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Markovian forward–backward stochastic differential equations and stochastic flows
Systems & Control Letters 61 (2012) 1017–1022
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Markovian forward–backward stochastic differential equations and stochastic flows Robert J. Elliott a,b,∗ , Tak Kuen Siu c a
School of Mathematical Sciences, University of Adelaide, SA 5005, Australia
b
Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada
c
Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, NSW 2109, Sydney, Australia
article
info
Article history: Received 16 November 2011 Accepted 16 April 2012 Available online 5 September 2012 Keywords: Markovian forward–backward stochastic differential equations Stochastic flows Martingale representation Special semimartingale Convex risk measures
abstract Markovian forward–backward stochastic differential equations, (MFBSDEs), are discussed by exploiting techniques of stochastic flows. Using martingale representation, a differentiation rule, stochastic flows of diffeomorphisms and the unique decomposition of special semimartingales, we identify the solution of the backward system of the FBSDE. Applications of the result to convex risk measures are discussed. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Backward stochastic differential equations, (BSDEs), first appeared in the work of Bismut [1], where he discussed a maximum principle for stochastic optimal control using linear BSDEs. In a later paper [2], he established some existence and uniqueness results for BSDEs. The theory of general BSDEs was established in the work of Pardoux and Peng [3], where the existence and uniqueness results for general BSDEs were established. The link between BSDEs and partial differential equations (PDEs) was investigated by Peng [4] and Pardoux and Peng [5]. The study of forward–backward stochastic differential equations (FBSDEs) began in the early 1990s. Some examples include Antonelli [6], Ma et al. [7] and Hu and Peng [8]. An FBSDE is a coupled system consisting of a forward stochastic differential equation and a backward one, where the coefficient of the backward equation depends on the solution of the forward one. Nowadays, both BSDEs and FBSDEs have many important applications in Mathematical Finance, Stochastic Control and Partial Differential Equations. For an excellent account of FBSDEs and their applications, please refer to Ma and Yong [9]. Stochastic flows are an important concept in stochastic differential equations. They describe how the trajectories of the
∗ Corresponding author at: Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada. Tel.: +1 403 220 5540; fax: +1 403 770 8104. E-mail addresses: [email protected] (R.J. Elliott), [email protected] (T.K. Siu). 0167-6911/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2012.04.013
solutions of stochastic differential equations change with respect to their initial conditions. The study of stochastic flows was initiated by Gihman and Skorohod. Some early works on stochastic flows include Ikeda and Watanabe [10], Bismut [11], Le Jan and Watanabe [12] and Kunita [13]. Stochastic flows have been applied to Malliavin calculus, mathematical finance, stochastic filtering and control. Elliott and Kohlmann [14,15] applied stochastic flows to Malliavin calculus in the diffusion and the jumpdiffusion cases. In particular, they derived an integration-byparts formula using stochastic flows and discussed the existence and smoothness of the density functions of the solutions of stochastic differential equations. Baras et al. [16], Elliott [17] and Elliott and Yang [18] applied the concept of stochastic flows to derive minimum principles for stochastic optimal control in fully and partially observed diffusion models. Colwell et al. [19], Colwell and Elliott [20], Elliott et al. [21] and Elliott and Siu [22] applied stochastic flows to discuss the pricing and hedging of contingent claims. Elliott and van der Hoek [23] and Elliott and Siu [24] applied stochastic flows to derive exponential affine formulae for bond valuation under stochastic interest rate models. It appears interesting to further explore the link and application of stochastic flows to other important areas of stochastic analysis and mathematical finance from both theoretical and practical perspectives. In this paper, an approach based on stochastic flows is used to discuss the solution of a Markovian forward–backward stochastic differential equation, (MFBSDE), driven by a standard Brownian motion. More specifically we consider the situation where the
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solutions of the forward and backward systems in the MFBSDE are of Markovian type. In this case, the martingale component in the backward system of the MFBSDE can be related to a Markov process with respect to the filtration generated by a standard Brownian motion. Applying a martingale representation, a differentiation rule and a unique decomposition of a special semimartingale, we identify the solution of the backward system of the MFBSDE. In particular, the first component of the solution is related to a solution of a partial differential equation, and a representation of the control component of the solution is obtained using the Jacobians of stochastic flows of diffeomorphisms. By noting the relationship between a convex risk measure and the initial value of the solution of the backward system of the MFBSDE, we apply the martingale representation result to evaluate a convex risk measure in a Markovian environment. The paper is organized as follows. The next section discusses the concepts of MFBSDEs and some related results. In Section 3, we discuss stochastic flows of diffeomorphisms. Section 4 establishes the main result which gives a representation of the solution of the backward system of the MFBSDE. In Section 5, we apply the result of Section 4 to discuss convex risk measures. 2. Markovian forward–backward stochastic differential equations Consider a complete filtered probability space (Ω , F , F, P ), where T denotes the time interval [0, T ], T < ∞, and F := {F (t )|t ∈ T } is a filtration satisfying the usual conditions, (i.e. right continuity and P-completeness). Let W := {W (t )|t ∈ T } be an (F, P ) standard Brownian motion, which describes the randomness of both the forward and backward systems in a Markovian forward–backward stochastic differential equation, (MFBSDE), to be defined later. A forward–backward stochastic differential equation, (FBSDE), consists of a forward system and a backward system. The forward system is described by a forward stochastic differential equation, (FSDE), while the backward one is described by a backward stochastic differential equation, (BSDE), whose data, namely, the driver function g and the terminal condition Φ , depend on the solution of the FSDE. If we take into account the initial state (t , x) of the FSDE, (i.e., the state process), the solution (Y , Z ) of the BSDE is regarded as the solution of a parametrized BSDE whose data (g , Φ ) are parametrized by the initial state (t , x). This dependence is used to discuss the regularity properties of the solutions of the BSDE which follow from the regularity properties of the coefficients of the FBSDE. See El Karoui et al. [25]. Since this is not our focus, we shall not discuss this dependence further. Here we consider the situation when the solution of the FSDE is Markovian with respect to the filtration F. The key property of the FBSDE is that the solution (Y , Z ) of the BSDE can be written as, a (deterministic), function of time t and the value of the state process at that time. Consequently, from the Markovian assumption for the solution of the FSDE, the solution (Y , Z ) of the BSDE is Markovian with respect to F. This property is important when we use a stochastic flow approach to discuss the solution of the BSDE. To simplify our discussion and notation, we consider real-valued FSDE and BSDE. The results in this paper can be extended to the multivariate case. Let µ and σ be real-valued and positive-valued functions defined on T × ℜ. Suppose g is a real-valued, square-integrable, Borel-measurable function on T × ℜ3 and Φ is a real-valued, square-integrable, Borel-measurable function on ℜ. As usual, we impose the following standard assumptions of Lipschitz continuity for the coefficients in the FSDE and BSDE. These are required for the existence, uniqueness and regularity of the solutions of the BSDE and FSDE. Suppose there is a positive constant K such that
1. for each x1 , x2 , y1 , y2 , z1 , z2 ∈ ℜ and each t ∈ T ,
|µ(t , x1 ) − µ(t , x2 )| + |σ (t , x1 ) − σ (t , x2 )| ≤ K (1 + |x1 − x2 |), |g (t , x, y1 , z1 ) − g (t , x, y2 , z2 )| ≤ K (|y1 − y2 | + |z1 − z2 |) 2. for each t ∈ T , (x, y, z ) ∈ ℜ3 and each k ≥
1 , 2
|µ(t , x)| + |σ (t , x)| ≤ K (1 + |x|), |g (t , x, y, z )| + |Φ (x)| ≤ K (1 + |x|k ). Suppose X := {X (t )|t ∈ T } is the unique strong solution of the following FSDE: dX (t ) = µ(t , X (t ))dt + σ (t , X (t ))dW (t ),
t ∈T
X (0) = x0 ∈ ℜ.
