Bull Braz Math Soc, New Series DOI 10.1007/s00574-017-0031-2
Pontryagin’s Risk-Sensitive Stochastic Maximum Principle for Backward Stochastic Differential Equations with Application Adel Chala1
Received: 8 April 2016 / Accepted: 6 February 2017 © Sociedade Brasileira de Matemática 2017
Abstract This paper studies the risk-sensitive optimal control problem for a backward stochastic system. More precisely, we set up a necessary stochastic maximum principle for a risk-sensitive optimal control of this kind of equations. The control domain is assumed to be convex and the generator coefficient of such system is allowed to be depend on the control variable. As a preliminary step, we study the risk-neutral problem for which an optimal solution exists. This is an extension of initial control system to this type of problem, where the set of admissible controls is convex. An example to carried out to illustrate our main result of risk-sensitive control problem under linear stochastic dynamics with exponential quadratic cost function. Keywords Backward stochastic differential equation · Risk-sensitive · Stochastic maximum principle · Adjoint equation · Variational principle · Logarithmic transformation Mathematics Subject Classification 93 E20 · 60 H30 · 60H10 · 91B28
1 Introduction In this paper, we will establish a necessary optimality condition, of the Pontryagin’s maximum principle type, for risk-sensitive performance functionals. We solve the
This work is Partially supported by The CNEPRU project N: C00L03UN070120140029.
B 1
Adel Chala
[email protected];
[email protected] Laboratory of Applied Mathematics, Mohamed Khider University, P.O. Box 145, 07000 Biskra, Algeria
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problem by using the approach developed by Djehiche et al. (2015). Their contribution can be summarized as follows. They have established a stochastic maximum principle for a class of risk-sensitive mean-field type control problems, where the distribution enters only through the mean of state process, this means that the drift, diffusion, and terminal cost functions depend on the state, the control and the means of state process. This work extends the results of Lim and Zhou (2005) to risk-sensitive control problems for dynamics that are non-Markovian and of mean-field type. In the risk-sensitive control problem, the system is governed by the nonlinear backward stochastic differential equation
dytv = − f (t, ytv , z tv , vt )dt + z tv dWt , yTv = a,
We define the criterion to be minimized, with initial and final risk-sensitive cost, as follows J θ (v) = E[eθ{(y
v (0))+ T 0
f (t,ytv ,z tv ,vt )dt}
],
A control u is called optimal if it solves J θ (u) = inf v∈U J θ (v). A stochastic maximum principle (SMP) for risk-sensitive optimal control problems for Markov diffusion processes with an exponential of integral performance functional was obtained in Lim and Zhou (2005) by making the relationship between the SMP and the Dynamic Programming Principle, the authors have used the first order adjoint process as the gradient of the value function of the control problem. This relationship holds only when the value function is smooth (see Assumption (B4) in Lim and Zhou 2005). By using the smoothness assumption the two papers of Shi and Wu (2011, 2012), have been used the approach used above, but extended it to jump processes. The existence of an optimal solution for this problem has been solved to achieve the objective of this paper, and establish necessary optimality conditions for these model. We give the optimality conditions for risk-sensitive controls. The idea is to reformulate in the first the risk-sensitive control problem in terms of an augmented state process and terminal payoff problem. An intermediate stochastic maximum principle is then obtained by applying the SMP of Yong (2010) (Theorem 3.1) for loss functionals without running cost. Then, we transform the intermediate first order adjoint processes to a simpler form by using the fact that set of controls is convex. Then, we establish necessary optimality conditions by using the logarithmic transform established by ElKaroui and Hamadène (2003). More precisely, the method of Lim and Zhou (2005), shows in fact that it suffices to use a generic square-integral martingale to transform p1 (t), 0), where the process p1 (t) is still a the pair ( p1 , q1 ) into the adjoint process ( p1 (T ), and is equal to square-integrable martingale, which would mean that p1 (t) = the constant E[ p1 (T )]. But, this generic martingale needs not be related to the adjoint → process − p (t) as in Lim and Zhou (2005). Instead, it will be part of the adjoint equation associated with the risk-sensitive SMP (see Theorem 3.1, below). The paper is organized as follows: in Sect. 2, we give the precise formulation of the problem and introduce the risk-sensitive model, we formulate the problem and give the various assumptions used throughout the paper. In Sect. 3, we shall study our system of
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Pontryagin’s Risk-Sensitive Stochastic Maximum Principle for. . .
backward SDE and give the new method of the transformation of the adjoint process, which will be the aim result in next section, and give our main result, the necessary optimality conditions for risk-sensitive control problem under additional hypothesis. In Sect. 4 we finished the paper by the application to Linear quadratic stochastic control problem.
