Optimal investment and risk control for an insurer under inside information

Accepted Manuscript Optimal investment and risk control for an insurer under inside information Xingchun Peng, Wenyuan Wang PII: DOI: Reference:

S0167-6687(16)30185-8 http://dx.doi.org/10.1016/j.insmatheco.2016.04.008 INSUMA 2205

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Insurance: Mathematics and Economics

Received date: November 2014 Revised date: March 2016 Accepted date: 25 April 2016 Please cite this article as: Peng, X., Wang, W., Optimal investment and risk control for an insurer under inside information. Insurance: Mathematics and Economics (2016), http://dx.doi.org/10.1016/j.insmatheco.2016.04.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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OPTIMAL INVESTMENT AND RISK CONTROL FOR AN INSURER UNDER INSIDE INFORMATION∗ XINGCHUN PENG† AND WENYUAN WANG‡ Abstract. This paper is devoted to the study of the optimal investment and risk control strategy for an insurer who has some inside information on the financial market and the insurance business. The insurer’s risk process and the risky asset process in the financial market are assumed to be very general jump diffusion processes. The two processes are supposed to be correlated. Under the criterion of logarithmic utility maximization of the terminal wealth, we solve our problem by using forward integral approach. Some interesting particular cases are studied in which the explicit expressions of the optimal strategy are derived by using enlargement of filtration techniques. Key words: investment, risk control, inside information, forward integral, enlargement of filtration 2010 Mathematics Subject classification: 97M30, 91B70, 60H30

1. Introduction As the insurer can invest in the financial market to avoid risk, optimal investment problems have attracted much attention for scholars in actuarial science in recent years. However, the risk of insurance cannot be avoided completely by only investing in the financial market where risk-free and risky asset are available. In addition to investment, the insurer may also take other business to reduce insurance risk. One popular and effective approach to avoid insurance risk is to take reinsurance business. Optimal investment and reinsurance for an insurer has drawn much concern in the actuarial literature recently. See for example, Schmidli (2002), Bai and Guo (2008), Liu and Ma (2009), Zeng and Li (2011), Peng and Hu (2013), Bi and Guo (2013), Guan and Liang (2014), Peng et al. (2014) and the references therein. Recently, Zou and Cadenillas (2014) considered an optimal investment and risk control problem for an insurer. In their paper, the insurer avoid the insurance risk through managing the insurance policies, not by taking reinsurance business as mentioned above. In their model, the insurer’s risk process is modeled by a jump diffusion process and is negatively correlated with the capital gains in the financial market. Under the criterion of maximizing the expected utility of the terminal wealth, they obtained explicit solutions of optimal strategies for various utility functions. In the present paper, we will consider a more general model in which both the insurer’s risk process and the risky asset process in the financial market are assumed to be very general jump diffusion processes. All the parameter processes in our model are assumed to ∗

Supported in part by grants of the National Natural Science Foundation of China (Nos.11371284 and 11401498) and the Fundamental Research Funds for the Central Universities (WUT:2015IVA066). † Department of Statistics, Wuhan University of Technology, Wuhan, P.R.China email: [email protected]. ‡ School of Mathematical Sciences, Xiamen University, Xiamen, P.R.China email:[email protected]. 1

be stochastic in order to reflect many uncertainties exist in the insurance and financial markets more properly. As Zou and Cadenillas (2014) suggested, we also incorporate the correlation between the insurer’s liabilities and the capital gains in financial market. The main contribution of our model is that we assume the insurer has some inside information on the financial assets and the insurance liabilities. Most of the exist works in the actuarial literature supposed that the insurer made her decisions relying on all the information which is generated by the events in financial markets and insurance business up to the time in which her decisions to be taken. However, it should to be noted that some insurers have more detailed information about future events about the financial markets and insurance business at present time. Since Karatzas and Pikovsky (1996) considered anticipative portfolio optimization problem in the financial market, many researchers have studied the insider trading problems in finance. See for example Le´on et al. (2003), Biagini and Øksendal (2005), Kohatsu–Higa and Sulem (2006), Hu and Øksendal (2007), Danilora et al. (2010), Kohatsu–Higa and Yamazato (2011), Pamen et al. (2013), Kchia and Protter (2014) and the references therein. However, up to now, few works are devoted to the study of optimization problem for an insurer under inside information. Recently, Batlas at al. (2012) investigated an optimal investment and proportional reinsurance problem for an insurer under inside information. In their paper, it was assumed that the insurer has some extra information at her disposal concerning the future realizations of its claims process, available from the beginning of the trading interval and hidden from the reinsurer. They assumed that the claims process was a Brownian motion with drift and one can invest in a Black-Scholes-type market. With the aim of maximizing the expected utility of the terminal wealth, they obtained the optimal investment and proportional reinsurance strategy for insurer and reinsurer by techniques of initial enlargement of filtration and the HJB equation. In our model, we assume the insurer not only has some inside information about the claims process, but also own some inside information about the financial risky asset process. Unlike in Batlas et al. (2012), the inside information in our model is not necessarily represented by a random variable, but with a very general form instead. Therefore, we can not use the enlargement of filtration technique to transform our problem into an ordinary stochastic control problem that can be solved by the HJB equation method (See Bai and Guo (2008), Batles et al. (2012) and Guan and Liang (2014)) or by the martingale method (See Perera (2010), Wang et al. (2007) and Zou and Cadenillas (2014)). Motivated by Di Nunno et al. (2006) who studied an optimal investment problem for an insurer in a financial market driven by L´evy processes, we use the forward integral with respect to L´evy process approach to solve our problem. The original forward integral was introduced with respect to the Brownian Motion only (see Russo and Vallois (1993)). Then it was extended by Di Nunno et al. (2005) to the form with respect to Poisson random measure. Now, the forward integrals are widely used to model inside trading problems in finance. See for example Bigina and Øksendal (2005), Kohatsu-Higa and Sulem (2006), Hu and Øksendal (2007), Di Nunno et al. (2006), Pamen et al. (2013) and so on. As pointed out in Di Nunno et al. (2006), the forward integrals have some advantages in modeling inside trading problems. First, the forward integral may be regarded as the limit of the natural Riemann sums. Second, the forward integral provides the natural interpretation of the gains from the trade process. Third, if the integrator processes (Brownian motion or Poisson random measure) happen to be semimartingales under given filtrations and the integrand process are adapted, then the forward integrals coincide with the ordinary stochastic integrals with respect to semimartingales. The present paper considers a more difficult problem than that in Di Nunno et al. (2006). Not only optimal investment in financial market but also optimal control of the insurance policies are considered. Moreover, the 2

insurer may have inside information on both the financial market and the insurance liabilities. Under the criterion of logarithm utility maximization of the terminal wealth we use the forward integral approach to give a characterization of the optimal strategy. Then, by using enlargement of filtration techniques, we study some interesting particular cases in which the explicit expressions of the optimal investment and risk control strategies can be derived. It turns out that the inside information about the future events about the financial market and insurance liabilities does have some impact on the optimal strategies. When no inside information is considered, our result reduces to that in Zou and Cadenillas (2014))(see Remark 5.9). The rest of the paper is organized as follows. Section 2 gives a short review of forward integrals with respect to Brownian motions and Poisson random measures. Section 3 describes the model and formulate the optimal investment and risk control problem under inside information. Section 4 is devoted to the characterization of the optimal strategies. Section 5 studies some interesting particular cases where the explicit expressions of the optimal strategies are derived. Finally, Section 6 concludes this paper. 2. A short review of forward integrals Let (Ω, F , P) be a complete probability space. On this space there are two Brownian motions W and W 2 and two compensated Poisson stochastic measures N˜ 1 and N˜ 2 which have, respectively, the expressions N˜ i (dt, dz) = N i (dt, dz) − vi (dz)dt, i = 1, 2, where N i (dt, dz), i = 1, 2, are Poisson stochastic measures, and vi (dz), i = 1, 2, are their L´evy measures. We assume that R vi (dz) satisfy R z2 vi (dz) < ∞, where R0 = R\ {0}, i = 1, 2. All the processes W 1 , W 2 , N˜ 1 and 0 N˜ 2 are assumed to be mutually independent. For any t, let Ft represent the σ-field generated by random variables W1 (s), W2 (s), N˜ 1 (s, z), N˜ 2 (s, z), z ∈ R0 , 0 6 s 6 t, augmented for all the sets of P-zero probability. We equip the given probability space (Ω, F , P) with the corresponding filtration (Ft )t>0 . Fix a constant T > 0 as the time horizon. We consider two types of forward integrals. They are the forward integral with respect to Brownian motion and the forward integral with respect to Poisson random measure. In this section, we only review the results, for their proofs, see Di Nunno et al. (2009). 1

Definition 2.1. We say that a stochastic process ϕ = ϕ(t), t ∈ [0, T ], is forward integrable over the interval [0, T ] with respect to Wi if there exists a process Ii = Ii (t), t ∈ [0, T ], such that Z t W i (s + ε) − W i (s) sup ϕ(s) ds − Ii (t) → 0, ε → 0+ , ε t∈[0,T ] 0 in probability, i = 1, 2. In this case we write Z t Ii (t) , ϕ(s)d −W i (s), 0

t ∈ [0, T ],

and call Ii (t) the forward integral of ϕ with respect to W i on [0, T ], i = 1, 2.

The following proposition gives a more intuitive interpretation of the above defined forward integral as a limit of Riemann sums. Proposition 2.2. Suppose ϕ is caglad and forward integrable with respect to W i , i = 1, 2. Then Z T n X   − i ϕ(s)d W (s) = lim ϕ(t j−1 ) W i (t j ) − W i (t j−1 ) , i = 1, 2, 0

△t→0

j=1

3

with convergence in probability. Here  the limit  is taken over the partitions 0 = t0 < t1 < · · · < tn = T of [0, T ] with △t = max16 j6n t j − t j−1 → 0, n → ∞. The following result is an immediate consequence of Definition 2.1.