(1)
Consider now an associated BSDE defined as follows:
−dY (t ) = g (t , X (t ), Y (t ), Z (t ))dt − Z (t )dW (t ), Y (T ) = Φ (X (T )).
t ∈T (2)
The solution of the BSDE is the pair {(Y (t ), Z (t ))|t ∈ T }. We suppose that {(Y (t ), Z (t ))|t ∈ T } is square integrable. The coupled system consisting of (1) and (2) is called an FBSDE and its solution is given by the triple {(X (t ), Y (t ), Z (t ))|t ∈ T }. For the existence and uniqueness of the solution of the FBSDE, please refer to the monograph by Ma and Yong [9]. It was noted in [25] that the solution {(Y (t ), Z (t ))|t ∈ T } in the backward system of the FBSDE is Markovian with respect to the filtration F. That is, there are real-valued, (deterministic), functions Ψ1 and Ψ2 on T × ℜ such that for each t ∈ T , Y (t ) = Ψ1 (t , X (t )), Z (t ) = Ψ2 (t , X (t )). The object of this paper is to identify explicitly the process {Z (t )|t ∈ T } in the backward system of the FBSDE by exploiting stochastic flows of diffeomorphisms. 3. Stochastic flows of diffeomorphisms In this section we present the key ideas and results of stochastic flows which will be used to identify the control process {Z (t )|t ∈ T }. To apply the results in stochastic flows, we must impose some additional differentiability and growth conditions on the coefficients of the FSDE. In particular, we assume that the coefficients µ(t , x) and σ (t , x) of the FSDE are three times differentiable in x, and, together with their derivatives, have linear growth in x. Suppose, for each t , s ∈ T with t ≥ s, ξs,t (x) is the unique strong solution of (1) which has initial condition:
ξs,s (x) = x ∈ ℜ. From some standard results in [11,13], there is a set N ⊂ Ω of measure zero such that if ω ̸∈ N, there is a version of ξs,t (x) which is twice differentiable in x and continuous in t and s. The map x → ξs,t (x) is called a stochastic flow. Write, for each t , s ∈ T with t ≥ s, Ds,t :=
∂ξs,t (x) , ∂x
for the derivative of the map x → ξs,t (x) with respect to x. Suppose µx (t , x) and σx (t , x) are the derivatives of µ(t , x) and σ (t , x) with respect to x, respectively. Then D is the solution of the linearized equation: dDs,t = µx (t , X (t ))Ds,t dt + σx (t , X (t ))Ds,t dW (t ), with initial condition Ds,s = 1.
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1019
t
1 It is known that the inverse D− s,t of Ds,t exists and satisfies the following stochastic differential equation:
Y¯ (t ) =
1 2 dD− s,t = −Ds,t µx (t , X (t )) − σx (t , X (t )) dt
where γ := {γ (t )|t ∈ T } is an F-predictable process such that
−1
1 − D− s,t σx (t , X (t ))dW (t ),
(4)
This section presents the main result of the paper which gives a partial differential equation, (PDE), representation for the first component of the solution of the backward system (2) and an expectation representation for the control component of the solution of the backward system (2) using stochastic flows. Firstly, we assume the following differentiability and growth conditions: 1. g (t , x, y, z ) and Φ (x) are twice continuously differentiable in x, and, together with their derivatives, have linear growth in x, for each (t , y, z ) ∈ T × ℜ2 ; 2. g (t , x, y, z ) is continuously differentiable in t, for each (x, y, z ) ∈ ℜ3 ; 3. Ψ1 (t , x) and Ψ2 (t , x) are continuously differentiable in t and twice continuously differentiable in x. Note that the backward system (2) can be written in the following integral form: T t
Z (u)dW (u) = Φ (X (T )).
(5)
t
Conditioning on F (t ) under P and using the martingale property of the stochastic integral with respect to the Brownian motion gives: T
Y (t ) = E Φ (X (T )) +
g (u, X (u), Y (u), Z (u))du|F (t ) .
(6)
t
As X (t ) = x ∈ ℜ, using the Markovian property of X with respect to the filtration F as well as the fact that Y (t ) = Ψ1 (t , X (t )) and Z (t ) = Ψ2 (t , X (t )) we have: Y (t ) = Ψ1 (t , x) T
= E Φ (X (T )) +
g (u, X (u), Y (u), Z (u))du|X (t ) = x . t
Write, for each t ∈ T , V (t ) :=
1. g and Φ satisfy the three differentiability and growth conditions given at the top of this Section 4; 2. the derivatives ∂x Y t ,x (u) and ∂x Z t ,x (u) of Y t ,x (u) and Z t ,x (u) with respect to x exist and are bounded. Then for each t ∈ T ,
γ (t ) = E
T
gx (u, ξ0,u (x0 ), Ψ1 (u, ξ0,u (x0 )),
t
Ψ2 (u, ξ0,u (x0 )))D0,u (x0 )du
1 + Φx (ξ0,T (x0 ))D0,T (x0 )|F (t ) D− 0,t (x0 )σ (t , ξ0,t (x0 )) T
+E
(gy (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Y t ,x (u)
t
Furthermore, Ψ1 (t , x) satisfies the following partial differential equation, (PDE):
∂ Ψ1 ∂ Ψ1 1 ∂ 2 Ψ1 + µ(t , x) + σ 2 (t , x) 2 ∂t ∂x 2 ∂x + g (t , x, Ψ1 (t , x), Ψ2 (t , x)) = 0, with terminal condition:
Ψ1 (T , X (T )) = Φ (X (T )). Proof. For each t ∈ T , write x := ξ0,t (x0 ). From the uniqueness of the strong solution of the FSDE (1), ξ0,T (x0 ) satisfies the following semigroup property:
ξ0,T (x0 ) = ξt ,T (ξ0,t (x0 )) = ξt ,T (x). Differentiating this with respect to x gives:
t
Theorem 1. Suppose the following conditions are satisfied:
× σ (t , ξ0,t (x0 )).
T
+
|γ (t )|2 dt < ∞.
+ gz (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Z t ,x (u))du|F (t )
g (u, X (u), Y (u), Z (u))du
T
Our goal is to determine the integrand process γ . The following theorem presents the main result which gives an explicit representation for the integrand process γ .
4. The solution of the backward system
E 0
1 with initial condition D− s,s = 1.
Y (t ) −
γ (u)dW (u), 0
D0,T (x0 ) = Dt ,T (x)D0,t (x0 ).
g (u, X (u), Y (u), Z (u))du. 0
Again because Y (t ) = Ψ1 (t , X (t )) and Z (t ) = Ψ2 (t , X (t )), the process V := {V (t )|t ∈ T } is F-adapted. Now define another process Y¯ := {Y¯ (t )|t ∈ T } by putting:
Let {⟨X , X ⟩(t )|t ∈ T } be the predictable quadratic variation of the process X . Applying Itô’s differentiation rule to Y¯ (t ) := Ψ (t , x, v) gives:
Ψ (t , X (t ), V (t )) = Ψ (0, x0 , v0 ) +
Y¯ (t ) := Y (t ) + V (t ) T
= E Φ (X (T )) +
g (u, X (u), Y (u), Z (u))du|F (t ) . 0
This is a square-integrable, (F, P )-martingale. Now if V (t ) = v ∈ ℜ, Y¯ (t ) = Ψ1 (t , x) + v := Ψ (t , x, v),
say.