2 Formulation of the Problem Let (, F, (Ft(W ) )t≥0 , P) be a probability space satisfying the usual conditions, in which a one−dimensional Brownian motion W = (Wt : 0 ≤ t ≤ T ) is defined. (W ) (W ) = σ (W (r ); 0 ≤ r ≤ t) ∨N , We assume that (Ft ) is defined by ∀t ≥ 0, Ft where N denotes the totality of P−null sets. Let M2 (0, T ; R) denote the set of one dimensional measurable random processes {ϕt , t ∈ [0, T ]} which satisfy: jointly T
(i) E 0
(W )
|ϕt |2 dt < ∞, (ii) ϕt is (Ft
) measurable, for any t ∈ [0, T ].
We denote similarly by S 2 ([0, T ]; R) the set of continuous one dimensional random processes which satisfy: (i) E
(W )
sup |ϕt |2 < ∞, (ii) ϕt is (Ft
) measurable, for any t ∈ [0, T ].
0≤t≤T
Let T be a strictly positive real number and U is a non empty subset of R. Definition 2.1 An admissible control v is a process with values in U such that sup |vt |2 < ∞. We denote by U the set of all admissible controls. 0≤t≤T
The set of all admissible control should be convex. For any v ∈ U, we consider the following backward stochastic differential equation system v dyt = −g(t, ytv , z tv , vt )dt + z tv dWt , (2.1) yTv = a, where g : [0, T ] × R × R × U → R, and the terminal condition a ∈ R is a random variable FT -measurable. We define the criterion to be minimized, with initial and final risk-sensitive cost, as follows ⎫⎤ ⎡ ⎧ ⎢ J θ (v) = E ⎢ ⎣e
⎨ θ (y v (0))+ ⎩
T
f (t,ytv ,z tv ,vt )dt 0
⎬
⎭⎥
⎥, ⎦
(2.2)
where θ is the risk-sensitive index and : R → R, f : [0, T ] × R × R × U → R. The control problem is to minimize the functional J θ over U, if u ∈ U is an optimal solution, that is (2.3) J θ (u) = inf J θ (v). v∈U
(H1 ): We assume that g(t, 0, 0) ∈
M2 (0, T ; R),
and there exists c > 0, such that
|g(t, y1 , z 1 ) − (t, y2 , z 2 )| ≤ c(|y1 − y2 | + z 1 − z 2 ).
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Proposition 2.1 For any given admissible control v(.), we assume that (H1 ) holds. Then the backward SDE (2.1) has a unique solution (ytv , z tv ) ∈ M2 (0, T ; R) × S 2 (0, T ; R). The proof can be seen in Pardoux and Peng (1990). A control that solves the problem {(2.1), (2.2), (2.3)} is called optimal. Our goal is to establish risk-sensitive necessary conditions of optimality, satisfied by a given optimal control, in the form of risk-sensitive stochastic maximum principle. We also assume (H2 ): ⎧ (i) g, f and are continuously differentiable with respect to ⎪ ⎪ ⎪ ⎪ ⎨ (y, z). (ii) The derivatives of g and f are bounded by ⎪ ⎪ C(1 + |y| + |z| + |v|). ⎪ ⎪ ⎩ (iii) The derivative of is bounded by C(1 + |y|). Under the above assumption, for every v ∈ U Eq. (2.1) has a unique strong solution and the cost function J θ is well defined from U into R. For more details the reader can see the famous paper Yong (2010).
3 Risk-Sensitive Stochastic Maximum Principle of Backward Type Control We introduce an auxiliary state process ξ v(t) which is solution of the following forward SDE: dξtv = f (t, ytv , z tv , vt )dt, ξ v (0) = 0 The backward type control problem is equivalent to ⎧ v ⎪ inf v∈U E[eθ{(y (0))+ξ(T )} ], ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ subject to dξtv = f (t, ytv , z tv , vt )dt, ⎪ ⎪ ⎪ dytv = −g(t, ytv , z tv , vt )dt + z tv dWt , ⎪ ⎪ ⎪ ⎩ v ξ (0) = 0, yTv = a. Recall that ⎧ ⎨
A T := e
123
θ (y u (0))+ ⎩
T f (t,,ytu ,z tu ,u t )dt
0
⎫ ⎬ ⎭
.