Proposition 2.3. Suppose ϕ is forward integrable with respect to Wi , i = 1, 2, and G is a random variable. Then the product Gϕ is forward integrable and Z T Z T − i Gϕ(t)d W (t) = G ϕ(t)d − W i (t), i = 1, 2. 0

0

The next result implies that the forward integral in Definition 2.1 is an extension of the stochastic integral with respect to a semimartingale.

Proposition 2.4. Let H , {Ht , t ∈ [0, T ]} be a given filtration. Suppose that (1) W i is a semimartingale with respect to the R T filtration H, i = 1, 2, (2) ϕ is H–predictable and the Itˆo integral 0 ϕ(t)dW i (t) with respect to W i , i = 1, 2, exists. Then ϕ is forward integrable and Z T Z T − i ϕ(t)d W (t) = ϕ(t)dW i (t), i = 1, 2. 0

0

In what follows, we give the definition of the forward integral with respect to Poisson random measure and state some related properties. Definition 2.5. The forward integral i

J (θ) ,

Z

T 0

Z

θ(t, z)N˜ i (d − t, dz) R0

with respect to the Poisson random measure N˜ i of a random field θ(t, z), t ∈ [0, T ], z ∈ R0 , with θ(t, z) , θ(w, t, z), w ∈ Ω, and caglad with respect to t, is defined as Z TZ i J (θ) , lim θ(t, z)1Um N˜ i (dt, dz), i = 1, 2, m→∞

0

R0

if the limit exists in L2 (P). Here, Um , m = 1, 2, · · · , is an increasing sequence of compact sets Um ⊆ R0 with vi (Um ) < ∞ , i = 1, 2, such that limm→∞ Um = R0 .

Similar to the forward integral with respect to Brownian motion, we have the following properties about the forward integral with respect to Poisson random measure. Proposition 2.6. Suppose θ(t, z), t ∈ [0, T ], z ∈ R0 , is forward integrable with respect to N˜ i , i = 1, 2, and G is a random variable. Then the product Gθ is also forward integrable and Z TZ Z TZ i − G θ(t, z)N˜ (d t, dz) = Gθ(t, z)N˜ i (d − t, dz), i = 1, 2. 0

R0

0

R0

Proposition 2.7. If H , {Ht , t ∈ [0, T ]} is a filtration such that (1) Ft ⊆ Ht for all t ∈ [0, R tTR]. i (2) The process η (t) = 0 R zN˜ i (ds, dz), t ∈ [0, T ], is a semimartingale with respect to H, i = 0 1, 2. (3) The random field θ = θ(t, z), t ∈ [0, T ], z ∈ R0 , is H–predictable. 4

RTR (4) The integrable 0 R θ(t, z)N˜ i (dt, dz) exists as a classical Itˆo integral, i = 1, 2. 0 Then the forward integral of θ with respect to N˜ i also exists and we have Z TZ Z TZ i − θ(t, z)N˜ i (dt, dz), i = 1, 2. θ(t, z)N˜ (d t, dz) = 0

0

R0

R0

At the end of this section, we give the very useful Itˆo formula for the forward integrals.

Proposition 2.8. Let X(t), t ∈ [0, T ], be a forward process of the form Z t 2 Z t 2 Z tZ X X − i α(s)ds + ϕi (s)d W (s) + θi (s, z)N˜ i (d − s, dz) X(t) = x + 0

i=1

0

i=1

0

R0

where θi (s, z), s ∈ [0, T ], z ∈ R0 , is locally bounded in z near z = 0 such that Z

T 0

Z

R0

|θi (s, z)|2 vi (dz)ds < ∞,

P − a.s.,

i = 1, 2.

Also suppose that ϕi (s) and |θi (s, z)|, s ∈ [0, T ], z ∈ R0 , are forward integrable, i = 1, 2. For any function f ∈ C 2 (R), we have Z t 2 Z t 2 Z X 1 X t ′′ ′ ′ − i f (X(t)) = f (X(s)) α(s)ds + f (X(s)) ϕi (s)d W (s) + f (X(s)) ϕ2i (s)ds 2 0 0 0 i=1 i=1 ! Z Z 2 X t
f X(s−) + θi (s, z) − f (X(s−)) N˜ i (d − s, dz) + +

i=1

0

i=1

0

2 Z t X

R0

Z

R0

!
′ f X(s−) + θi (s, z) − f (X(s−)) − f (X(s−)) θi (s, z) vi (dz)ds. 3. Model formulation

In our model, we assume that all uncertainties come from the complete probability space (Ω, F , P) given in Section 2. The processes W 1 , W 2 , N˜ 1 , and N˜ 2 and the completed filtration (Ft )t>0 are defined as in Section 2. Also we fix T > 0 as the terminal time. In the financial market, there are two assets available for an insurer to invest in: • a risk free asset with price dynamics ( dA(t) = r(t)A(t)dt, A0 = 1, • a risky asset with dynamics h i R ( dS (t) = S (t−) µ(t)dt + σ(t)dWt1 + R γ1 (t, z)N˜ 1 (dt, dz) , 0 S (0) > 0.

Here, r(t), µ(t), σ(t) and γ1 (t, z) are assumed to be bounded exogenous parameter processes that are caglad and adapted to (Ft )06t6T . We assume that γ1 (t, z) > −1, dt × v1 (dz) − a.e. The insurer’s risk (per policy) is given by R ( dRt = p(t)dt + q(t)dW t + R γ2 (t, z)N˜ 2 (dt, dz), 0 R0 = 0. 5

where p(t), q(t), γ2(t, z) are supposed to be bounded exogenous parameter processes that are caglad and adapted to (Ft )06t6T . We assume that γ2 (t, z) > −1, dt × v2 (dz) − a.e.. As Zou and Cadenillas (2014) suggested, here W t , t ∈ [0, T ], is a Brownian motion given by W t = ρW 1 (t) + p 1 − ρ2 W 2 (t) with ρ ∈ (−1, 1) to describe the correlation between the insurer’s liabilities and her capital gains in the financial market. In reality, ρ usually takes negative values (see Stein (2012)). At the time t, the insurer chooses π˜ (t), the dollar amount invested in the risky asset, and total number of insurance policies L(t). We assume that the insurer can short sell the risky asset and can also borrow money for investment in the risky asset. Thus, π˜ (t) can take any real values. Moreover, we suppose that L(t) can take negative values, which means that the insurer can buy some insurance policies from other insurers and act as a policyholder. When claims occur, the insurer can get the compensation. We recall that Ft is the σ-field generated by random variables 2 W1 (s), W2 (s), N˜ 1 (s, z), N˜p (s, z), z ∈ R0 , 0 6 s 6 t, and all the P-null sets of F . From the 1 relation W t = ρW (t) + 1 − ρ2 W 2 (t), ρ ∈ (−1, 1), we can easily see that Ft is equal to the σ-field generated by random variables W1 (s), W(s), N˜ 1 (s, z), N˜ 2 (s, z), z ∈ R0 , 0 6 s 6 t, and all the P-null sets of F . In other words, (Ft )t>0 is the completed flow of information that is generated by the noise events from financial markets and actuarial claims. This represents the full information at disposal to all honest insurers. In this paper, we study the situation in which the insurer has access to larger information modeled by a general filtration H = (Ht ⊂ F )06t6T larger than F = (Ft )06t6T , that is, Ft ⊆ Ht ⊂ F , for each t ∈ [0, T ]. The σ field Ht represents the information obtained by the insurer at time t which may contain some inside information about the financial risky asset process and the insurance claims process. The insurer can make her decisions based on this larger filtration. So we assume that the strategy processes π˜ (t) and L(t) are all adapted to H = (Ht )06t6T . In this case, we can use the forward integrals with respect to Brownian motions and Poisson random measures introduced in Section 2 to model the wealth process (surplus process) X u˜ (t) corresponding to the strategy u˜ = (˜π, L). It is given by the following SDE: Z   u˜ − 1 dX (t) = π˜ (t)µ(t)dt + π˜ (t)σ(t)d Wt + π(t) ˜ γ1 (t, z)N 1 (d − t, dz) + X u˜ (t) − π˜ (t) r(t)dt R0 Z   p − 1 2 2 − L(t)p(t)dt − L(t)q(t)d ρW (t) + 1 − ρ W (t) − L(t) γ1 (t, z)N 2 (d − t, dz) + λ(t)L(t)dt R0

u˜

= [(µ(t) − r(t)) π(t) ˜ + r(t)X (t) + (λ(t) − p(t))L(t)]dt + (σ(t)˜π(t) − ρq(t)L(t)) d − Wt1 Z Z p − 2 1 − 2 ˜ ˜ γ1 (t, z)N (d t, dz) − L(t) γ2 (t, z)N˜ 2 (d − t, dz), − 1 − ρ q(t)L(t)d Wt + π(t) R0

(3.1)

R0

with initial wealth X u˜ (0) = x > 0. Here λ(t) is the premium per policy for the insurer at time t which is assumed to be a bounded caglad process adapted to (Ft )06t6T satisfying λ(t) > p(t) > 0, P − a.s., for each t ∈ [0, T ]. The revenue from selling insurance policies over the time period (t, t + dt) is given by λ(t)L(t)dt. L(t) As in Zou and Cadenillas (2014), we define the ratio of liability over surplus as κ(t) = X(t) (which is called the liability ratio). Let π(t) be the proportion of wealth invested in the risky asset at time t. Then for a control u(t) , (π(t), κ(t)), we have u˜ (t) = X(t)u(t). We then rewrite (3.1) as: dX u (t) = [(µ(t) − r(t)) π(t) + r(t) + (λ(t) − p(t)) κ(t)]dt + (σ(t)π(t) − ρq(t)κ(t)) d − Wt1 X u (t−) 6

p

−

ρ2 q(t)κ(t)d − Wt2

1−

+ π(t)