From the differentiability conditions for g, Φ , Ψ1 and Ψ2 , Ψ (t , x, v) is continuously differentiable in t and twice continuously differentiable in x. It is obvious from the definition that Ψ (t , x, v) is continuously differentiable in v . By the martingale representation theorem, Y¯ has the following integral representation:
t
0
∂Ψ du + ∂u
0
t
∂Ψ dV (t ) ∂v
∂Ψ 1 t ∂ 2Ψ d⟨X , X ⟩(t ) + dX (t ) + ∂x 2 0 ∂ x2 0 t ∂Ψ ∂Ψ = Ψ (0, x0 , v0 ) + + µ(u, ξ0,u (x0 )) ∂u ∂x 0 1 2 ∂ 2Ψ + σ (u, ξ0,u (x0 )) 2 2 ∂x
t
+ g (u, ξ0,u (x0 ), Ψ1 (u, ξ0,u (x0 )), Ψ2 (u, ξ0,u (x0 ))) du t ∂Ψ + σ (u, ξ0,u (x0 ))dW (u). ∂x 0
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R.J. Elliott, T.K. Siu / Systems & Control Letters 61 (2012) 1017–1022
This is a special semimartingale. Note that Ψ (t , x, v) is also a martingale. By the unique decomposition of a special semimartingale, the bounded variation term in the above decomposition must be identical to zero. This implies that Ψ (t , x, v) satisfies the following PDE:
∂Ψ ∂Ψ 1 ∂ Ψ + µ(t , x) + σ 2 (t , x) 2 ∂t ∂x 2 ∂x + g (t , x, Ψ1 (t , x), Ψ2 (t , x)) = 0, 2
with terminal condition:
Ψ1 (T , x) = Φ (x). Furthermore, for each t ∈ T ,
with terminal condition:
Ψ1 (T , X (T )) = Φ (X (T )).
u, ξ0,u (x0 ), Ψ1 (u, ξ0,u (x0 )),
∂ Ψ1 σ (u, ξ0,u (x0 )) (u, ξ0,u (x0 )) D0,u (x0 )du ∂x 1 + Φx (ξ0,T (x0 ))D0,T (x0 )|F (t ) D− 0,t (x0 )σ (t , ξ0,t (x0 )).
Ψ (T , X (T ), V (T )) = Φ (X (T )) + V (T ).
∂ Ψ1 1 ∂ 2 Ψ1 ∂ Ψ1 + µ(t , x) + σ 2 ( t , x) 2 ∂t ∂x 2 ∂x + g (t , x, Ψ1 (t , x), Ψ2 (t , x)) = 0,
gx
t
with terminal condition: Consequently, Ψ1 (t , x) satisfies the following PDE:
T
Z (t ) = E
Proof. The first statement follows directly from the second ∂Ψ statement in Theorem 1 and by noting that Ψ2 = σ (t , x) ∂ x1 . It remains to prove the second statement. By the martingale representation, Y (t ) +
t
g (u, X (u), Y (u), Z (u))du = Y¯ (t ) 0
∂Ψ σ (t , ξ0,t (x0 )) ∂x ∂ Ψ1 σ (t , ξ0,t (x0 )). = ∂x Note ξt ,T (x) = ξ0,T (x0 ), so from the differentiability and growth conditions of g and Φ , T ∂ Ψ1 =E gx (u, ξt ,u (x), Ψ1 (u, ξt ,u (x)), Ψ2 (u, ξt ,u (x))) ∂x t × Dt ,u (x)du + Φx (ξt ,T (x))Dt ,T (x)|F (t )
γ (t ) =
T
+E
(gy (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Y t ,x (u)
t
+ gz (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Z t ,x (u))du|F (t )
t
This proves the second statement. Furthermore,
γ (u)dW (u).
=
(7)
0
Putting t = T gives:
Φ (X (T )) +
T
g (u, X (u), Y (u), Z (u))du = 0
T
γ (u)dW (u).
(8)
0
Subtracting (8) from (7) gives: Y (t ) −
T
g (u, X (u), Y (u), Z (u))du t T
γ (u)dW (u) = Φ (X (T )).
+ t
By the uniqueness of the solution of the MFBSDE, we must have: Z (t ) = γ (t ),
t ∈T.
The second statement then follows from the first statement in ∂Ψ Theorem 1 by noting again that Ψ2 = σ (t , x) ∂ x1 . 5. Convex risk measures
T
gx (u, ξ0,u (x0 ), Ψ1 (u, ξ0,u (x0 )),
=E t
Ψ2 (u, ξ0,u (x0 )))D0,u (x)du
1 + Φx (ξ0,T (x0 ))D0,T (x)|F (t ) D− 0,t (x0 ) T
+E
(gy (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Y t ,x (u)
t
+ gz (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Z t ,x (u))du|F (t ) . This proves the first statement, completing the proof of the theorem. The following theorem identifies the solution (Y , Z ) of the backward system of the MFBSDE. Theorem 2. For each t ∈ T , Y (t ) = Ψ1 (t , X (t )), which is a probabilistic representation for the solution Ψ1 (t , x) of the following semi-linear PDE:
∂ Ψ1 ∂ Ψ1 1 ∂ 2 Ψ1 + µ(t , x) + σ 2 ( t , x) 2 ∂t ∂x 2 ∂x ∂ Ψ1 + g t , x, Ψ1 (t , x), σ (t , x) (t , x) = 0, ∂x
The object of this section is to apply the results in the last section to evaluate a convex risk measure for a contingent claim with maturity at time T which is written on an underlying risky asset S whose price dynamics are governed by a geometric Brownian motion. Let α : T × ℜ+ → ℜ and β : T × ℜ+ → ℜ be measurable functions of (t , s) ∈ T × ℜ+ , where ℜ+ is the set of positive real numbers. We assume that µ(t , s) and σ (t , s) are three times differentiable in s and which, together with their derivatives, are bounded, for each t ∈ T . For each t ∈ T , µ(t , s) and σ (t , s) represent the appreciation rate and volatility of the underlying risky asset at time t when the current value of the risky S (t ) = s. Suppose, under the real-world probability measure P, the price process of the underlying risky asset S := {S (t )|t ∈ T } is governed by the following geometric Brownian motion: dS (t ) = α(t , S (t ))S (t )dt + β(t , S (t ))S (t )dW (t ),
t ∈T
S (0) = s0 > 0. To simplify our analysis and notation, we assume that the riskfree rate of interest is zero. The case of non-zero rate can be incorporated easily by considering S as the discounted price of the risky asset. Putting µ(t , S (t )) := α(t , S (t ))S (t ) and
β(t , S (t )) := β(t , S (t ))S (t ), we write the price dynamics of the underlying risky asset in the
R.J. Elliott, T.K. Siu / Systems & Control Letters 61 (2012) 1017–1022
form of the FSDE (1): dS (t ) = µ(t , S (t ))dt + σ (t , S (t ))dW (t ).