(3.1)
Pontryagin’s Risk-Sensitive Stochastic Maximum Principle for. . .
We can put T = (y u (0)) + tional is given by
T 0
f (t, , ytu , z tu , u t )dt, the risk-sensitive loss func-
T 1 log E exp (y u (0)) + f (t, , ytu , z tu , u t )dt θ 0 1 = log(E{exp θ T }). θ
θ =:
When the risk-sensitive index θ is small, the loss functional θ can be expanded as E(T ) +
θ V ar (T ) + O(θ 2 ), 2
where, V ar (T ) denotes the variance of T . If θ < 0, the variance of T , as a measure of risk, improves the performance θ , in which case the optimizer is called risk seeker. But, when θ > 0, the variance of T worsens the performance θ , in which case the optimizer is called risk averse. The risk-neutral loss functional E(T ) can be seen as a limit of risk-sensitive functional θ when θ → 0, for more details the reader can see the paper Djehiche et al. (2015), Tembine et al. (2014). Notation 3.1 We will use the following notations throughout the paper. For every φ ∈ f, g, respectively, we define ⎧ u u ⎨ [c]lφ(t) = φ(t, yt , z t , u t ), u ∂φ(t) = φ(t, yt , z tu , vt ) − φ(t, ytu , z tu , u t ), ⎩ u u φζ (t) = δφ δζ (t, yt , z t , u t ), ζ = y, z. where u t in an admissible control from U. We assume that (H1 ), and (H2 ) hold, the adjoint equation can be found by using the stochastic maximum principle for risk-neutral of forward-backward type control from Yong (2010) to augmented state dynamics (ξ, y, z). There exists a unique Fadapted pair of processes (( p1 , q1 ), ( p2 , q2 )), which solves the following system of forward-backward SDEs: ⎧ dp1 (t) 0 0 p1 (t) ⎪ − → ⎪ d p (t) = = − dt ⎪ ⎪ (t) dp2 (t) p2 (t) ⎨ f y (t) g y q (t) + 1 θ dWt , ⎪ ⎪ − Hz (t) ⎪ ⎪ ⎩ p2 (0) = −θ y (y u (0))AθT , p1 (T ) = −θ AθT ,
(3.2)
with E
i=2
T
sup | pi (t)| +
i=1 0≤t≤T
2
|q1 (t)| dt < ∞. 2
0
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θ be the Hamiltonian associated with the optimal state We suppose here that H → → dynamics (ξ, y, z), and the pair of adjoint process (− p (t), − q (t)) given by: → → θ (t, y u , z u , u, − p (t), − q (t)) := H
f (t) − → p (t). g(t)
The following theorem called the stochastic maximum principle for risk-neutral of forward-backward type control from Theorem 3.1 Assume that (H1 ), (H2 ) hold. If (ξ, y u , z u ) is an optimal solution of the risk-neutral control problem (3.1), then there is a pair of Ft -adapted processes (( p1 , q1 ), ( p2 , q2 )) that satisfies (3.2), such that θ (t) ≤ 0, ∂H
(3.3)
for all u ∈ U, almost every t and P-almost surely, where → → → → θ (t, ξ u , y u , z u , v, − θ (t, ξ u , y u , z u , u, − θ (t) := H p (t), − q (t)) − H p (t), − q (t)). ∂H 3.1 Finding the New Adjoint Equation As we say the Theorem 3.1 is a good stochastic maximum principle for the risk-neutral forward backward control problem. We follow the new approach of Djehiche et al. (2015) and suggest a transformation of the adjoint processes ( p1 , q1 ), and ( p2 , q2 ) in such a way to get rid of the first component ( p1 , q1 ) in (3.2) and to obtain the stochastic q2 ). maximum principle in terms of only one adjoint process that we denote ( p2 , Noting that dp1 (t) = q1 (t)dWt and p1 (T ) = −θ AθT , the explicit solution of this backward SDE is (3.4) p1 (t) = −θ E[AθT Ft ] = −θ V θ (t), where
V θ (t) := E[AθT Ft ], 0 ≤ t ≤ T.
