Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations

Abstract

In this paper we study the fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and we prove that the value function of the optimal stochastic control problem is the maximal viscosity solution of the associated stochastic HJB equation. For the superparabolic cases when the diffusion coefficients are deterministic functions of time, states and controls, the uniqueness is addressed as well. (2010): 49L20, 49L25, 93E20, 35D40, 60H15 Keywords: stochastic Hamilton-Jacobi-Bellman equation, optimal stochastic control, backward stochastic partial differential equation, viscosity solution Mathematics Subject Classificationwhere T ∈ (0, ∞) is a fixed deterministic terminal time. Let U ⊂ R n be a nonempty compact set and U the set of all the U -valued and F t -adapted processes. The process (X t ) t∈[0,T ] is the state process. It is governed by the control θ ∈ U . We sometimes write X r,x;θ t for 0 ≤ r ≤ t ≤ T

Uniqueness of Viscosity Solutions of Stochastic Hamilton-Jacobi Equations

“…No.x Qiu & Wei: VISCOSITY SOLUTIONS OF STOCHASTIC HJ EQUATIONS 5 the space C 2 F in [28], by Definition 2.1, we have in fact characterized the two linear operators d t and d ω which is consistent with the two differential operators w.r.t. the paths of Wiener process W in the sense of [17], defined via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales; in particular, for the operator d ω u, an earlier discussion may be found in [4,Section 5.2].…”

“…Remark 2.1 Instead of C 2 F defined in [28] which requires D 2 u and Dd ω u to be lying in L 2 (C(R d )), we use C 1 F which imposes no requirement on D 2 u or Dd ω u. This is basically because the two terms D 2 u and Dd ω u are not involved in the first-order BSPDE (1.1).…”

“…The stochastic HJ equation (1.1) is a particular case of [28,Theorem 4.2] with vanishing diffusion coefficients. Therefore, as a straightforward consequence, we have the following existence of viscosity solution of BSPDE (1.1).…”

“…We note that even though the test function space used in [28] is C 2 F instead of C 1 F , the proof of Theorem 2.1 follows exactly the same as that of [28,Theorem 4.2] as there would be no term involving D 2 u or Dd ω u in the proof.…”

“…In this paper, we shall drop the strong assumptions on regularity of coefficients (see Remark 2.2) and prove the uniqueness of viscosity solution to stochastic HJ equation (1.1) corresponding to a degenerate fully nonlinear case of [28]. 3 Recalling heuristically the notion of viscosity solution proposed in [28], we may think of the concerned random fields like the first unknown variable u and the value function V as stochastic differential equations (SDEs) of the following form:…”