(9)
Suppose, for each t , s ∈ T with t ≥ s, ξs,t (x) is the unique strong solution of the FSDE (9) such that ξs,s (x) = x. Then, as in Section 3, we define a stochastic flow of diffeomorphism of the map x → ξs,t (x) whose derivative satisfies the linearized Eq. (3). Let P (S (T )) be the payoff of the contingent claim at the maturity time T , where P (S (T )) ∈ L2 (Ω , F (T ), P ), the space of squareintegrable, F (T )-measurable, random variables with respect to the measure P. Our goal is to evaluate a convex risk measure for the claim P (S (T )). Indeed, the risk measurement of derivative securities has been discussed in a number of papers, for example, Boyle et al. [26] using a binomial approach, Siu and Yang [27] and Siu et al. [28,29] using a Bayesian approach and Elliott et al. [30] using a partial differential equation approach. Different risk measures have been considered in these works such as Value at Risk, Expected Shortfall, Coherent Risk Measure pioneered by Artzner et al. [31] and Convex Risk Measures proposed by Föllmer and Schied [32] and Frittelli and Rosazza-Gianin [33]. Here we consider a convex risk measure and adopt the approach based on a MFBSDE. We recall some essential concepts and results for convex risk measures. Write X for the space Lp (Ω , F (T ), P ) of p-integrable, F (T )-measurable, random variables, where p ∈ [1, ∞]. The space X is the set of random variables representing the future values of risky financial positions which will be realized at time T . Then a convex risk measure is defined as follows: Definition 1. A convex risk measure is a functional ρ : X → ℜ which satisfies the following three conditions: 1. (Translation invariance) For each k ∈ ℜ and X ∈ X,
ρ(X + k) = ρ(X ) − k. 2. (Monotonicity) For each X1 , X2 ∈ X, if X1 (ω) ≤ X2 (ω) for each ω ∈ Ω,
ρ(X1 ) ≥ ρ(X2 ). 3. (Convexity) For each X1 , X2 ∈ X and each λ ∈ (0, 1),
ρ(λX1 + (1 − λ)X1 ) ≤ λρ(X1 ) + (1 − λ)ρ(X2 ). The following theorem is due to Frittelli and RosazzaGianin [33] and Föllmer and Schied [32]. It is a central result in the analysis of convex risk measures and gives a representation for a, (generic), convex risk measure. Theorem 3. A functional ρ : X → ℜ is a convex risk measure if and only if there is a family of probability measures Ma absolutely continuous with respect to P and a convex ‘‘penalty’’ function η : Ma → ℜ with η(Q ) < ∞ for each Q ∈ Ma such that for each X ∈ X,
ρ(X ) = sup {E Q [−X ] − η(Q )}. Here E Q is the expectation under Q . An important example of convex risk measures is the entropic risk measure which is related to an expected exponential utility function. Suppose Ma represents the space of all probability measures absolutely continuous with respect to P and η is the relative entropy defined by:
dQ dP
ln
dQ dP
:= R(Q , P ),
Q ∈ Ma .
Then an entropic risk measure ρ E is defined by:
ρ E (X ) := sup {E Q (−X ) − R(Q , P )}, Q ∈Ma
Besides the representation in Theorem 3, a convex risk measure can also be represented as the g-expectation. The definition of the g-expectation was given in [34], where it was defined as the solution of a BSDE. We state the definition of the conditional gexpectation by Peng [34] and Rosazza-Gianin [35] by adapting it to our context and notation here. Definition 2. For any X ∈ L2 (Ω , F (T ), P ), let {(Y −X (t ), −X Z (t ))|t ∈ T } be the square-integrable solution of the following BSDE with terminal condition −X :
−dY (t ) = g (t , X (t ), Y (t ), Z (t ))dt − Z (t )dW (t ), Y (T ) = −X . Then the conditional g-expectation of −X , denoted by
Eg (−X |F (t )), is defined as:
Eg (−X |F (t )) = Y −X (t ). If the function g is convex, a convex risk measure ρ(X ) := ρ g (X ), X ∈ L2 (Ω , F , P ), can be represented as the g-expectation of −X as follows: ρ g (X ) = Eg (−X |F (0)) = Y −X (0) = Eg (−X ). In our case, we are interested in evaluating the convex risk measure ρ(P (S (T ))) of the contingent claim with payoff P (S (T )). Suppose P (S (T )) = −Φ (S (T )) and g is the driver function of the BSDE (2). Then the convex risk measure ρ(P (S (T ))) := ρ g (P (S (T ))) associated with the function g can be represented as the following g-expectation:
ρ g (P (S (T ))) = Eg (Φ (S (T ))). Consequently,
ρ g (Φ (S (T ))) = Eg (Φ (S (T ))) = Y (0), where Y (0) is the initial value of the first component Y of the solution of the BSDE (2). Since the price process S satisfies the FSDE (1), the evaluation of the convex risk measure ρ g (P (S (T ))) can be formulated as the problem of solving the backward system in the MFBSDE described by the coupled system consisting of (1) and (2). Applying Theorems 1 and 2, we obtain the following corollary which gives a way to evaluate convex risk measure ρ g (P (S (T ))). Corollary 1. Suppose Ψ1 ∈ C 1,2 (T × ℜ+ ) satisfies the following semi-linear PDE:
∂ Ψ1 ∂ Ψ1 1 ∂ 2 Ψ1 + µ(t , s)s + σ 2 (t , s)s2 2 ∂t ∂s 2 ∂s ∂ Ψ1 + g t , s, Ψ1 (t , s), σ (t , s) (t , s) = 0, ∂s with terminal condition:
Ψ1 (T , s) = −P (s). Then ρ g (P (S (T ))) = Ψ1 (0, s0 ). This generalizes the PDE result for coherent risk measures in [27].
Q ∈Ma
η(Q ) := E
1021
X ∈ X.
6. Conclusion We adopted techniques from stochastic flows to discuss the solution of MBSDEs. The solution of the backward system in an MBSDE was identified using a martingale representation, a differentiation rule, a unique decomposition of special semimartingales and stochastic flows. The first component of the solution was represented as the solution of a semilinear PDE while the second component was represented as an expectation of an integral with an integrand given by the derivative of a stochastic flow. An application to evaluate a convex risk measure for a contingent claim was discussed.
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Acknowledgment We wish to acknowledge the Discovery Grant from the Australian Research Council (ARC), (Project No.: DP1096243). References [1] J.M. Bismut, Conjugate convex functions in optimal stochastic control, Journal of Mathematical Analysis and Applications 44 (1973) 384–404. [2] J.M. Bismut, Contröle des Systèmes Linéaires Quadratiques: Applications de l’Intégrale Stochastique, in: Sém. Probab. XII., Lecture Notes in Math., vol. 649, Springer-Verlag, Berlin and New York, 1994, pp. 180–264. [3] E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation, Systems and Control Letter 14 (1990) 55–61. [4] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics 37 (1991) 61–74. [5] E. Pardoux, S. Peng, Backward stochastic differential equations and quaslinear parabolic partial differential equations, in: Lecture Notes in CIS 176, SpringerVerlag, Berlin, Heidelberg, New York, 1992, pp. 200–217. [6] F. Antonelli, Backward–forward stochastic differential equations, Annals of Applied Probability 3 (1993) 777–793. [7] J. Ma, P. Protter, J. Yong, Solving forward–backward stochastic differential equations explicitly—a four step scheme, Probability Theory and Related Fields 98 (1994) 339–359. [8] Y. Hu, S. Peng, Solution of forward–backward stochastic differential equations, Probability Theory and Related Fields 103 (1995) 273–283. [9] J. Ma, J. Yong, Forward–Backward Stochastic Differential Equations and Their Applications, Springer-Verlag, Berlin and New York, 1999. [10] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Kodansha, 1981. [11] J.M. Bismut, Martingales, the Malliavin calculus and hypoellipticity under general Hörmanders conditions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 56 (1981) 469–505. [12] Y. Le Jan, S. Watanabe, Stochastic flows of diffeomorphisms, in: K. Itô (Ed.), Stochastic Analysis, North-Holland, Amsterdam, 1984, pp. 307–332. [13] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, United Kingdom, 1990. [14] R.J. Elliott, M. Kohlmann, Integration by parts, homogeneous chaos expansions and smooth densities, Annals of Probability 17 (1989) 194–207. [15] R.J. Elliott, M. Kohlmann, Integration by parts and densities for a jump process, Stochastics 27 (1989) 83–97.