(3.5) → → As a good look of (3.4), it would be natural to choose a transformation of (− p ,− q)
into an adjoint process ( p, q ), where p1 (t) = We consider the following transform p (t) =
p1 (t) p2 (t)
:=
1 p (t) θ V θ (t) 1
1 θ V θ (t)
= −1.
− → p (t), 0 ≤ t ≤ T.
(3.6)
By using (3.2) and (3.6), we have p (T ) := p1 (T ) = −1,
and
p2 (0) = − y (y u (0)).
The following properties of the generic martingale V θ are essential in order to investigate the properties of these new processes ( p (t), q (t)).
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Pontryagin’s Risk-Sensitive Stochastic Maximum Principle for. . .
First, we assume that (H2 ) holds, f and are bounded by constant C > 0, we have (3.7) 0 < e−(1+T )Cθ ≤ AθT ≤ e(1+T )Cθ . Therefore, V θ is a uniformly bounded F-martingale satisfying 0 < e−(1+T )Cθ ≤ Vtθ ≤ e(1+T )Cθ , 0 ≤ t ≤ T.
(3.8)
The sufficient conditions of the Logarithmic transform established in (El-Karoui and Hamadène 2003, Proposition 3.1) can be applied in the martingale V θ as following: V θ (t) = exp θ t + θ
t
0
f (s, ysu , z su , u s )ds , 0 ≤ t ≤ T.
and V θ (0) = E[θ 0 ] = E[AθT ], it is very easy to see from (3.8) and the boundedness of f that sup | t |2
E
≤ CT ,
(3.9)
0≤t≤T.
where, C T is a positive constant that depends only on T and the boundedness of f, and . The process is the first component of the F-adapted pair of processes ( , l) which is the unique solution to the following quadratic backward SDE:
d t = −{ f (t, ytu , z tu , u t ) + θ2 |l(t)|2 }dt + l(t)dWt ,
T = (y0u ),
(3.10)
where
T
E
|l(t)| dt < ∞. 2
0
In this paragraph, we will state and prove the necessary conditions of optimality for the system driven by stochastic differential equation with a risk sensitive performance type. For this end, let us summarize and prove some of lemmas that we will use thereafter. And for short notation let use the Notation 3.1. Lemma 3.1 In particular, V θ solves the following linear backward SDE d V θ (t) = θl(t)V θ (t)dWt , V θ (T ) = AθT .
(3.11)
Hence, the process defined on (, F, (Ft(W ) )t≥0 , P) by L θt , where V θ (t) = exp V θ (0)
0
t
θl(s)dWs −
θ2 2
0
t
|l(s)|2 ds
:= L θt , 0 ≤ t ≤ T.
(3.12)
is a uniformly bounded F-martingale.
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Proof First of all, t we may find the explicit form of (3.11) , by applying Itô’s formula to Vtθ = exp(θ 0 f (s)ds) exp(θ θt ). t f (s)ds . exp(θ θt ) d(Vtθ ) = d exp θ 0 t f (s)ds = θ f (t) exp θ θt + θ 0 t θ θ f (s)ds + θ (d t ). exp θ t + θ 0 t 1 22 θ f (s)ds dt + θ l (t). exp θ t + θ 2 0 = θl(t)V θ (t)dWt . Now, we can prove (3.12), by starting from the integral form of (3.11), such that t V θ (t) = θ 0 l(s)V θ (s)dWs . On the other hand, we have V θ (t) = exp θ
t 0
f (s)ds . exp(θ θt ).
By replacing θt in (3.10), we have V θ (t) = exp V θ (0)
t
θl(s)dWs −
0
θ2 2
t
|l(s)|2 ds
0
:= L θt .
In view of (3.9), the last expression (3.12) is a uniformly bounded F-martingale.
