(s), w(s) and m(s). Now we discretize the system of equations and solve them numerically. Define wn = w(nh), un = u(nh), we have a √ 1 b ¯ n + λ¯ wn+1 = wn + 4h wn [β¯ 2 (1 − ρ2 ) + wn ]−1 wn −λm nh + b un aba 1 1 1 1 ¯2 2 ¯ ¯ (0) ¯ − λV β − r ¯ + (1 − ρ ) + (¯ r nh + α − ρ β) , (5.17) − √ 0 0 (nh + b)a+1 un 2 2 wn
Fig. 5. Pareto claim-size with ρ = −0.1.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
632
nh
mn = 0
un+1 − un 1 =− , √ wn hun
un+1
a a n b b 1 un−i dy ≈ h, √ y+b wn−i un ih + b i=1 h . = un 1 − √ wn
1 u(nh − y) √ un w(nh − y)
(5.18)
(5.19)
The optimal investment policy can be found by using A∗n
=
µ − r0 σ
2
√
wn −
βρ . σ
The survival probability can be obtained by V (s) =
s 0
u(y) dy or Vn ≈
n
i=1 ui h.
Example 3. For Pareto claim-size distribution, we choose µ = 0.1, r0 = 0.04, σ = 0.3, n = 2, k = 1, λ = 3, a = 3, b = 2, safety loading θ = 0.2, and h = 0.01, then α = (1 + θ)λE(Y ) = 3.6. The results are shown in Figs. 5 and 6. In Fig. 5, we let ρ = −0.1, and let β take values of 0.1, 0.5 and 0.9. It can be seen from the figure that, unlike exponential and gamma cases, the larger the value of β, the less optimal investment amount in risky asset. This is due to the fact that Pareto is heavy-tailed. For the same β, the optimal investment amount in the risky asset increases as the surplus increases. This is very different from cases where the claim-size is exponential or gamma. In Fig. 6, we let β = 0.1, and let ρ = −0.1, −0.5, −0.9. It can be seen that the larger the coefficient between the two Brownian motions, the larger the optimal investment amount in risky asset. For the same ρ, the optimal investment amount in risky asset increases as the surplus increases.
Fig. 6. Pareto claim-size with β = 0.1.
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
633
Comparing Figs. 5 and 6, we can see that the optimal investment amount in risky asset is more sensitive to β than it is to ρ. From the figures one can see that the optimal strategy is asymptotically linear. For the compound Poisson model, this can be shown mathematically, see Gaier and Grandits (2002, 2004) and Schmidli (2005).
Acknowledgments The authors would like to thank the referee for careful reading of the paper and helpful suggestions. The work described in this paper was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7239/04H) and the Natural Science Foundation of China (Grant No. 70371002).
References Asmussen, S., Taksar, M., 1997. Controlled diffusion models for optimal dividend pay-out. Insurance: Mathematics and Economics 20, 1–15. Browne, Sid, 1995. Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Mathematics of Operations Research 20, 937–958. Browne, Sid, 1997. Survival and growth with a liability: optimal portfolio strategies in continuous time. Mathematics of Operations Research 22, 468–493. Browne, Sid, 1999. Beating a moving target: optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastics 3, 275–294. B¨uhlmann, H., 1970. Mathematical Methods in Risk Theory. Springer-Verlag, Berlin. Cai, J., Yang, H., 2005. Ruin in the perturbed compound Poisson risk process under interest force. Advances in Applied Probability, 37, in press. Cont, R., Tankov, P., 2003. Financial Modelling With Jump Processes. Chapman & Hall/CRC Financial Mathematics Series. Crandall, M.G., Lions, P.L., 1983. Viscosity solutions of Hamilton–Jacobi equations. Transactions of the American Mathematical Society 277, 1–42. Crandall, M.G., Ishii, H., Lions, P.L., 1992. User’s guide to viscosity solutions of second order partial differential equations. Bulletin (new series) of the American Mathematical Society 27, 1–67. Dayananda, P.W.A., 1970. Optimal reinsurance. Journal of Applied Probability 7, 134–156. Embrechts, P., Kl¨uppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer-Verlag, NY. Fleming, W.H., Rishel, R.W., 1975. Deterministic and Stochastic Optimal Control. Springer-Verlag, NY. Fleming, W.H., Soner, H.M., 1993. Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, NY. Gaier, J., Grandits, P., 2002. Ruin probabilities in the presence of regularly varying tails and optimal investment. Insurance: Mathematics and Economics 30, 211–217. Gaier, J., Grandits, P., 2004. Ruin probabilities and investment under interest force in the presence of regularly varying tails. Scandinavian Actuarial Journal, 256–278. Gerber, H., 1969. Entscheidigungskriterien f¨urden zusammengesetzten Poisson Prozess. Mitteilungen der Vereinigung Schweizer Versicherungsmathematiker 19, 185–228. Gerber, H., 1979. An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph No. 8. Gerber, H., Shiu, E.S.W., 2004. Optimal dividends: analysis with Brownian motion. North American Actuarial Journal 8 (1), 1–20. Goovaerts, M.J., Kass, R., van Heerwarrden, A.E., Bauwelinckx, T., 1990. Effective Actuarial Methods. North-Holland, Amsterdam. Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215–228. Hipp, C., Plum, M., 2003. Optimal investment for investors with state dependent income, and for insurers. Finance and Stochastics 7, 299–321. Hipp, C., Taksar, M., 2000. Stochastic control for optimal new business. Insurance: Mathematics and Economics 26, 185–192. Huang, C.H., Litzenberger, R.H., 1987. Foundations for Financial Economics. Prentice-Hall, NJ. Højgaard, B., Taksar, M., 1998. Optimal proportional reinsurance policies for diffusion models. Scandinavian Actuarial Journal 2, 166–180. Højgaard, B., Taksar, M., 1998. Optimal proportional reinsurance policies for diffusion models with transaction costs. Insurance: Mathematics and Economics 22, 41–51. Højgaard, B., Taksar, M., 2000. Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Preprint. Irgend, C., Paulsen, J., 2004. Optimal control of risk exposure, reinsurance and investments for insurance portfolios. Insurance: Mathematics and Economics 35, 21–51.
634
H. Yang, L. Zhang / Insurance: Mathematics and Economics 37 (2005) 615–634
Jeanblanc-Picqu´e, M., Shiryaev, A.N., 1995. Optimization of the flow of dividends. Russian Mathematical Surveys 50, 257–277. Karatzas, I., Shreve, S., 1988. Brownian Motion and Stochastic Calculus. Springer-Verlag, NY. Krylov, N.V., 1980. Controlled Diffusion Processes. Springer-Verlag, NY. Liu, C.S., Yang, H., 2004. Optimal investment for a insurer to minimize its probability of ruin. North American Actuarial Journal 8 (2), 11–31. Martin-l¨of, A., 1973. A method for finding the optimal decision rule for a policy holder of an insurance with a bonus system. Scandinavian Actuarial Journal 23–39. Martin-l¨of, A., 1983. Premium control in an insurance system, an approach using linear control theory. Scandinavian Actuarial Journal, 1–27. Martin-l¨of, A., 1994. Lectures on the use of control theory in insurance. Scandinavian Actuarial Journal, 1–25. Moore, K.S., Young, V.R., 2004. Optimal insurance in a continuous-time model. Preprint. Milevsky, M.A., Moore, K.S., Young, V.R., 2004. Optimal asset allocation and ruin minimization annuitization strategies. Preprint. Paulsen, J., 2003. Optimal dividend payouts for diffusions with solvency constraints. Finance and Stochastics 4, 457–474. Paulsen, J., Gjessing, H.K., 1997. Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance: Mathematics and Economics 20, 215–223. Schmidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Annals of Applied Probability 12, 890–907. Schmidli, H., 2005. On optimal investment and subexponential claims. Insurance: Mathematics and Economics 36, 25–35. Taksar, M., 2000. Optimal risk and dividend distribution control for an insurance company. Mathematical Methods of Operations Research 1, 1–42. Taksar, M., Markussen, C., 2003. Optimal dynamic reinsurance policies for large insurance portfolios. Finance and Stochastics 7, 97–121. Wang, G., Wu, R., 2001. Distributions for risk process with a stochastic return on investment. Stochastic Processes and their Application 95, 329–341.