[16] J. Baras, R.J. Elliott, M. Kohlmann, The partially observed stochastic minimum principle, SIAM Journal of Control and Optimization 27 (1989) 1279–1292. [17] R.J. Elliott, The optimal control of diffusions, Applied Mathematics and Optimization 22 (1990) 229–240. [18] R.J. Elliott, H. Yang, The control of partially observed diffusions, Journal of Optimization Theory and Applications 71 (1991) 485–501. [19] D. Colwell, R.J. Elliott, P.E. Kopp, Martingale representations and hedging policies, Stochastic Processes and Applications 38 (1991) 335–345. [20] D. Colwell, R.J. Elliott, Discontinuous asset prices and nonattainable contingent claims, Mathematical Finance 3 (1993) 295–308. [21] R.J. Elliott, T.K. Siu, H. Yang, Martingale representation for contingent claims with regime switching, Communications on Stochastic Analysis 1 (2) (2007) 279–292. [22] R.J. Elliott, T.K. Siu, Pricing and hedging contingent claims with regime switching risk, Communications in Mathematical Sciences 9 (2) (2011) 477–498. [23] R.J. Elliott, J. van der Hoek, Stochastic flows and the forward measure, Finance and Stochastics 5 (2001) 511–525. [24] R.J. Elliott, T.K. Siu, On Markov-modulated exponential-affine bond price formulae, Applied Mathematical Finance 16 (1) (2009) 1–15. [25] N. El Karoui, S. Peng, M.C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance 7 (1997) 1–71. [26] P.P. Boyle, T.K. Siu, H. Yang, Risk and probability measures, Risk 15 (7) (2002) 53–57. [27] T.K. Siu, H. Yang, A P.D.E. approach for measuring risk of derivatives, Applied Mathematical Finance 7 (3) (2000) 211–228. [28] T.K. Siu, H. Tong, H. Yang, Bayesian risk measures for derivatives via random Esscher transform, North American Actuarial Journal 5 (3) (2001) 78–91. [29] T.K. Siu, H. Tong, H. Yang, On Bayesian value at risk: from linear to non-linear portfolios, Asia-Pacific Financial Markets 11 (2) (2004) 161–184. [30] R.J. Elliott, T.K. Siu, L. Chan, A P.D.E. approach for risk measures for derivatives with regime switching, Annals of Financev 4 (1) (2008) 55–74. [31] P. Artzner, F. Delbaen, J. Eber, D. Heath, Coherent measures of risk, Mathematical Finance 9 (3) (1999) 203–228. [32] H. Föllmer, A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics 6 (2002) 429–447. [33] M. Frittelli, E. Rosazza-Gianin, Putting order in risk measures, Journal of Banking and Finance 26 (2002) 1473–1486. [34] S. Peng, BSDE and related g-expectation, in: N. El Karoui, L. Mazliak (Eds.), Backwards Stochastic Differential Equations: Pitman Research Notes in Math. Series, Vol. 364, Longman, England, 1997, pp. 141–159. [35] E. Rosazza-Gianin, Risk measures via g-expectations, Insurance: Mathematics and Economics 39 (2006) 19–34.
Contents lists available at SciVerse ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Markovian forward–backward stochastic differential equations and stochastic flows Robert J. Elliott a,b,∗ , Tak Kuen Siu c a
School of Mathematical Sciences, University of Adelaide, SA 5005, Australia
b
Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada
c
Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, NSW 2109, Sydney, Australia
article
info
Article history: Received 16 November 2011 Accepted 16 April 2012 Available online 5 September 2012 Keywords: Markovian forward–backward stochastic differential equations Stochastic flows Martingale representation Special semimartingale Convex risk measures
abstract Markovian forward–backward stochastic differential equations, (MFBSDEs), are discussed by exploiting techniques of stochastic flows. Using martingale representation, a differentiation rule, stochastic flows of diffeomorphisms and the unique decomposition of special semimartingales, we identify the solution of the backward system of the FBSDE. Applications of the result to convex risk measures are discussed. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Backward stochastic differential equations, (BSDEs), first appeared in the work of Bismut [1], where he discussed a maximum principle for stochastic optimal control using linear BSDEs. In a later paper [2], he established some existence and uniqueness results for BSDEs. The theory of general BSDEs was established in the work of Pardoux and Peng [3], where the existence and uniqueness results for general BSDEs were established. The link between BSDEs and partial differential equations (PDEs) was investigated by Peng [4] and Pardoux and Peng [5]. The study of forward–backward stochastic differential equations (FBSDEs) began in the early 1990s. Some examples include Antonelli [6], Ma et al. [7] and Hu and Peng [8]. An FBSDE is a coupled system consisting of a forward stochastic differential equation and a backward one, where the coefficient of the backward equation depends on the solution of the forward one. Nowadays, both BSDEs and FBSDEs have many important applications in Mathematical Finance, Stochastic Control and Partial Differential Equations. For an excellent account of FBSDEs and their applications, please refer to Ma and Yong [9]. Stochastic flows are an important concept in stochastic differential equations. They describe how the trajectories of the
∗ Corresponding author at: Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada. Tel.: +1 403 220 5540; fax: +1 403 770 8104. E-mail addresses: [email protected] (R.J. Elliott), [email protected] (T.K. Siu). 0167-6911/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2012.04.013
solutions of stochastic differential equations change with respect to their initial conditions. The study of stochastic flows was initiated by Gihman and Skorohod. Some early works on stochastic flows include Ikeda and Watanabe [10], Bismut [11], Le Jan and Watanabe [12] and Kunita [13]. Stochastic flows have been applied to Malliavin calculus, mathematical finance, stochastic filtering and control. Elliott and Kohlmann [14,15] applied stochastic flows to Malliavin calculus in the diffusion and the jumpdiffusion cases. In particular, they derived an integration-byparts formula using stochastic flows and discussed the existence and smoothness of the density functions of the solutions of stochastic differential equations. Baras et al. [16], Elliott [17] and Elliott and Yang [18] applied the concept of stochastic flows to derive minimum principles for stochastic optimal control in fully and partially observed diffusion models. Colwell et al. [19], Colwell and Elliott [20], Elliott et al. [21] and Elliott and Siu [22] applied stochastic flows to discuss the pricing and hedging of contingent claims. Elliott and van der Hoek [23] and Elliott and Siu [24] applied stochastic flows to derive exponential affine formulae for bond valuation under stochastic interest rate models. It appears interesting to further explore the link and application of stochastic flows to other important areas of stochastic analysis and mathematical finance from both theoretical and practical perspectives. In this paper, an approach based on stochastic flows is used to discuss the solution of a Markovian forward–backward stochastic differential equation, (MFBSDE), driven by a standard Brownian motion. More specifically we consider the situation where the