Lemma 3.2 The risk-sensitive for the adjoint equation satisfied by ( p2 , q2 ) and (V θ , l) becomes ⎧ [c]ld p2 (t) = −Hyθ (t)dt − Hzθ (t)dWtθ , ⎪ ⎪ ⎨ θ d V (t) = θl(t)V θ (t)dWt , (3.13) ⎪ V θ (T ) = Aθ (T ), ⎪ ⎩ p2 (0) = − y (y (0)) . The solution ( p, q , V θ , l) of the system (3.13) is unique, such that E
θ
T
sup | p (t)| + sup |V (t)| + 2
0≤t≤T
0≤t≤T
2
(| q (t)| + |l(t)| )dt < ∞, 2
2
(3.14)
0
where p2 (t) , V θ (t), l(t) = (g(t) + z t θl(t)) p2 − f (t). H θ t, yt , z t , q2 (t)
123
(3.15)
Pontryagin’s Risk-Sensitive Stochastic Maximum Principle for. . .
Proof We want to identify the processes α and q such that d p (t) = − α (t)dt + q (t)dWt . → p (t), and using the expression By applying Itô’s formula to the process − p (t) = θ V θ (t) θ of V in (3.11), we obtain
0 0 p1 (t) dt − θl(t) q (t)dt p2 (t) θ V θ (t) f y (t) g y (t) 1 q1 (t) p (t)dWt . dWt − θl(t) − θ V θ (t) −Hz (t) 1
d p (t) = −
By identifying the coefficients, we get the diffusion term q (t) =
1 θ V θ (t)
q1 (t) −Hz (t)
+ θl(t) p (t),
and the drift term of the process p (t): α (t) =
1 θ V θ (t)
Using the relation p (t) =
0 0 f y (t) g y (t)
→ 1 − p (t), θ V θ (t)
q (t) =
p1 (t) p2 (t)
+ θl(t) q (t).
the coefficient q (t) will be as:
q1 (t) z (t) −H
+ θl(t) p (t),
and the drift term of the process p (t): α (t) =
0 0 f y (t) g y (t)
p1 (t) p2 (t)
+ θl(t) q (t).
We finally obtain
0 0 d p (t) = − f y (t) g y (t)
p1 (t) dt − θl(t) q (t)dt p2 (t)
+ q (t)dWt . It is easily verified that q1 (t)[−θl(t)dt + dWt ], p1 (T ) = −1. d p1 (t) = In view of (3.12), the use of Girsanov’s Theorem, it lets us to write q1 (t)dWtθ , Pθ -as. p1 (T ) = −1, d p1 (t) =
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A. Chala
where, dWtθ = −θl(t)dt + dWt . is a Pθ -Brownian motion, where, t dPθ θ2 t θ 2 |l (s)| ds , 0 ≤ t ≤ T. := L t = exp θl (s) dWs − dP Ft 2 0 0 In view of (3.7) and (3.12), the probability measures Pθ and P are equivalent. p1 (t) = Hence, noting that p1 (t) := θ V 1θ (t) p1 (t) is square-integrable, we get that θ
p1 (T ) | Ft ] = −1. Its easy to find that the process q1 (t) is quadratic variation EP [ T such that 0 | q1 (t)|2 dt = 0. This implies that, for almost every 0 ≤ t ≤ T, q1 (t) = 0, Pθ and P-a.s.
0 0 d p (t) = − f y (t) g y (t) By using the relation q (t) =
q1 (t) z (t) −H
p1 (t) dt + q (t)dWtθ . p2 (t)
+ θl(t) p (t), in the above equation, we
obtain p1 (t) 0 0 dt d p (t) = − p2 (t) f y (t) g y (t) q1 (t) + − θl(t) p (t) dWtθ . p1 − gz (t) p2 − f z (t)
(3.16)
The second component of p2 given by (3.16), is p1 (t) + g y (t) p2 (t)}dt d p2 (t) = −{ f y (t) p1 (t) + gz (t) p2 (t) + θl(t) p2 (t)}dWtθ . − { f z (t) q2 ) and The main risk-sensitive for the second adjoint equation satisfied by ( p2 , (V θ , l) becomes ⎧ [c]ld p2 (t) = −Hyθ (t)dt − Hzθ (t)dWtθ , ⎪ ⎪ ⎨ θ d V (t) = θl(t)V θ (t)dWt , (3.17) ⎪ V θ (T ) = Aθ (T ), ⎪ ⎩ p2 (0) = − y (y(0)). The solution ( p, q , V θ , l) of the system (3.13) is unique, such that E
sup | p (t)|2 + sup |V θ (t)|2 + 0≤t≤T
123
0≤t≤T
0
T
(| q (t)|2 + |l(t)|2 )dt < ∞,
(3.18)
Pontryagin’s Risk-Sensitive Stochastic Maximum Principle for. . .