Z

R0

γ1 (t, z)N˜ 1 (d − t, dz) − κ(t)

Z

γ2 (t, z)N˜ 2 (d − t, dz)

(3.2)

R0

with X u˜ (0) = x > 0. By the Itˆo formula for forward integrals (Proposition 2.8), we have (Z t h 1 u X (t) = x exp (µ(s) − r(s))π(s) + r(s) + (λ(s) − p(s))κ(s) − σ2 (s)π2 (s) 2 0 Z t i 1 (σ(s)π(s) − ρq(s)κ(s))d − W s1 + ρσ(s)q(s)π(s)κ(s) − q2 (s)κ2 (s) ds + 2 0 Z tZ Z tp 1 − ρ2 q(s)κ(s)d − W s2 + log(1 + π(s)γ1 (s, z))N˜ 1 (d − s, dz) − 0 R0 0 Z tZ Z tZ 2 − ˜ + log(1 − κ(s)γ2 (s, z))N (d s, dz) + [log(1 + π(s)γ1 (s, z)) − π(s)γ1 (s, z)]v1 (dz)ds 0 R0 0 R0 ) Z tZ + [log(1 − κ(s)γ2 (s, z)) + κ(s)γ2 (s, z)]v2 (dz)ds (3.3) 0

R0

Based on the criterion of logarithm utility maximization of the terminal wealth, the optimization problem for the insurer can be formulated as ∗

U(x) , sup E[log X u (T )] = E[log X u (T )]

(3.4)

sup E (J u (T ))

(3.5)

u∈AH

This is equivalent to solve u∈AH

where

Z

T

h

1 (µ(s) − r(s)) π(s) + (λ(s) − p(s)) κ(s) − σ2 (s)π2 (s) + ρσ(s)q(s)π(s)κ(s) 2 0 Z T p Z T i 1 2 − 1 2 (σ(s)π(s) − ρq(s)κ(s)) d W s − 1 − ρ2 q(s)κ(s)d − W s2 − q (s)κ (s) ds + 2 0 0 Z TZ Z TZ 1 − + log(1 + π(s)γ1 (s, z))N˜ (d s, dz) + log(1 − κ(s)γ2 (s, z))N˜ 2 (d − s, dz) u

J (T ) =

0

+

Z

T

0

+

Z

T

0

Denote

Z

Z

0

R0

R0

[log(1 + π(s)γ1 (s, z)) − π(s)γ1 (s, z)]v1 (dz)ds

R0

[log(1 − κ(s)γ2 (s, z)) + κ(s)γ2 (s, z)]v2 (dz)ds

M2u (t)

Z th

Z t i = µ(s) − r(s) − σ (s)π(s) + ρσ(s)q(s)κ(s) ds + σ(s)dW s1 0 0 Z tZ Z tZ 2 γ1 (s, z)π(s) γ1 (s, z) N˜ 1 (d − s, dz) − v1 (dz)ds + 0 R0 1 + π(s)γ1 (s, z) 0 R0 1 + π(s)γ1 (s, z)

M1u (t)

and

R0

2

Z th Z t i 2 = λ(s) − p(s) + ρσ(s)q(s)π(s) − q (s)κ(s) ds − ρq(s)dW s1 0

7

0

(3.6)

−

Z

0

t

p

1−

ρ2 q(s)κ(s)dW s2

−

Z

0

t

γ2 (s, z) N˜ 2 (d − s, dz) − 1 − κ(s)γ2 (s, z)

Z tZ 0

R0

κ(s)γ22 (s, z) v2 (dz)ds, 1 − κ(s)γ2 (s, z) (3.7)

with t ∈ [0, T ]. Definition 3.1. We define AH as the set of all admissible strategies with initial condition X(0) = x that satisfy the following conditions. (1) u is caglad and adapted to the filtration H. (2) σ(s)π(s), s ∈ [0, T ], is forward integrable with respect to d − W 1 (s), and q(s)κ(s), s ∈ [0, T ], is forward integrable with respect to d −W 1 (s) and d − W 2 (s). γ1 (s,z) , s ∈ [0, T ], z ∈ R, are forward integrable (3) π(s)γ1 (s, z), log(1 + π(s)γ1 (s, z)) and 1+π(s)γ 1 (s,z) γ2 (s,z) with respect to N˜ 1 (d − t, dz). κ(s)γ2 (s, z), log(1 − κ(s)γ2 (s, z)) and 1−κ(s)γ , s ∈ [0, T ], z ∈ R, 2 (s,z) 2 − ˜ are forward integrable with respect to N (d t, dz). (4) π(s)γ1 (s, z) > −1 + επ for a.e. (t, z) with respect to dt × v1 (dz) for some επ ∈ (0, 1) depending on π. κ(s)γ2 (s, z) < 1 − ε′κ for a.e. (t, z) with respect to dt × v2 (dz) for some ε′κ ∈ (0, 1) depending on κ. (5) Z T   E |µ(s) − r(s)| |π(s)| + |λ(s) − p(s)| |κ(s)| + 1 + σ2 (s) π2 (s) + |ρσ(s)q(s)π(s)κ(s)| 0 Z Z    2 2 2 2 2 + 1 + q (s)κ (s) + π (s) γ1 (s, z)v1 (dz) + κ (s) γ22 (s, z)v2 (dz) ds < ∞ R0

R0

(6) For all u = (π, κ), β = (β1 , β2 ) ∈ AH with β1 and β2 bounded, there exists a constant ζ > 0 such that the families {M1(π+δβ1 ,κ)}δ∈(−ζ,ζ) and {M2(π,κ+δβ2 ) }δ∈(−ζ,ζ) are uniformly integrable. Note that, for (π, κ) ∈ AH and (β1 , β2) ∈ AH with β1 and β2 bounded, (π + δβ1 , κ + δβ2 ) ∈ AH for any δ ∈ (−ζ, ζ) with ζ > 0 small enough. Remark 3.2. Condition (6) can be verified under some mild conditions. Observe that,  since  both (π, κ) and (β1 , β2 ) belong to AH , we have 1 + (π(s) + δβ1 (s)) γ1 (s, z) > επ + δ εβ1 − 1 ,   dt × v1 (dz) –a.e. and 1 − (κ(s) + δβ2 (s)) γ2 (s, z) > ε′π + δ ε′β2 − 1 , dt × v2 (dz) –a.e., Thus, we can find ζ > 0 small enough such that for all δ ∈ (−ζ, ζ), we have 1 + (π(s) + δβ1 (s)) γ1(s, z) > επ − ζ1 , dt × v1 (dz)–a.e. for some ζ1 ∈ (0, επ ) and 1 − (κ(s) + δβ2 (s)) γ2 (s, z) > ε′κ − ζ2 , dt × v2 (dz) –a.e. for some ζ2 ∈ (0, ε′κ ). Moreover, for each positive integer n, we have 2 Z T Z γ1 (s, z) 1 ˜ N (ds, dz) E 1 1 + (π(s) + δβ1 (s)) γ1 (s, z) 0 n 6|z|6n  Z T Z 1 2 γ (s, z)v (dz)ds 6 E 1 1 1 (επ − ζ1 )2 0 n 6|z|6n Z T Z  1 2 6 E γ (s, z)v (dz)ds 1 1 (επ − ζ1 )2 0 R0

and

E

Z

T 0

Z

1 n 6|z|6n

2 γ2 (s, z) N˜ 2 (ds, dz) 1 − (κ(s) + δβ2 (s)) γ2 (s, z) 8

6 6 Suppose that E

R R T 0

R0

1 ε′κ − ζ2 1 ε′κ − ζ2


2 E
2 E

Z Z

T 0

0

Z

T Z



1 n 6|z|6n

R0

γ22 (s, z)v2 (dz)ds

 γ22 (s, z)v2 (dz)ds .



γi2 (s, z)vi (dz)ds (i = 1, 2) are finite and Condition (3) holds. Letting

n → ∞, by the definition of forward integral (Definition 2.5), we have 2 Z T Z γ1 (s, z) 1 − ˜ N (d s, dz) E 0 R0 1 + (π(s) + δβ1 (s)) γ1 (s, z) Z T Z  1 2 6 E γ (s, z)v (dz)ds <∞ 1 1 (επ − ζ1 )2 0 R0 and Z T Z 2 γ2 (s, z) E N˜ 2 (d − s, dz) 0 R0 1 − (κ(s) + δβ2 (s)) γ2 (s, z) Z T Z  1 2 E γ (s, z)v (dz)ds <∞ 6 2
2 ε′κ − ζ2 2 0 R0 2 2   Therefore, we have E M1(π+δβ1 ,κ) < ∞ and E M2(π,κ+δβ2 ) < ∞ uniformly in δ ∈ (−ζ, ζ) if,   and for example, the coefficients µ, r, σ, γ1, γ2 are bounded, consequently, M1(π+δβ1 ,κ) δ∈(−ζ,ζ)  (π,κ+δβ )  2 M2 are uniformly integrable. δ∈(−ζ,ζ)