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solutions of the forward and backward systems in the MFBSDE are of Markovian type. In this case, the martingale component in the backward system of the MFBSDE can be related to a Markov process with respect to the filtration generated by a standard Brownian motion. Applying a martingale representation, a differentiation rule and a unique decomposition of a special semimartingale, we identify the solution of the backward system of the MFBSDE. In particular, the first component of the solution is related to a solution of a partial differential equation, and a representation of the control component of the solution is obtained using the Jacobians of stochastic flows of diffeomorphisms. By noting the relationship between a convex risk measure and the initial value of the solution of the backward system of the MFBSDE, we apply the martingale representation result to evaluate a convex risk measure in a Markovian environment. The paper is organized as follows. The next section discusses the concepts of MFBSDEs and some related results. In Section 3, we discuss stochastic flows of diffeomorphisms. Section 4 establishes the main result which gives a representation of the solution of the backward system of the MFBSDE. In Section 5, we apply the result of Section 4 to discuss convex risk measures. 2. Markovian forward–backward stochastic differential equations Consider a complete filtered probability space (Ω , F , F, P ), where T denotes the time interval [0, T ], T < ∞, and F := {F (t )|t ∈ T } is a filtration satisfying the usual conditions, (i.e. right continuity and P-completeness). Let W := {W (t )|t ∈ T } be an (F, P ) standard Brownian motion, which describes the randomness of both the forward and backward systems in a Markovian forward–backward stochastic differential equation, (MFBSDE), to be defined later. A forward–backward stochastic differential equation, (FBSDE), consists of a forward system and a backward system. The forward system is described by a forward stochastic differential equation, (FSDE), while the backward one is described by a backward stochastic differential equation, (BSDE), whose data, namely, the driver function g and the terminal condition Φ , depend on the solution of the FSDE. If we take into account the initial state (t , x) of the FSDE, (i.e., the state process), the solution (Y , Z ) of the BSDE is regarded as the solution of a parametrized BSDE whose data (g , Φ ) are parametrized by the initial state (t , x). This dependence is used to discuss the regularity properties of the solutions of the BSDE which follow from the regularity properties of the coefficients of the FBSDE. See El Karoui et al. [25]. Since this is not our focus, we shall not discuss this dependence further. Here we consider the situation when the solution of the FSDE is Markovian with respect to the filtration F. The key property of the FBSDE is that the solution (Y , Z ) of the BSDE can be written as, a (deterministic), function of time t and the value of the state process at that time. Consequently, from the Markovian assumption for the solution of the FSDE, the solution (Y , Z ) of the BSDE is Markovian with respect to F. This property is important when we use a stochastic flow approach to discuss the solution of the BSDE. To simplify our discussion and notation, we consider real-valued FSDE and BSDE. The results in this paper can be extended to the multivariate case. Let µ and σ be real-valued and positive-valued functions defined on T × ℜ. Suppose g is a real-valued, square-integrable, Borel-measurable function on T × ℜ3 and Φ is a real-valued, square-integrable, Borel-measurable function on ℜ. As usual, we impose the following standard assumptions of Lipschitz continuity for the coefficients in the FSDE and BSDE. These are required for the existence, uniqueness and regularity of the solutions of the BSDE and FSDE. Suppose there is a positive constant K such that
1. for each x1 , x2 , y1 , y2 , z1 , z2 ∈ ℜ and each t ∈ T ,
|µ(t , x1 ) − µ(t , x2 )| + |σ (t , x1 ) − σ (t , x2 )| ≤ K (1 + |x1 − x2 |), |g (t , x, y1 , z1 ) − g (t , x, y2 , z2 )| ≤ K (|y1 − y2 | + |z1 − z2 |) 2. for each t ∈ T , (x, y, z ) ∈ ℜ3 and each k ≥
1 , 2
|µ(t , x)| + |σ (t , x)| ≤ K (1 + |x|), |g (t , x, y, z )| + |Φ (x)| ≤ K (1 + |x|k ). Suppose X := {X (t )|t ∈ T } is the unique strong solution of the following FSDE: dX (t ) = µ(t , X (t ))dt + σ (t , X (t ))dW (t ),
t ∈T
X (0) = x0 ∈ ℜ.
(1)
Consider now an associated BSDE defined as follows:
−dY (t ) = g (t , X (t ), Y (t ), Z (t ))dt − Z (t )dW (t ), Y (T ) = Φ (X (T )).
t ∈T (2)
The solution of the BSDE is the pair {(Y (t ), Z (t ))|t ∈ T }. We suppose that {(Y (t ), Z (t ))|t ∈ T } is square integrable. The coupled system consisting of (1) and (2) is called an FBSDE and its solution is given by the triple {(X (t ), Y (t ), Z (t ))|t ∈ T }. For the existence and uniqueness of the solution of the FBSDE, please refer to the monograph by Ma and Yong [9]. It was noted in [25] that the solution {(Y (t ), Z (t ))|t ∈ T } in the backward system of the FBSDE is Markovian with respect to the filtration F. That is, there are real-valued, (deterministic), functions Ψ1 and Ψ2 on T × ℜ such that for each t ∈ T , Y (t ) = Ψ1 (t , X (t )), Z (t ) = Ψ2 (t , X (t )). The object of this paper is to identify explicitly the process {Z (t )|t ∈ T } in the backward system of the FBSDE by exploiting stochastic flows of diffeomorphisms. 3. Stochastic flows of diffeomorphisms In this section we present the key ideas and results of stochastic flows which will be used to identify the control process {Z (t )|t ∈ T }. To apply the results in stochastic flows, we must impose some additional differentiability and growth conditions on the coefficients of the FSDE. In particular, we assume that the coefficients µ(t , x) and σ (t , x) of the FSDE are three times differentiable in x, and, together with their derivatives, have linear growth in x. Suppose, for each t , s ∈ T with t ≥ s, ξs,t (x) is the unique strong solution of (1) which has initial condition:
ξs,s (x) = x ∈ ℜ. From some standard results in [11,13], there is a set N ⊂ Ω of measure zero such that if ω ̸∈ N, there is a version of ξs,t (x) which is twice differentiable in x and continuous in t and s. The map x → ξs,t (x) is called a stochastic flow. Write, for each t , s ∈ T with t ≥ s, Ds,t :=
∂ξs,t (x) , ∂x
for the derivative of the map x → ξs,t (x) with respect to x. Suppose µx (t , x) and σx (t , x) are the derivatives of µ(t , x) and σ (t , x) with respect to x, respectively. Then D is the solution of the linearized equation: dDs,t = µx (t , X (t ))Ds,t dt + σx (t , X (t ))Ds,t dW (t ), with initial condition Ds,s = 1.
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1019
t
1 It is known that the inverse D− s,t of Ds,t exists and satisfies the following stochastic differential equation:
Y¯ (t ) =
1 2 dD− s,t = −Ds,t µx (t , X (t )) − σx (t , X (t )) dt
where γ := {γ (t )|t ∈ T } is an F-predictable process such that
−1
1 − D− s,t σx (t , X (t ))dW (t ),
(4)
This section presents the main result of the paper which gives a partial differential equation, (PDE), representation for the first component of the solution of the backward system (2) and an expectation representation for the control component of the solution of the backward system (2) using stochastic flows. Firstly, we assume the following differentiability and growth conditions: 1. g (t , x, y, z ) and Φ (x) are twice continuously differentiable in x, and, together with their derivatives, have linear growth in x, for each (t , y, z ) ∈ T × ℜ2 ; 2. g (t , x, y, z ) is continuously differentiable in t, for each (x, y, z ) ∈ ℜ3 ; 3. Ψ1 (t , x) and Ψ2 (t , x) are continuously differentiable in t and twice continuously differentiable in x. Note that the backward system (2) can be written in the following integral form: T t
Z (u)dW (u) = Φ (X (T )).
(5)
t
Conditioning on F (t ) under P and using the martingale property of the stochastic integral with respect to the Brownian motion gives: T
Y (t ) = E Φ (X (T )) +
g (u, X (u), Y (u), Z (u))du|F (t ) .
(6)
t
As X (t ) = x ∈ ℜ, using the Markovian property of X with respect to the filtration F as well as the fact that Y (t ) = Ψ1 (t , X (t )) and Z (t ) = Ψ2 (t , X (t )) we have: Y (t ) = Ψ1 (t , x) T
= E Φ (X (T )) +
g (u, X (u), Y (u), Z (u))du|X (t ) = x . t
Write, for each t ∈ T , V (t ) :=
1. g and Φ satisfy the three differentiability and growth conditions given at the top of this Section 4; 2. the derivatives ∂x Y t ,x (u) and ∂x Z t ,x (u) of Y t ,x (u) and Z t ,x (u) with respect to x exist and are bounded. Then for each t ∈ T ,
γ (t ) = E
T
gx (u, ξ0,u (x0 ), Ψ1 (u, ξ0,u (x0 )),
t
Ψ2 (u, ξ0,u (x0 )))D0,u (x0 )du
1 + Φx (ξ0,T (x0 ))D0,T (x0 )|F (t ) D− 0,t (x0 )σ (t , ξ0,t (x0 )) T
+E
(gy (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Y t ,x (u)
t
Furthermore, Ψ1 (t , x) satisfies the following partial differential equation, (PDE):
∂ Ψ1 ∂ Ψ1 1 ∂ 2 Ψ1 + µ(t , x) + σ 2 (t , x) 2 ∂t ∂x 2 ∂x + g (t , x, Ψ1 (t , x), Ψ2 (t , x)) = 0, with terminal condition:
Ψ1 (T , X (T )) = Φ (X (T )). Proof. For each t ∈ T , write x := ξ0,t (x0 ). From the uniqueness of the strong solution of the FSDE (1), ξ0,T (x0 ) satisfies the following semigroup property:
ξ0,T (x0 ) = ξt ,T (ξ0,t (x0 )) = ξt ,T (x). Differentiating this with respect to x gives:
t
Theorem 1. Suppose the following conditions are satisfied:
× σ (t , ξ0,t (x0 )).