where p2 (t) θ , V (t), l(t) H t, yt , z t , q2 (t) θ
p2 − f (t). = (g(t) + z t θl(t))
Theorem 3.2 (Risk-sensitive stochastic maximum principle) We assume that (H1 ) holds, if (x(.), u(.)) is an optimal solution of the risk-sensitive control problem {2.1), (2.2), (2.3)}, then there exists pair of F-adapted processes (V θ , l), ( p, q) that satisfy (3.17), (3.18) such that ∂ H θ (t) ≤ 0, for all u ∈ U, almost every 0 ≤ t ≤ T and P-almost surely, where the Hamiltonian θ associated with (3.1), given by H p2 (t) → → θ (t, ξtu , ytu , z tu , − p t, − q t , u t ) = {θ Vtθ }H θ t, ytu , z tu , , V θ (t), l(t), u t , H q2 (t) and H θ is the risk-sensitive Hamiltonian given by (3.15).
4 Application to the Linear Quadratic Risk-Sensitive Control Problem We provide a concrete example of risk-sensitive backward stochastic LQ problem, and give the explicit optimal control and validate our major theoretical results in Theorem 3.2 (Risk-sensitive stochastic maximum principle). First let the control domain be U = [−1, 1]. Consider the following linear quadratic risk-sensitive control problem ⎧ ! 1 T 2 1 2 ⎪ inf E exp v dt + y (0) , ⎪ v∈ U 2 0 t 2 ⎪ ⎪ ⎪ ⎨ subject to d Vtθ = θl(t)V θ (t)dWt , ⎪ ⎪ ⎪ dy v = (Aytv + Bvt + C)dt + CdWt , ⎪ ⎪ ⎩ θt V (0) = A T , yTv = a.
(4.1)
where A, B, and C are real constants. T u u u Recall that A T := eθ{(y (0))+ 0 f (t,,yt ,z t ,u t )dt} . Instantly, we give the Hamiltonian H θ defined by 1 H θ (t, y, z, u, p, q, l) = (Ay + Bu + C + zθl) p − u 2 2
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We have Hyθ (t, y, z, u, p, q, l) = Ap, Hzθ (t, y, z, u, p, q, l) = p(θl + C), and Huθ (t, y, z, u, p, q, l) = Bp − u, and maximizing the Hamiltonian yields u = Bp.
(4.2)
Then, the optimal state dynamics is given by
dyt = (Ay + B 2 p + C)dt + CdWt , yT = a.
According to (3.10), the BSDE satisfied by (V θ , l) is equivalent to the solution ( , l) of the backward SDE d t = − 21 (u 2 + θl 2 (t))dt + l(t)dWt ,
T = 21 y 2 (0). As a technique of such a problem, we use a solution of the form
t =
1 α(t)2 y(t) + β(t), 2
where, α and β are deterministic functions such that α(T ) = 1, and β(T ) = 0. By using Ito’s formula and identifying the coefficients, we obtain l(t) = α(t)C y(t), and 1 1 · (u(t) + Bα(t)y(t))2 + (α(t) + (θC 2 + B)α 2 (t) + 2 Aα(t))y 2 (t) 2 2 · 1 2 + β(t) + C α(t) = 0. 2 Hence
B u(t) = −Bα(t)y(t) = − l(t), C
(4.3)
and ·
α(t) + (θC 2 + B)α 2 (t) + 2 Aα(t) = 0, α(T ) = 1. · 1 β(t) + C 2 α(t) = 0, β(T ) = 0. 2 We compare the equalities (4.2), and (4.3), we find the relation between the adjoint process p and l, 1 p(t) = − l(t). C
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Pontryagin’s Risk-Sensitive Stochastic Maximum Principle for. . .
Conclusion 4.1 The optimal control is u = −Bα(t)y(t), where α solves the risksensitive Riccati equation
·
α(t) + (θC 2 + B)α 2 (t) + 2 Aα(t) = 0, α(T ) = 1.
The optimal dynamics solves the linear backward SDE
d y(t) = ((A − Bα(t))y(t) + C)dt + CdWt , y(T ) = a.
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