4. Characterization of the optimal strategy under inside information

In this section, we will give some necessary conditions that an optimal strategy should satisfy. The following theorem is the main result of this section. Theorem 4.1. If the stochastic process u∗ = (π∗ , κ∗ ) ∈ AH is optimal for the problem (3.5), then ∗ ∗ ∗ ∗ the stochastic processes {M1(π ,κ ) (t)} and {M2(π ,κ ) (t)} are (H, P)–martingales (i.e. martingales with respect to the filtration H and under the probability measure P).  ∗   ∗  ∗ Proof. Suppose that u∗ = (π∗ , κ∗ ) ∈ AH is optimal. Then we have E J u (T ) > E J (π +δβ1 ,κ ) (T )  ∗   ∗ ∗  and E J u (T ) > E J (π ,κ +δβ2 ) (T ) , for all (β1 , ·), (·, β2) ∈ AH bounded and |δ| < ζ small enough,  ∗  ∗ which implies that δ = 0 is a local maximum point of the functions δ 7→ E J (π +δβ1 ,κ ) (T ) and  ∗   ∗ ∗   ∗ ∗  ∗ d d E J (π +δβ1 ,κ ) (T ) = 0 and dδ E J (π ,κ +δβ2 ) (T ) = 0. Since the δ 7→ E J (π ,κ +δβ2 ,) (T ) . Thus, dδ ∗

∗

∗

∗

families {M1(π +δβ1 ,κ ) }δ∈(−ζ,ζ) and {M2(π ,κ +δβ2 ) }δ∈(−ζ,ζ) are uniformly integrable, we can deduce that (Z T d  (π∗ +δβ1 ,κ∗ )  E J (T ) = E [µ(s) − r(s) − σ2 (s)π(s) + ρσ(s)q(s)κ(s)]β1 (s)ds dδ δ=0 0 Z T Z TZ γ1 (s, z)β1(s) ˜ 1 − + σ(s)β1 (s)d − W s1 + N (d s, dz) 0 0 R0 1 + π(s)γ1 (s, z) ) Z TZ γ12 (s, z)π(s)β1 (s) − v1 (dz)ds = 0 0 R0 1 + π(s)γ1 (s, z) 9

and

(Z T d  (π∗ ,κ∗ +δβ2 )  E J (T ) = E [λ(s) − p(s) + ρσ(s)q(s)π(s) − q2 (s)κ(s)]β2 (s)ds dδ δ=0 0 Z T Z T Z T p γ2 (s, z)β2 (s) ˜ 2 − − 2 − 1 2 N (d s, dz) 1 − ρ q(s)β2(s)d W s − − ρq(s)β2 (s)d W s − 0 1 − κ(s)γ2 (s, z) 0 0 ) Z TZ κ(s)γ22 (s, z)β2(s) − v2 (dz)ds = 0 0 R0 1 − κ(s)γ2 (s, z)

Let us choose β1 to be of the following form:

β1 (t) = α1 I(u,u+ε] (t), for α1 ∈ Hu , u ∈ [0, T ], and ε > 0 such that u + ε < T. Then we obtain ( Z E α1

u

u+ε



Z  µ(s) − r(s) − σ (s)π(s) + ρσ(s)q(s)κ(s) ds + 2

u+ε

u 2 γ1 (s, z)π(s)

σ(s)dW s1 +

) γ1 (s, z) 1 − ˜ N (d s, dz) − v1 (dz)ds = 0. u R0 1 + π(s)γ1 (s, z) u R0 1 + π(s)γ1 (s, z) i h  ∗ ∗ Equivalently, E α1 M1u (u + ε) − M1u (u) = 0 for all α1 ∈ Hu . Therefore,  i h ∗ ∗ E M1u (u + ε) − M1u (u) Hu = 0, u ∈ [0, T ]. Z

u+ε

Z

Z

u+ε

Z

∗

Hence, the process M1u is an (H, P)–martingale. By similar arguments, we can conclude that ∗  M2u is also an (H, P)–martingale. We recall that the integer valued random measure N 1 (dt, dz) and N 2 (dt, dz) have unique pre∗ dictable compensators v1H (dt, dz) and v2H (dt, dz), respectively, with respect to H. Then M1u (t) can be rewritten as Z t Z tZ   γ1 (s, z) 1 u∗ 1 1 σ(s)dW s + M1 (t) = N − v H (ds, dz) ∗ 0 0 R0 1 + π (s)γ1 (s, z) Z t Z tZ     γ1 (s, z) 1 1 2 ∗ ∗ v − v (ds, dz) + µ(s) − r(s) − σ (s)π (s) + ρσ(s)q(s)κ (s) ds + H F ∗ 0 0 R0 1 + π (s)γ1 (s, z) Z tZ γ12 (s, z)π∗ (s) − v1 (dz)ds, (4.1) ∗ 0 R0 1 + π (s)γ1 (s, z) with t ∈ [0, T ]. Here, v1F (ds, dz) = v1 (dz)ds. ∗ ∗ Further, we see that the orthogonal decomposition of {M1u (t)} into a continuous part {c M1u (t)} ∗ and a discontinuous part {d M1u (t)}, is given by Z t Z t 1 c u∗ σ(s)dW s + M1 (t) = σ(s)α1 (s)ds (4.2) 0

and

d

∗ M1u (t)

=

Z tZ 0

R0

0

  γ1 (s, z) 1 1 N − v (ds, dz), H 1 + π∗ (s)γ1 (s, z) 10

(4.3)

where {α1 (s)} is an H–adapted process. By uniqueness of the semimartingale decomposition of ∗ ∗ the H–semimartingale {M1u (t)}, we conclude that the finite variation part of {M1u (t)} must be 0. Consequently, we obtain the following result Z t Z tZ   γ1 (s, z) 1 1 σ(s)α1 (s)ds = v − v (ds, dz) H F ∗ 0 0 R0 1 + π (s)γ1 (s, z) Z t Z tZ γ12 (s, z)π∗ (s)   2 ∗ ∗ + µ(s) − r(s) − σ (s)π (s) + ρσ(s)q(s)κ (s) ds − v1 (dz)ds, (4.4) ∗ 0 0 R0 1 + π (s)γ1 (s, z)

with t ∈ [0, T ]. Similarly, we can write Z tZ Z tp Z t   γ2 (s, z) 2 2 2 1 u∗ 2 N − v (ds, dz) 1 − ρ q(s)dW s − ρq(s)dW s − M2 (t) = − H ∗ 0 R0 1 − κ (s)γ2 (s, z) 0 0 Z tZ Z t     γ2 (s, z) 2 2 + vF − vH (ds, dz) + λ(s) − p(s) + ρσ(s)q(s)π∗ (s) − q2 (s)κ∗ (s) ds ∗ 0 R0 1 − κ (s)γ2 (s, z) 0 Z tZ ∗ 2 κ (s)γ2 (s, z) − v2 (dz)ds, (4.5) ∗ 0 R0 1 − κ (s)γ2 (s, z) with t ∈ [0, T ]. Here, v2F (ds, dz) = v2 (dz)ds. ∗ Further, we can deduce that the orthogonal decomposition of {M2u (t)} into a continuous part ∗ ∗ {c M2u (t)} and a discontinuous part {d M2u (t)}, is given by Z tp Z t Z t Z tp 2 c u∗ 1 2 1 − ρ q(s)dW s − 1 − ρ2 q(s)α2 (s)ds M2 (t) = − ρq(s)dW s − ρq(s)α1 (s)ds − 0

0

0

0

(4.6)

and d

∗ M2u (t)

=−

Z tZ 0

R0

  γ2 (s, z) 2 2 N − v H (ds, dz), 1 − κ∗ (s)γ2 (s, z)

(4.7)

where {α2 (s)} is also an H–adapted process. ∗ By uniqueness of the semimartingale decomposition of the H–semimartingale {M2u (t)}, we ∗ deduce that the finite variation part of {M2u (t)} must be 0. Therefore, we have the following result Z t Z tp Z tZ   γ2 (s, z) 2 2 2 1 − ρ q(s)α2 (s)ds + ρq(s)α1 (s)ds + v − v F H (ds, dz) ∗ 0 0 0 R0 1 − κ (s)γ2 (s, z) Z t Z tZ γ22 (s, z)κ∗ (s)   ∗ 2 ∗ v2 (dz)ds = 0, + λ(s) − p(s) + ρσ(s)q(s)π (s) − q (s)κ (s) ds − ∗ 0 0 R0 1 − κ (s)γ2 (s, z) (4.8) for t ∈ [0, T ]. In summary, we have the following theorem. Theorem 4.2. Suppose that u∗ ∈ AH is an optimal strategy for problem (3.4). Then u∗ = (π∗ , κ∗ ) solves equations (4.4) and (4.8), where {α1 (s)} and {α2 (s)} are H–adapted processes, v1H and v2H are the H compensators of N 1 and N 2 respectively, and viF (ds, dz) = vi (dz)ds, i = 1, 2. In particular, we get 11

Corollary 4.3. Suppose H = F, That is, the insurer has no inside information. Then the necessary conditions for u∗ = (π∗ , κ∗ ) to be optimal are Z t Z tZ γ12 (s, z)π∗ (s)   2 ∗ ∗ µ(s) − r(s) − σ (s)π (s) + ρσ(s)q(s)κ (s) ds − v1 (dz)ds = 0, ∗ 0 0 R0 1 + π (s)γ1 (s, z) (4.9) and Z

t 0

  λ(s) − p(s) + ρσ(s)q(s)π∗ (s) − q2 (s)κ∗ (s) ds −

Z tZ 0

R0

κ∗ (s)γ22 (s, z) v2 (dz)ds = 0, 1 − κ∗ (s)γ2 (s, z) (4.10)

for a.e. t ∈ [0, T ].

From the argument that leads to Theorem 4.2, we can also deduce the following interesting result. Proposition 4.4. Assume u∗ = (π∗ , κ∗ ) foroour problem (3.4). n strategy o n R t R that there exists an optimal RtR , {W 1 (t)}06t6T , 0 R γ2 (s, z)N˜ 2 (ds, dz) Then the processes 0 R γ1 (s, z)N˜ 1 (ds, dz) 0 0 06t6T 06t6T and {W 2 (t)}06t6T are all H–semimartingales.