T
+
|γ (t )|2 dt < ∞.
+ gz (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Z t ,x (u))du|F (t )
g (u, X (u), Y (u), Z (u))du
T
Our goal is to determine the integrand process γ . The following theorem presents the main result which gives an explicit representation for the integrand process γ .
4. The solution of the backward system
E 0
1 with initial condition D− s,s = 1.
Y (t ) −
γ (u)dW (u), 0
D0,T (x0 ) = Dt ,T (x)D0,t (x0 ).
g (u, X (u), Y (u), Z (u))du. 0
Again because Y (t ) = Ψ1 (t , X (t )) and Z (t ) = Ψ2 (t , X (t )), the process V := {V (t )|t ∈ T } is F-adapted. Now define another process Y¯ := {Y¯ (t )|t ∈ T } by putting:
Let {⟨X , X ⟩(t )|t ∈ T } be the predictable quadratic variation of the process X . Applying Itô’s differentiation rule to Y¯ (t ) := Ψ (t , x, v) gives:
Ψ (t , X (t ), V (t )) = Ψ (0, x0 , v0 ) +
Y¯ (t ) := Y (t ) + V (t ) T
= E Φ (X (T )) +
g (u, X (u), Y (u), Z (u))du|F (t ) . 0
This is a square-integrable, (F, P )-martingale. Now if V (t ) = v ∈ ℜ, Y¯ (t ) = Ψ1 (t , x) + v := Ψ (t , x, v),
say.
From the differentiability conditions for g, Φ , Ψ1 and Ψ2 , Ψ (t , x, v) is continuously differentiable in t and twice continuously differentiable in x. It is obvious from the definition that Ψ (t , x, v) is continuously differentiable in v . By the martingale representation theorem, Y¯ has the following integral representation:
t
0
∂Ψ du + ∂u
0
t
∂Ψ dV (t ) ∂v
∂Ψ 1 t ∂ 2Ψ d⟨X , X ⟩(t ) + dX (t ) + ∂x 2 0 ∂ x2 0 t ∂Ψ ∂Ψ = Ψ (0, x0 , v0 ) + + µ(u, ξ0,u (x0 )) ∂u ∂x 0 1 2 ∂ 2Ψ + σ (u, ξ0,u (x0 )) 2 2 ∂x
t
+ g (u, ξ0,u (x0 ), Ψ1 (u, ξ0,u (x0 )), Ψ2 (u, ξ0,u (x0 ))) du t ∂Ψ + σ (u, ξ0,u (x0 ))dW (u). ∂x 0
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R.J. Elliott, T.K. Siu / Systems & Control Letters 61 (2012) 1017–1022
This is a special semimartingale. Note that Ψ (t , x, v) is also a martingale. By the unique decomposition of a special semimartingale, the bounded variation term in the above decomposition must be identical to zero. This implies that Ψ (t , x, v) satisfies the following PDE:
∂Ψ ∂Ψ 1 ∂ Ψ + µ(t , x) + σ 2 (t , x) 2 ∂t ∂x 2 ∂x + g (t , x, Ψ1 (t , x), Ψ2 (t , x)) = 0, 2
with terminal condition:
Ψ1 (T , x) = Φ (x). Furthermore, for each t ∈ T ,
with terminal condition:
Ψ1 (T , X (T )) = Φ (X (T )).
u, ξ0,u (x0 ), Ψ1 (u, ξ0,u (x0 )),
∂ Ψ1 σ (u, ξ0,u (x0 )) (u, ξ0,u (x0 )) D0,u (x0 )du ∂x 1 + Φx (ξ0,T (x0 ))D0,T (x0 )|F (t ) D− 0,t (x0 )σ (t , ξ0,t (x0 )).
Ψ (T , X (T ), V (T )) = Φ (X (T )) + V (T ).
∂ Ψ1 1 ∂ 2 Ψ1 ∂ Ψ1 + µ(t , x) + σ 2 ( t , x) 2 ∂t ∂x 2 ∂x + g (t , x, Ψ1 (t , x), Ψ2 (t , x)) = 0,
gx
t
with terminal condition: Consequently, Ψ1 (t , x) satisfies the following PDE:
T
Z (t ) = E
Proof. The first statement follows directly from the second ∂Ψ statement in Theorem 1 and by noting that Ψ2 = σ (t , x) ∂ x1 . It remains to prove the second statement. By the martingale representation, Y (t ) +
t
g (u, X (u), Y (u), Z (u))du = Y¯ (t ) 0
∂Ψ σ (t , ξ0,t (x0 )) ∂x ∂ Ψ1 σ (t , ξ0,t (x0 )). = ∂x Note ξt ,T (x) = ξ0,T (x0 ), so from the differentiability and growth conditions of g and Φ , T ∂ Ψ1 =E gx (u, ξt ,u (x), Ψ1 (u, ξt ,u (x)), Ψ2 (u, ξt ,u (x))) ∂x t × Dt ,u (x)du + Φx (ξt ,T (x))Dt ,T (x)|F (t )
γ (t ) =
T
+E
(gy (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Y t ,x (u)
t
+ gz (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Z t ,x (u))du|F (t )
t
This proves the second statement. Furthermore,
γ (u)dW (u).
=
(7)
0
Putting t = T gives:
Φ (X (T )) +
T
g (u, X (u), Y (u), Z (u))du = 0
T
γ (u)dW (u).
(8)
0
Subtracting (8) from (7) gives: Y (t ) −
T
g (u, X (u), Y (u), Z (u))du t T
γ (u)dW (u) = Φ (X (T )).
+ t
By the uniqueness of the solution of the MFBSDE, we must have: Z (t ) = γ (t ),
t ∈T.
