Existence and Uniqueness of Viscosity Solutions for Nonlinear Variational Inequalities Associated with Mixed Control
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
Existence and Uniqueness of Viscosity Solutions for Nonlinear Variational Inequalities Associated with Mixed Control Shipei HU1
Abstract The author investigates the nonlinear parabolic variational inequality derived from the mixed stochastic control problem on finite horizon. Supposing that some sufficiently smooth conditions hold, by the dynamic programming principle, the author builds the Hamilton-Jacobi-Bellman (HJB for short) variational inequality for the value function. The author also proves that the value function is the unique viscosity solution of the HJB variational inequality and gives an application to the quasi-variational inequality. Keywords Optimal stopping, Mixed control, Variational inequality, Viscosity solution 2000 MR Subject Classification 49J20, 49L25, 60G40, 93E20
1 Introduction Let T > 0 and (Ω, F , (Ft )t≥0 , P) be a given complete filtered probability space, where {Ft ; t ∈ [0, T ], T < ∞} satisfies the usual conditions. Let Wt be an n-dimensional standard Brownian motion. Let S[0, T ] be the set of all {Ft }0≤t≤T -stopping times taking values in [0, T ]. For any τ1 , τ2 ∈ S[0, T ] with τ1 ≤ τ2 almost surely, P{τ1 < τ2 } > 0. For any s ∈ [0, T ) and x ∈ Rn , consider the following stochastic differential equation (SDE for short): dXt = [b(t, Xt ) + Ct ]dt + σ(t, Xt )dWt , t ∈ [s, T ], (1.1) Xs = x, where the mappings b(t, x) and σ(t, x) are two Lipschitz continuous functions and take value in Rn and Rn ⊗ Rn , respectively. Let A denote the class of all n-dimensional Ft -progressively RT measurable processes C = (Ct ) and there exists an M > 0 such that E 0 |Cs |2 ds < M . Thus, SDE (1.1) has a unique strong solution X· := X(·; s, x). The cost functional is given by J(C(·), τ ; s, x) i hZ τ =E e−α(t−s) {f (t, Xt ) + a−1 |Ct |2 }dt + e−α(τ −s) g(τ, Xτ ) , s
τ ∈ S[s, T ],
(1.2)
where a > 0, mappings f, g : [0, T ]×Rn → [0, ∞) are non-negative and satisfy proper conditions and α > 0 is a discount rate. In this case, the value function V : [0, T ] × Rn → R is defined as Manuscript received April 11, 2017. Revised October 9, 2018. of Mathematics, Jiaxing University, Jiaxing 314001, Zhejiang, China. E-mail: [email protected]
1 Department
794
S. P. Hu
follows: V (s, x) =
inf
(C(·),τ )∈A×S[s,T ]
J(C(·), τ ; s, x).
(1.3)
We call τ ∈ S[s, T ] is an optimal stopping time if the cost functional J defined by (1.2) has attained its infimum value, and the smallest one is referred to as the smallest optimal stopping time. The above optimal stopping problem over a finite time horizon can be reduced to the following variational inequality: ∂V a − LV + f − |DV |2 ≥ 0 in [0, T ) × Rn , ∂t 4 V ≤ g in [0, T ] × Rn , ∂V a − LV + f − |DV |2 (V − g)− = 0 in [0, T ) × Rn , ∂t 4 V (T, x) = g(T, x) on Rn .
(1.4)
Here L = L0 + α, L0 denotes the second order differential operator 1 L0 = − tr(σσ ∗ D2 ) − bD. 2 ∂ , · · · , ∂x∂n . Here |·| is the Euclidean norm, σ ∗ is the tran
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