Proof. Suppose u∗ =n R(π∗R, κ∗ ) is an optimal strategy for oproblem equations (4.4)   n R t R(3.4). γBy   and o t γ1 (s,z) 2 (s,z) 1 1 2 2 (4.8) we deduce that 0 R 1+π∗ (s)γ1 (s,z) vH − vF (ds, dz) and 0 R 1−κ∗ (s)γ2 (s,z) vH − vF (ds, dz) 0  n R t R0  o are of tinite variation. From item (4) in Definition 3.1, we have 0 R γ1 (s, z) v1H − v1F (ds, dz) 0   o nR t R 2 2 and 0 R γ2 (s, z) vH − vF (ds, dz) are of finite variation.    RtR  R t0 R RtR Since 0 R γ1 (s, z)N˜ 1 (ds, dz) equals to 0 R γ1 (s, z) N 1 − v1H (ds, dz)+ 0 R v1H − v1F (ds, dz) 0 0  nR t R o  o n R t R0 and 0 R γ1 (s, z) N 1 − v1H (ds, dz) is an H–martingale, we conclude that 0 R γ1 (s, z)N˜ 1 (ds, dz) 0 0 nR t R o is an H–semimartingale. Similarly, we have 0 R γ2 (s, z)N˜ 2 (ds, dz) is also an H–semimartingale. 0 By (4.2) we have Z t Z t 1 c u∗ 1 d M1 (s) − α1 (s)ds, 0 6 t 6 T. Wt = 0 0 σ(s) Then {Wt1 }06t6T is an H–semimartingale. Moreover, from (4.6) we have Z t Z t Z t ρWt1 1 ρ 2 c u∗ Wt = − + α1 (s)ds + d M2 (s) + p α2 (s)ds, p p 0 0 0 1 − ρ2 q(s) 1 − ρ2 1 − ρ2

with t ∈ [0, T ]. Then {Wt2 }06t6T is also an H–semimartingale.



5. Optimal strategies for some particular cases In this section, we want to analyze the optimization problem in which the insurer has some inside information represented by the values of the underlying driving processes W1 (T 0 ), W2 (T 0 ), η1 (T 0 ), η2 (T 0 ) at some time T 0 > T. This means that the information flow the insurer owns is n o H , Ht , Ft ∨ σ (W1 (T 0 ), W2 (T 0 ), η1 (T 0 ), η2 (T 0 )) : t ∈ [0, T ] (5.1)

that can be used to make her decisions. We refer to Protter (2005, page 364) for the following result. 12

Proposition 5.1. Let H be given by (5.1). Then Z t Wi (T 0 ) − Wi (s) Wi (t) − ds T0 − s 0

and

ηi (t) −

Z

t 0

ηi (T 0 ) − ηi (s) ds = ηi (t) − T0 − s

i = 1, 2, are (H, P)–martingales.

Z tZ 0

T0 s

Z

R0

(5.2)

z ˜i N (dr, dz)ds, T0 − s

(5.3)

Proposition 5.1 implies that in the pesent situationn of enlargement filtrationo the processes o nof W2 (T 0 )−W2 (s) W1 (T 0 )−W1 (s) and , respectively. {α1 (s)} and {α1 (s)} in (4.2) and (4.6) are of the form T 0 −s T 0 −s i Moreover, we can give the explicit expressions of the H compensators vH of N i , i = 1, 2, by using Proposition 5.1. In fact, we have the following result. Proposition 5.2. The compensating measure viH of the jump measure N i with respect to H is given by Z T0 1 i vH (ds, dz) = vi (dz)ds + N˜ i (dr, dz)ds (5.4) T0 − s s Z T0 1 = N i (dr, dz)ds, i = 1, 2. (5.5) T0 − s s Proof. It is sufficient to show that if viH is the right hand side of (5.4), then Z tZ 0

R0

f (z)(N i − viH )(ds, dz)

is an H–martingale for all f , which are bounded deterministic functions on R, zero around zero, and that determine a measure on R with weight zero in zero. The same argument holds for f invertible functions that are integrable with respect to viH (note that this implies also integrability with respect to vi ). Let f (z) be such a function, Then η¯ i (t), t ∈ [0, T ], given by Z tZ η¯ i (t) , f (z)N˜ i (ds, dz) 0

R0

is a pure jump L´evy process. Denote F¯t , t ∈ [0, T ], as the completed filtration generated by the L´evy process W1 (t) + η¯ 1 (t) 06t6T and W2 (t) + η¯ 2 (t) 06t6T . Since f is invertible, we have F¯t = Ft and H¯ t = Ht , where H¯ t , F¯t ∨ σ for t ∈ [0, T ]. R t(W R 1 (T 0 ), W2 (T 0 ), η1 (TR0 ),t RηT20(TR 0 )),f (z) i ˜ ¯ From Proposition 5.1 we get that Mi (t) , 0 R f (z)N (ds, dz) − 0 s R T 0 −s N˜ i (dr, dz)ds, 0 0 i = 1, 2, are H–martingales. This proves (5.4), and (5.5) holds by a straight forward algebraic transformation.  Using the measures given by (5.4), the necessary conditions (4.4) and (4.8) that an optimal strategy u∗ = (π∗ , κ∗ ) must satisfy in our present context become Z t Z t   W1 (T 0 ) − W1 (s) 2 ∗ ∗ ds µ(s) − r(s) − σ (s)π (s) + ρσ(s)q(s)κ (s) ds + σ(s) T0 − s 0 0 Z t Z Z T0 γ1 (s, z) N˜ 1 (dr, dz)ds + ∗ (1 + π (s)γ1 (s, z)) (T 0 − s) 0 R0 s 13

− and Z

Z tZ 0

R0

γ12 (s, z)π∗ (s) v1 (dz)ds = 0 1 + π∗ (s)γ1 (s, z)

(5.6)

Z t   W1 (T 0 ) − W1 (s) ∗ 2 ∗ ρq(s) λ(s) − p(s) + ρσ(s)q(s)π (s) − q (s)κ (s) ds − ds T0 − s 0 0 Z t Z Z T0 Z tp γ2 (s, z) W2 (T 0 ) − W2 (s) 2 1 − ρ q(s) N˜ 2 (dr, dz)ds ds − − ∗ (1 − κ (s)γ (s, z)) (T − s) T − s 2 0 0 0 R0 s 0 Z tZ γ22 (s, z)κ∗ (s) v2 (dz)ds = 0, (5.7) − ∗ 0 R0 1 − κ (s)γ2 (s, z) R respectively. When R |γi (s, z)| vi (dz) < ∞, i = 1, 2, by item (4) of Definition 3.1, we have 0 Z Z 1 |γ1 (s, z)| v1 (dz) 6 |γ1 (s, z)| v1 (dz) < ∞ ∗ επ∗ R0 R0 |1 + π (s)γ1 (s, z)| t

and

Z

Z |γ2 (s, z)| 1 v2 (dz) 6 |γ2 (s, z)| v2 (dz) < ∞. ∗ εκ∗ R0 R0 |1 − κ (s)γ2 (s, z)| Thus, (5.6) and (5.7) can be rewritten as Z t W1 (T 0 ) − W1 (s)  ds µ(s) − r(s) − σ2 (s)π∗ (s) + ρσ(s)q(s)κ∗ (s) + σ(s) T0 − s 0 Z t Z Z T0 Z tZ γ1 (s, z) 1 + N (dr, dz)ds − γ1 (s, z)v1 (dz)ds = 0 (1 + π∗ (s)γ1 (s, z)) (T 0 − s) 0 R0 s 0 R0

and Z

(5.8)

Z t   W1 (T 0 ) − W1 (s) ∗ 2 ∗ ds λ(s) − p(s) + ρσ(s)q(s)π (s) − q (s)κ (s) ds − ρq(s) T0 − s 0 0 Z t Z Z T0 Z tp γ2 (s, z) W2 (T 0 ) − W2 (s) 2 N 2 (dr, dz)ds ds − − 1 − ρ q(s) ∗ (1 − κ (s)γ2 (s, z)) (T 0 − s) T0 − s 0 R0 s 0 Z tZ + γ2(s, z)v2 (dz)ds = 0, (5.9) t

0

R0

respectively. The following theorem shows that under some additional assumptions, (5.8) and (5.9) are also sufficient conditions for the strategy (π∗ , κ∗ ) to be optimal. R Proposition 5.3. Suppose that R |γi (s, z)| vi (dz) < ∞, i = 1, 2, the strategy (π∗ , κ∗ ) is optimal 0 for the insurer with the information flow H given by (5.1) if and only if (π∗ , κ∗ ) ∈ AH , and for a.e. (ω, s), (π∗ , κ∗ ) solves the equations W1 (T 0 ) − W1 (s) µ(s) − r(s) − σ2 (s)π∗ (s) + ρσ(s)q(s)κ∗ (s) + σ(s) T0 − s Z Z T0 Z γ1 (s, z) + N 1 (dr, dz) − γ1 (s, z)v1 (dz) = 0 ∗ (1 + π (s)γ (s, z)) (T − s) 1 0 R0 s R0 14

(5.10)

and W1 (T 0 ) − W1 (s) λ(s) − p(s) + ρσ(s)q(s)π∗ (s) − q2 (s)κ∗ (s) − ρq(s) T0 − s Z Z T0 p γ2 (s, z) W2 (T 0 ) − W2 (s) N 2 (dr, dz) − 1 − ρ2 q(s) − ∗ (1 − κ (s)γ (s, z)) (T − s) T0 − s 2 0 R0 s Z + γ2 (s, z)v2 (dz) = 0. (5.11) R0

Proof. By Proposition 5.1 and Proposition 5.2, the optimization problem (5.3) can be rewritten as sup E (J u (T )) u∈AH