The second statement then follows from the first statement in ∂Ψ Theorem 1 by noting again that Ψ2 = σ (t , x) ∂ x1 . 5. Convex risk measures
T
gx (u, ξ0,u (x0 ), Ψ1 (u, ξ0,u (x0 )),
=E t
Ψ2 (u, ξ0,u (x0 )))D0,u (x)du
1 + Φx (ξ0,T (x0 ))D0,T (x)|F (t ) D− 0,t (x0 ) T
+E
(gy (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Y t ,x (u)
t
+ gz (u, X t ,x (u), Y t ,x (u), Z t ,x (u))∂x Z t ,x (u))du|F (t ) . This proves the first statement, completing the proof of the theorem. The following theorem identifies the solution (Y , Z ) of the backward system of the MFBSDE. Theorem 2. For each t ∈ T , Y (t ) = Ψ1 (t , X (t )), which is a probabilistic representation for the solution Ψ1 (t , x) of the following semi-linear PDE:
∂ Ψ1 ∂ Ψ1 1 ∂ 2 Ψ1 + µ(t , x) + σ 2 ( t , x) 2 ∂t ∂x 2 ∂x ∂ Ψ1 + g t , x, Ψ1 (t , x), σ (t , x) (t , x) = 0, ∂x
The object of this section is to apply the results in the last section to evaluate a convex risk measure for a contingent claim with maturity at time T which is written on an underlying risky asset S whose price dynamics are governed by a geometric Brownian motion. Let α : T × ℜ+ → ℜ and β : T × ℜ+ → ℜ be measurable functions of (t , s) ∈ T × ℜ+ , where ℜ+ is the set of positive real numbers. We assume that µ(t , s) and σ (t , s) are three times differentiable in s and which, together with their derivatives, are bounded, for each t ∈ T . For each t ∈ T , µ(t , s) and σ (t , s) represent the appreciation rate and volatility of the underlying risky asset at time t when the current value of the risky S (t ) = s. Suppose, under the real-world probability measure P, the price process of the underlying risky asset S := {S (t )|t ∈ T } is governed by the following geometric Brownian motion: dS (t ) = α(t , S (t ))S (t )dt + β(t , S (t ))S (t )dW (t ),
t ∈T
S (0) = s0 > 0. To simplify our analysis and notation, we assume that the riskfree rate of interest is zero. The case of non-zero rate can be incorporated easily by considering S as the discounted price of the risky asset. Putting µ(t , S (t )) := α(t , S (t ))S (t ) and
β(t , S (t )) := β(t , S (t ))S (t ), we write the price dynamics of the underlying risky asset in the
R.J. Elliott, T.K. Siu / Systems & Control Letters 61 (2012) 1017–1022
form of the FSDE (1): dS (t ) = µ(t , S (t ))dt + σ (t , S (t ))dW (t ).
(9)
Suppose, for each t , s ∈ T with t ≥ s, ξs,t (x) is the unique strong solution of the FSDE (9) such that ξs,s (x) = x. Then, as in Section 3, we define a stochastic flow of diffeomorphism of the map x → ξs,t (x) whose derivative satisfies the linearized Eq. (3). Let P (S (T )) be the payoff of the contingent claim at the maturity time T , where P (S (T )) ∈ L2 (Ω , F (T ), P ), the space of squareintegrable, F (T )-measurable, random variables with respect to the measure P. Our goal is to evaluate a convex risk measure for the claim P (S (T )). Indeed, the risk measurement of derivative securities has been discussed in a number of papers, for example, Boyle et al. [26] using a binomial approach, Siu and Yang [27] and Siu et al. [28,29] using a Bayesian approach and Elliott et al. [30] using a partial differential equation approach. Different risk measures have been considered in these works such as Value at Risk, Expected Shortfall, Coherent Risk Measure pioneered by Artzner et al. [31] and Convex Risk Measures proposed by Föllmer and Schied [32] and Frittelli and Rosazza-Gianin [33]. Here we consider a convex risk measure and adopt the approach based on a MFBSDE. We recall some essential concepts and results for convex risk measures. Write X for the space Lp (Ω , F (T ), P ) of p-integrable, F (T )-measurable, random variables, where p ∈ [1, ∞]. The space X is the set of random variables representing the future values of risky financial positions which will be realized at time T . Then a convex risk measure is defined as follows: Definition 1. A convex risk measure is a functional ρ : X → ℜ which satisfies the following three conditions: 1. (Translation invariance) For each k ∈ ℜ and X ∈ X,
ρ(X + k) = ρ(X ) − k. 2. (Monotonicity) For each X1 , X2 ∈ X, if X1 (ω) ≤ X2 (ω) for each ω ∈ Ω,
ρ(X1 ) ≥ ρ(X2 ). 3. (Convexity) For each X1 , X2 ∈ X and each λ ∈ (0, 1),
ρ(λX1 + (1 − λ)X1 ) ≤ λρ(X1 ) + (1 − λ)ρ(X2 ). The following theorem is due to Frittelli and RosazzaGianin [33] and Föllmer and Schied [32]. It is a central result in the analysis of convex risk measures and gives a representation for a, (generic), convex risk measure. Theorem 3. A functional ρ : X → ℜ is a convex risk measure if and only if there is a family of probability measures Ma absolutely continuous with respect to P and a convex ‘‘penalty’’ function η : Ma → ℜ with η(Q ) < ∞ for each Q ∈ Ma such that for each X ∈ X,
ρ(X ) = sup {E Q [−X ] − η(Q )}. Here E Q is the expectation under Q . An important example of convex risk measures is the entropic risk measure which is related to an expected exponential utility function. Suppose Ma represents the space of all probability measures absolutely continuous with respect to P and η is the relative entropy defined by:
dQ dP
ln
dQ dP
:= R(Q , P ),
Q ∈ Ma .
Then an entropic risk measure ρ E is defined by:
ρ E (X ) := sup {E Q (−X ) − R(Q , P )}, Q ∈Ma
Besides the representation in Theorem 3, a convex risk measure can also be represented as the g-expectation. The definition of the g-expectation was given in [34], where it was defined as the solution of a BSDE. We state the definition of the conditional gexpectation by Peng [34] and Rosazza-Gianin [35] by adapting it to our context and notation here. Definition 2. For any X ∈ L2 (Ω , F (T ), P ), let {(Y −X (t ), −X Z (t ))|t ∈ T } be the square-integrable solution of the following BSDE with terminal condition −X :
−dY (t ) = g (t , X (t ), Y (t ), Z (t ))dt − Z (t )dW (t ), Y (T ) = −X . Then the conditional g-expectation of −X , denoted by
Eg (−X |F (t )), is defined as:
Eg (−X |F (t )) = Y −X (t ). If the function g is convex, a convex risk measure ρ(X ) := ρ g (X ), X ∈ L2 (Ω , F , P ), can be represented as the g-expectation of −X as follows: ρ g (X ) = Eg (−X |F (0)) = Y −X (0) = Eg (−X ). In our case, we are interested in evaluating the convex risk measure ρ(P (S (T ))) of the contingent claim with payoff P (S (T )). Suppose P (S (T )) = −Φ (S (T )) and g is the driver function of the BSDE (2). Then the convex risk measure ρ(P (S (T ))) := ρ g (P (S (T ))) associated with the function g can be represented as the following g-expectation:
ρ g (P (S (T ))) = Eg (Φ (S (T ))). Consequently,
ρ g (Φ (S (T ))) = Eg (Φ (S (T ))) = Y (0), where Y (0) is the initial value of the first component Y of the solution of the BSDE (2). Since the price process S satisfies the FSDE (1), the evaluation of the convex risk measure ρ g (P (S (T ))) can be formulated as the problem of solving the backward system in the MFBSDE described by the coupled system consisting of (1) and (2). Applying Theorems 1 and 2, we obtain the following corollary which gives a way to evaluate convex risk measure ρ g (P (S (T ))). Corollary 1. Suppose Ψ1 ∈ C 1,2 (T × ℜ+ ) satisfies the following semi-linear PDE:
∂ Ψ1 ∂ Ψ1 1 ∂ 2 Ψ1 + µ(t , s)s + σ 2 (t , s)s2 2 ∂t ∂s 2 ∂s ∂ Ψ1 + g t , s, Ψ1 (t , s), σ (t , s) (t , s) = 0, ∂s with terminal condition:
Ψ1 (T , s) = −P (s). Then ρ g (P (S (T ))) = Ψ1 (0, s0 ). This generalizes the PDE result for coherent risk measures in [27].
Q ∈Ma
η(Q ) := E
1021
X ∈ X.
6. Conclusion We adopted techniques from stochastic flows to discuss the solution of MBSDEs. The solution of the backward system in an MBSDE was identified using a martingale representation, a differentiation rule, a unique decomposition of special semimartingales and stochastic flows. The first component of the solution was represented as the solution of a semilinear PDE while the second component was represented as an expectation of an integral with an integrand given by the derivative of a stochastic flow. An application to evaluate a convex risk measure for a contingent claim was discussed.
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