(Z

T

h

1 (µ(s) − r(s)) π(s) + (λ(s) − p(s)) κ(s) − σ2 (s)π2 (s) + ρσ(s)q(s)π(s)κ(s) 2 u∈AH 0 Z T i   1 (σ(s)π(s) − ρq(s)κ(s)) dW s1 + α1 (s)ds − q2 (s)κ2 (s) ds + 2 0 Z T p   Z T 2 2 (σ(s)π(s) − ρq(s)κ(s)) α1 (s)ds − 1 − ρ q(s)κ(s) dW s + α2 (s)ds − 0 0 Z T p Z TZ 2 1 − ρ q(s)κ(s)α2 (s)ds + + log(1 + π(s)γ1 (s, z))(N 1 − v1H )(ds, dz) = sup E

+

Z

Z

0

T

0 T

Z

Z

0

2

R0

log(1 − κ(s)γ2 (s, z))(N −

R0

v2H )(ds, dz) +

log(1 − κ(s)γ2 (s, z))v2H (ds, dz) + 0 R0 ) Z TZ + κ(s)γ2 (s, z)v2 (dz)ds

+

0

Z

T 0

Z

Z

0

T

Z

R0

log(1 + π(s)γ1 (s, z))v1H (ds, dz)

π(s)γ1 (s, z)v1 (dz)ds R0

R0

(Z

T

h

1 (µ(s) − r(s)) π(s) + (λ(s) − p(s)) κ(s) − σ2 (s)π2 (s) + ρσ(s)q(s)π(s)κ(s) 2 u∈AH 0 Z T i W1 (T 0 ) − W1 (s) 1 (σ(s)π(s) − ρq(s)κ(s)) ds − q2 (s)κ2 (s) ds + 2 T0 − s 0 Z T p Z T Z Z T0 W2 (T 0 ) − W2 (s) log(1 + π(s)γ1 (s, z)) 1 2 − 1 − ρ q(s)κ(s) ds + N (dr, dz)ds T0 − s T0 − s 0 0 R0 s Z T Z Z T0 log(1 − κ(s)γ2 (s, z)) 2 N (dr, dz)ds + T0 − s 0 R0 s ) Z TZ Z TZ − π(s)γ1 (s, z)v1 (dz)ds + κ(s)γ2 (s, z)v2 (dz)ds = sup E

0

R0

0

R0

We can solve this optimization problem pointwise for each fixed (ω, s). Define G(u) = G(π, κ) 1 = (µ(s) − r(s)) π + (λ(s) − p(s)) κ − σ2 (s)π2 + ρσ(s)q(s)πκ 2 15

W1 (T 0 ) − W1 (s) 1 − q2 (s)κ2 + (σ(s)π − ρq(s)κ) 2 T0 − s Z Z T0 p log(1 + πγ1 (s, z)) 1 W2 (T 0 ) − W2 (s) − 1 − ρ2 q(s)κ + N (dr, dz) T0 − s T0 − s R0 s Z Z T0 log(1 − κγ2 (s, z)) 2 N (dr, dz) + T0 − s R0 s Z Z − πγ1 (s, z)v1(dz) + κγ2 (s, z)v2 (dz). R0

R0

∗

∗

∗

Then a stationary point u = (π , κ ) of G is determined by W1 (T 0 ) − W1 (s) ∂G = µ(s) − r(s) − σ2 (s)π∗ (s) + ρσ(s)q(s)κ∗ (s) + σ(s) ∂π u=u∗ T0 − s Z Z Z T0 γ1 (s, z) N 1 (dr, dz) − γ1 (s, z)v1 (dz) = 0 + ∗ (1 + π (s)γ (s, z)) (T − s) 1 0 R0 R0 s and

W1 (T 0 ) − W1 (s) ∂G = λ(s) − p(s) + ρσ(s)q(s)π∗ (s) − q2 (s)κ∗ (s) − ρq(s) ∂κ u=u∗ T0 − s Z Z T 0 p γ2 (s, z) W2 (T 0 ) − W2 (s) − 1 − ρ2 q(s) − N 2 (dr, dz) ∗ (1 − κ (s)γ2 (s, z)) (T 0 − s) T0 − s R0 s Z + γ2 (s, z)v2 (dz) = 0. R0

That is to say, u∗ = (π∗ , κ∗ ) satisfies (5.10) and (5.11). Moreover, we have Z Z T0 γ12 (s, z) ∂2G 2 = −σ (s) − N 1 (dr, dz) < 0, 2 ∂π2 (1 + π(s)γ1 (s, z)) (T 0 − s) R0 s Z Z T0 2 γ22 (s, z) ∂G 2 = −q (s) − N 2 (dr, dz) < 0, 2 ∗ ∂κ2 (1 − κ (s)γ2 (s, z)) (T 0 − s) R0 s and

!2 ∂2G ∂2G ∂2G · − ∂π2 ∂κ2 ∂π∂κ Z Z   2 2 2 2 = 1 − ρ σ (s)q (s) + σ (s) + q2 (s)

Z Z R0

> 0.

Z Z

s

R0 2 γ1 (s, z)

2

s

(1 −

κ∗ (s)γ

2

2 (s, z))

(T 0 − s)

N 2 (dr, dz)

N 1 (dr, dz)+

(1 + π(s)γ1 (s, z)) (T 0 − s) Z Z T0 γ22 (s, z) γ12 (s, z) 1 N (dr, dz) · N 2 (dr, dz) ∗ (s)γ (s, z))2 (T − s) (1 + π(s)γ1 (s, z))2 (T 0 − s) (1 − κ R0 s 0 2 R0

T0

T0

γ22 (s, z)

T0

s

Hence, we can conclude that the strategy u∗ = (π∗ , κ∗ ) given by (5.10) and (5.11) is optimal. On the other hand, if u∗ = (π∗ , κ∗ ) is an optimal strategy, then (5.8) and (5.9) hold for a.e. (w, t) by the preceding arguments. Therefore, u∗ = (π∗ , κ∗ ) satisfies (5.10) and (5.11) for a.e. (w, t).  16

The nonlinear equations (5.10) and (5.11) are the sufficient and necessary conditions that an optimal strategy u∗ = (π∗ , κ∗ ) must satisfy. However, generally speaking, we can not get the concrete analytic expressions of the optimal strategies from equations (5.10) and (5.11). In what follows, we consider some particular cases in which we can obtain the expressions of the optimal investment and risk control strategies in explicit forms. Proposition 5.4. Assume that v1 (dz) = v2 (dz) = 0, that is, no jumps are considered in the risky asset process and the insurer’s risk process. Then the unique optimal investment and risk control strategy (π∗ , κ∗ ) for the insurer with the information flow  H = Ht = Ft ∨ σ (W1 (T 0 ), W2 (T 0 )) : t ∈ [0, T ] is given by π∗ (s) =

and

µ(s) − r(s) ρ (λ(s) − p(s)) W1 (T 0 ) − W1 (s) ρ W2 (T 0 ) − W2 (s) + + − p 2 2 2 (1 − ρ )σ (s) (1 − ρ )q(s)σ(s) (T 0 − s)σ(s) T0 − s 1 − ρ2 σ(s) (5.12) κ∗ (s) =

λ(s) − p(s) W2 (T 0 ) − W2 (s) ρ (µ(s) − r(s)) + − . p 2 2 2 (1 − ρ )q(s)σ(s) (1 − ρ )q (s) (T 0 − s) 1 − ρ2 q(s)

(5.13)

Proof. When v1 (dz) = v2 (dz) = 0, then N 1 (dr, dz) = N 2 (dr, dz) = 0. By Proposition 5.3, we get µ(s) − r(s) − σ2 (s)π∗ (s) + ρσ(s)q(s)κ∗ (s) + σ(s) and

W1 (T 0 ) − W1 (s) =0 T0 − s W1 (T 0 ) − W1 (s) T0 − s

λ(s) − p(s) + ρσ(s)q(s)π∗ (s) − q2 (s)κ∗ (s) − ρq(s) −

p W2 (T 0 ) − W2 (s) = 0. 1 − ρ2 q(s) T0 − s

By solving these two equations directly, (5.12) and (5.13) can be derived. Remark 5.5. (1) Denote ∆1 (s) = can be rewritten as

W1 (T 0 )−W1 (s) (T 0 −s)σ(s)

and ∆2 (s) = − √

ρ

1−ρ2 σ(s)

W2 (T 0 )−W2 (s) . T 0 −s

 Then (5.12)

µ(s) − r(s) ρ (λ(s) − p(s)) + + ∆1 (s) + ∆2 (s). 2 2 (1 − ρ )σ (s) (1 − ρ2)q(s)σ(s)

(5.14)

µ(s) − r(s) µ(s) − r(s) W1 (T 0 ) − W1 (s) + ∆ (s) = + , 1 σ2 (s) σ2 (s) (T 0 − s)σ(s)

(5.15)

π∗ (s) =

The terms ∆1 (s) and ∆2 (s) represent the effects of the inside information W1 (T 0 ) and W2 (T 0 ) on the optimal investment strategy respectively. When ρ = 0, then ∆2 (s) = 0. This implies that if the risky asset process and the risk process are irrelevant, then the inside information W2 (T 0 ) of the risk process actually has no influence on the optimal investment strategy. This phenomenon coincides with our intuition. Moreover, in the case ρ = 0, (5.14) simplifies to π∗ (s) =

which coincides with the result in Pikovsky and Karatzas (1996) who considered the optimal investment problem under inside information. 17

(2) Denote Π(s) = −

W2 (T 0 )−W2 (s) (T 0 −s)

√

1−ρ2 q(s)

. Then (5.13) can be rewritten as

ρ (µ(s) − r(s)) λ(s) − p(s) + + Π(s). (5.16) 2 (1 − ρ )q(s)σ(s) (1 − ρ2 )q2 (s) The term Π(s) reflects the effect of the inside information W2 (T 0 ) on the optimal risk control strategy. Different from (5.14), we can see from (5.16) that the inside information W1 (T 0 ) about the risky asset process has no impact on the optimal risk control κ∗ (s). When ρ = 0, that is, the risky asset process and the risk process are irrelevant, then Π(s) = 2 (T 0 )−W2 (s) − W(T and (5.16) simplifies to 0 −s)q(s) κ∗ (s) =

λ(s) − p(s) W2 (T 0 ) − W2 (s) − , (5.17) q2 (s) (T 0 − s)q(s) which is irrelevant with the risky asset process. Proposition 5.4 gives the optimal investment and risk control strategies when there are no jumps in the risky asset process and the risk process. Next, we will consider the cases that there exist jumps in the risky asset process or the risk process and the insurer has some inside information on the jump processes. κ∗ (s) =

Proposition 5.6. Assume that v1 (dz) = λ1 δ1 (dz), v2 (dz) = 0, and γ1 (t, z) = γ1 z, where λ1 , γ1 > 0 are constants. In this case, Rthere are no jumps in the risk process, and the jumps term in the risky asset process becomes R γ1 (t, z)N˜ 1 (dt, dz) = γ1 d(Q1 (t) − λ1 t), where {Q1 (t)} is a Poisson 0 ∗ ∗ process with intensity λ1 > 0. Then the optimal  investment and risk control strategy (π , κ ) for the insurer under the information flow H = Ht = Ft ∨ σ (W1 (T 0 ), W2 (T 0 ), Q1 (T 0 )) : t ∈ [0, T ] is given by ! q
2 1 ∗ 2 2 2 2 2 π (s) = Σ1 (s) − 2(1 − ρ )σ (s) − 4γ1 (1 − ρ )σ (s)l1 (s) , −Σ1 (s) + 2γ1 (1 − ρ2 )σ2 (s) (5.18) and λ(s) − p(s) ρα1 (s) ρσ(s) ∗ π (s) + + + κ∗ (s) = q(s) q2 (s) q(s)

p

1 − ρ2 α2 (s) , q(s)

(5.19)

1 (s) 2 (s) 1 (s) where α1 (s) = − W1 (TT00)−W , α2 (s) = − W2 (TT00)−W , l1 (s) = − Q1 (TT00)−Q and Σ1 (s) = γ1 (1 − −s −s −s p γ ρ(λ(s)−p(s))σ(s) 1 2 − γ1 (µ(s) − r(s) − λ1 γ1 ) + (1 − ρ2 )σ2 (s). ρ )σ(s)α1 (s) − γ1 ρ 1 − ρ2 σ(s)α2 (s) − q(s)

Proof. When v1 (dz) = λ1 δ1 (dz), v2 (dz) = 0, and γ1 (t, z) = γ1 . Then N 2 (dr, dz) = 0. By Proposition 5.3, we get γ1 l1 (s) µ(s) − r(s) − σ2 (s)π∗ (s) + ρσ(s)q(s)κ∗ (s) − σ(s)α1 (s) − λ1 γ1 − = 0, (5.20) 1 + π∗ (s)γ1 and p ρq(s)α1 (s) + 1 − ρ2 q(s)α2 (s) + λ(s) − p(s) + ρσ(s)q(s)π∗ (s) − q2 (s)κ∗ (s) = 0. (5.21)

From (5.21) we can easily deduce that (5.19) holds. Substituting (5.19) into (5.20) and by some computations we obtain 1 1 (5.22) γ1 (1 − ρ2 )σ2 (s) (π∗ (s))2 + Σ1 (s)π∗ (s) + Σ1 (s) − (1 − ρ2 )σ2 (s) + γ1 l1 (s) = 0. γ1 γ1 18

Since γ1 (1 − ρ2 )σ2 (s) > 0 and −l1 (s) > 0, we get

! 1 1 2 2 − 4γ1 (1 − ρ )σ (s) Σ1 (s) − (1 − ρ )σ (s) + γ1 l1 (s) γ1 γ1  2 = Σ1 (s) − 2(1 − ρ2 )σ2 (s) − 4γ12 (1 − ρ2 )σ2 (s)l1 (s) > 0,

Σ21 (s)

2

2

(5.22)

which implies that the discriminant of the quadratic equation (5.22) is positive. Thus, equation (5.21) has two solutions. One of them is given by ! q
1 2 ∗ −Σ1 (s) − π− (s) = Σ1 (s) − 2(1 − ρ2 )σ2 (s) − 4γ12 (1 − ρ2 )σ2 (s)l1 (s) 2γ1 (1 − ρ2 )σ2 (s)   1 1 2 2 6 , Σ (s) − 2(1 − ρ )σ (s) − Σ (s) − < 1 1 2γ1 (1 − ρ2 )σ2 (s) γ1 which is not included in the admissible set AH . The other solution is ! q
2 1 ∗ 2 2 2 2 2 π+ (s) = Σ1 (s) − 2(1 − ρ )σ (s) − 4γ1 (1 − ρ )σ (s)l1 (s) −Σ1 (s) + 2γ1 (1 − ρ2 )σ2 (s)   1 1 2 2 > . > Σ (s) − 2(1 − ρ )σ (s) − Σ (s) + 1 1 2γ1 (1 − ρ2 )σ2 (s) γ1 It is easy to verify that π∗+ (s) is admissible. Consequently, (π∗ (s), κ∗ (s)) given by (5.18) and (5.19) is the optimal strategy.



Remark 5.7. When ρ , 0, we can see from (5.18) and (5.19) that all the inside information W1 (T 0 ), W2 (T 0 ) and Q1 (T 0 ) have some influences on the optimal investment π∗ (s) and the risk control κ∗ (s). When ρ = 0, then (5.18) and (5.19) become  1 ∗ − γ1 σ(s)α1 (s) + γ1 (µ(s) − r(s) − λ1 γ1 ) − σ2 (s) π (s) = 2 2γ1 σ (s) q 
2 2 2 2 (µ(s) ) −γ1 σ(s)α1 (s) + γ1 − r(s) − λ1 γ1 + σ (s) − 4γ1 σ (s)l1 (s) (5.23) +

and

κ∗ (s) =

λ(s) − p(s) α2 (s) − , q2 (s) q(s)

(5.24)

respectively. In particular, if γ1 = 1, (5.23) coincides with expression (72) in Di Nunno et al. (2006) who considered the optimal investment problem under inside information. This implies the risk process has no effect on the optimal investment when the risky asset process and the risk process are irrelevant. Moreover, (5.24) coincides with (5.17), which implies that both the inside information Q1 (T 0 ) and W 1 (T 0 ) have no effect on the optimal risk control when ρ = 0. Proposition 5.8. Assume that v1 (dz) = 0, v2 (dz) = λ2 δ1 (dz) and γ2 (t, z) = γ2z, where λ2 , γ2 > 0 are constants. That is to say, there are no jumps in the risky asset process, and the jump term in R the risk process has the form R γ2 (t, z)N˜ 2 (dt, dz) = γ2 d(Q2 (t) − λ2 t), where {Q2 (t)} is a Poisson 0 process with intensity λ2 > 0. Then the optimal investment and risk control strategy (π∗ , κ∗ ) for  the insurer with the information flow H = Ht = Ft ∨ σ (W1 (T 0 ), W2 (T 0 ), Q2 (T 0 )) : t ∈ [0, T ] is 19

given by π∗ (s) = and 1 κ (s) = Σ2 (s) − 2γ2 (1 − ρ2 )q2 (s) ∗

µ(s) − r(s) α1 (s) ρq(s) ∗ κ (s) + − σ(s) σ2 (s) σ(s) q

Σ2 (s) − 2(1 −


ρ2 )q2 (s) 2

−

4γ22 (1

(5.25)

−

ρ2 )q2 (s)l

! 2 (s) , (5.26)

1 (s) 2 (s) 2 (s) where α1 (s) = − W1 (TT00)−W , α2 (s) = − W2 (TT00)−W as in Proposition 5.6, l2 (s) = − Q2 (TT00)−Q and −s −s −s p γ ρ(µ(s)−r(s))q(s) Σ2 (s) = γ2 1 − ρ2 q(s)α2 (s) + γ2 (λ(s) − p(s) + λ2 γ2 ) + 2 σ(s) + (1 − ρ2)q2 (s).

Proof. When v1 (dz) = 0, v2 (dz) = λ2 δ1 (dz), and γ2 (t, z) = γ2 z. Then N 1 (dr, dz) = 0. By Proposition 5.3, we have and

µ(s) − r(s) − σ2 (s)π∗ (s) + ρσ(s)q(s)κ∗ (s) − σ(s)α1 (s) = 0

(5.27)

p

1 − ρ2 q(s)α2 (s) + λ(s) − p(s) + ρσ(s)q(s)π∗ (s) γ2 l2 (s) = 0. (5.28) − q2 (s)κ∗ (s) + λ2 γ2 + 1 − κ∗ (s)γ2 From (5.27) we can easily deduce the equation (5.25). Substituting (5.25) into (5.28) and by some computations we obtain ρq(s)α1 (s) +

γ2 (1 − ρ2 )q2 (s) (κ∗ (s))2 − Σ2 (s)κ∗ (s) +

q2 (s)(1 − ρ2 ) 1 Σ2 (s) − + γ2 l2 (s) = 0. γ2 γ2

(5.29)

Since γ2 (1 − ρ2 )q2 (s) > 0 and −l2 (s) > 0, we get

! (1 − ρ2 )q2 (s) 1 − 4γ2(1 − ρ )q (s) Σ2 (s) − + γ2 l2 (s) γ2 γ2  2 = Σ2 (s) − 2(1 − ρ2 )q2 (s) − 4γ22 (1 − ρ2 )q2 (s)l2 (s) > 0.

Σ22 (s)

2

2

It means that the discriminant of the quadratic equation (5.29) is positive. Thus, equation (5.29) has two solutions. One of them is given by q
Σ2 (s) − 2(1 − ρ2 )q2 (s) 2 − 4γ22 (1 − ρ2 )q2 (s)l2 (s) Σ2 (s) + κ+∗ (s) = 2γ2 (1 − ρ2 )q2 (s) Σ2 (s) + Σ2 (s) − 2(1 − ρ2 )q2 (s) 1 > > , 2 2 2γ2 (1 − ρ )q (s) γ2 which is not included in the admissible set AH . The other solution is q
Σ2 (s) − Σ2 (s) − 2(1 − ρ2 )q2 (s) 2 − 4γ22 (1 − ρ2 )q2 (s)l2 (s) κ−∗ (s) = 2γ2 (1 − ρ2 )q2 (s) Σ2 (s) − Σ2 (s) − 2(1 − ρ2 )q2 (s) 1 6 . < 2 2 2γ2 (1 − ρ )q (s) γ2 ∗ It is easy to verify that κ− (s) is admissible. Consequently, (π∗ (s), κ∗ (s)) given by (5.25) and (5.26) is the optimal strategy. 20



Remark 5.9. When ρ , 0, we can see from (5.19) that the optimal risk control κ∗ (s) depends on the inside information W 2 (T 0 ) and Q2 (T 0 ), and it is independent of the inside information W 1 (T 0 ) of the risky asset process. However, the optimal investment π(s) may depend on all the inside information W 1 (T 0 ), W 2 (T 0 ) and Q2 (T 0 ). When ρ = 0, then (5.25) and (5.26) become µ(s) − r(s) α1 (s) π∗ (s) = − (5.30) σ2 (s) σ(s) and  1 κ∗ (s) = γ2 q(s)α2 (s) + γ2 (λ(s) − p(s) + λ2 γ2 ) + q2 (s) 2γ2 q2 (s) q 
γ2 q(s)α2 (s) + γ2 (λ(s) − p(s) + λ2 γ2 ) − q2 (s) 2 − 4γ22 q2 (s)l2 (s) , − respectively. (5.30) coincides with (5.15), which implies that both the inside information Q2 (T 0 ) and W 2 (T 0 ) have no effect on the optimal investment strategy when ρ = 0. Moreover, if H = F, that is, there is no inside information for the insurer. Then the processes {α1 (s)}, {α2 (s)} and {l2 (s)} from the compensators of {W1 (s)}, {W2 (s)} and {Q2 (s)} become α1 (s) = α2 (s) = 0 and l2 (s) = −λ2 , s ∈ [0, T ]. Thus, (5.25) and (5.26) become µ(s) − r(s) ρq(s) ∗ π∗ (s) = + κ (s) σ2 (s) σ(s) and "  ρ (µ(s) − r(s)) q(s)  1 ∗ γ2 λ(s) − p(s) + λ2 γ2 + + (1 − ρ2 )q2 (s) κ (s) = 2 2 2γ2(1 − ρ )q (s) σ(s) s !2 # γ2ρ(µ(s) − r(s))q(s) − − (1 − ρ2 )q2 (s) + 4λ2 γ22 q2 (s)(1 − ρ2 ) , γ2 (λ(s) − p(s) + λ2 γ2 ) + σ(s)

respectively, which coincide with the results (4) and (7) in Zou and Cadenillas (2014) with only notational differences. 6. Conclusion We have investigated the optimal investment and risk control problem for an insurer who has some inside information about the financial risky asset and the insurance policies. The insurer’s risk process and the financial risky asset process are assumed to be correlated jump diffusion processes with very general forms. With the aim of maximizing the logarithmic utility of the terminal wealth, we solve the problem by using forward stochastic calculus. In the general setting, we give some necessary conditions that the optimal strategy should satisfy under some mild technical conditions. Some interesting cases are studied where the optimal strategies are derived in explicit forms by using enlargement of filtration techniques. It turns out that the inside information does have some effects on the optimal strategies. Compared to Zou and Cadenillas (2014), the contributions of this paper include three aspects. First, in our optimization problem the insurer may have some inside information about the financial risky asset and the insurance policies that can be used for her to make decisions. When there is no insider information for the insurer, our results coincide with that in Zou and Cadenillas (2014). Second, the parameters in the financial risky asset process and the insurance policies are assumed to be general stochastic processes, not necessarily constants or deterministic constants. Third, we use 21

a new approach based on forward stochastic calculus and enlargement of filtration techniques to solve our problem. However, there are some interesting problems which deserve to be investigated further in future study. For example, only logarithmic utility maximization is considered here. One can also consider other utility maximization criteria, such as exponential utility maximization and power utility maximization. Under the criterion of exponential or power utility maximization, we could also give some necessary conditions that the optimal strategy must satisfy. But it is very difficult for us to derive the implicit expressions of the optimal strategy even in some particular cases studied in Section 5. Besides this point, it is valuable to consider the optimization problem with some constraints on the investment and risk control. When there are some constraints on the strategy, we could not derive the necessary conditions that the optimal strategy must satisfy by using the same perturbation argument as in the proof of Theorem 4.1. To overcome the difficulty about constraints, we should use some other perturbation arguments or very different methods. Acknowledgements The authors are very grateful to the Editor and the referee for their helpful comments and suggestions on the original version of the manuscript, which led to this improved version of the manuscript. References [1] Bai, L., Guo, J., 2008. Optimal proportioanl reinsurance and investment with multiple risky assets and no short constraint. Insurance: Mathematics and Economics 42, 968–957. [2] Baltas, I.D., Frangos, N.E., Yannacopoulos, A.N., 2012. Optimal investment and reinsurance policies in insurance markets under the effect of inside information. Applied Stochastic Models in Business and Industry 28 (6), 506–528. [3] Bi, J.N., Guo, J.Y.,, 2013. Optimal mean–variance problem with constrained controls in a jump–diffusion financial market for an insurer. Journal of Optimization Theory and Applications 157, 252–275. [4] Biagini, F., Øksendal, B., 2005. A general stochastic calculus aproach to insider trading. Applied Mathematics and Optimization 52, 167–181. [5] Di Nunno, G., Øksendal, B., Proske, F., 2009. Malliavin Calculus for L´evy Processes with Applications to Finance. Springer. [6] Di Nunno, G., Meyer–Brandis, T., Øksendal, B., Proske, F., 2005. Malliavin calculus and anticipate Itˆo formulae for L´evy processes. Inf. Dim. Anal. Quant. Probab. Rel. Topics 8, 235–258. [7] Di Nunno, G., Meyer–Brandis, T., Øksendal, B., Proske, F., 2006. Optimal portfolio for an insider in a market driven by L´evy processes. Quantitative Finance 6 (1), 83–94. [8] Danilova A., Monoyios, M., Ng, A., 2010. Optimal investment with inside information and parameter uncertainty. Mathematics and Financial Economics 3, 13–38. [9] Guan, G., Liang Z., 2014. Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks. Insurance: Mathematics and Economics 55, 105–115. [10] Hu, Y., Øksendal, B., 2007. Optimal smooth portfolio selection for an insider. Journal of Applied Probability 44 (3), 742–752. [11] Karatzas, I., Pikovsky, I., 1996. Anticipating portfolio optimization. Advances in Applied Probability 28, 1095–1122. [12] Kchia, Y., Protter, P., 2014. On Progressive Filtration Expansions with a Process: Applications to Insider Trading, arXiv: 1403.6323v2. [13] Kohatsu–Higa, A., Sulem, A., 2006. Utility maximization in an insider influenced market. Mathematical Finance 16 (1), 153–179. [14] Kohatsu-Higa, A., Yamazato, M., 2011. Insider Models with Finite Utility in Markets with Jumps. Appllied Mathematics and Optimization 64, 217–255. 22

[15] Liu, Y., Ma, J., 2009. Optimal reinsurance/investment problems for general insurance models. Annals of Applied Probability 19 (4), 1495–1528. [16] Le´on, J.A., Navarro, R., Nualart, D., 2003. An anticipating calculus approach to the utility maximization of an insider. Mathematical Finance 13, 171–185. [17] Menoukeu Pamen, O., Proske, F., Binti Salleh H., 2014. Stochastic Differential Games in Insider Markets via Malliavin Calculus. Journal of Optimization Theory and Applications 160 (1), 302–343. [18] Perera, R., 2010. Optimal consumption, investment and insurance with insurable risk for an investor an a L´evy market. Insurance: Mathematics and Economics 46, 479–484. [19] Protter, P., 2005. Stochastic integration and differential equations. Springer. [20] Peng, X., Hu, Y., 2013. Optimal proportional reinsurance and investment under partial information. Insurance: Mathematics and Economics 53, 416–428. [21] Peng, X., Wei, L., Hu, Y., 2014. Optimal investment, consumption and proportional reinsurance for an insurer with option type payoff. Insurance: Mathematics and Economics 59, 78–86. [22] Russo, F. Vallois, P., 1993. Forward, backward and symmetric stochastic integration. Probability Theory and Related Fields 97, 403–421. [23] Schimidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Annals of Applied Probability 12, 890–907. [24] Stein, J., 2012. Stochastic Control and the U.S. Financial Debt Crisis. Springer. [25] Wang, Z., Xia, J., Zhang, L., 2007. Optimal investment for an insurer: the martingale approach. Insurance: Mathematics and Economics 40, 322–334. [26] Zou, B., Cadenillas, A., 2014. Optimal investment and risk control policies for an insurer: Expected utility maximization. Insurance: Mathematics and Economics 58, 57–67. [27] Zeng, Y., Li, Z.F., 2011. Optimal time–consistent investment and reinsurance policies for mean–variance insurers. Insurance: Mathematics and Economics 49, 145–154.

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