Extended backward stochastic Volterra integral equations, Quasilinear parabolic equations, and Feynman–Kac formula

Abstract

In this paper, we establish the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. We first introduce the extended backward stochastic Volterra integral equations (EBSVIEs, for short). Under some mild conditions, we establish the well-posedness of EBSVIEs and obtain some regularity results of the adapted solution to the EBSVIEs via Malliavin calculus. We show that a given function expressed in terms of the solution to the EBSVIEs solves a certain system of non-local parabolic partial differential equations (PDEs, for short), which generalizes the famous nonlinear Feynman-Kac formula in Pardoux-Peng [21].Let us look at some special cases of EBSVIE (1.1). Suppose g(t, s, y, y ′ , z) = g(t, s, y, z), ∀(t, s, y, y ′ , z)then EBSVIE (1.1) is reduced to the following form:which is a family of so-called backward stochastic differential equations (BSDEs, for short) parameterized by t ∈ [0, T ]; see [20,12,16,39] for systematic discussions of BSDEs.On the other hand, if, then EBSVIE (1.1) is reduced to the following form:which is a so-called backward stochastic Volterra integral equation (BSVIE, for short). This is exactly why we call (1.1) an extended backward stochastic Volterra integral equation. BSVIEs of the form (1.3) was initially studied by Lin [15] and followed by several other researchers: Aman and NZi [3], Yong [35], Ren [24], Anh, Grecksch, and Yong [4], Djordjevi'c and Jankovi'c [6, 7], Hu and Øksendal [10], and the references therein. Recently, Wang, Sun, and Yong [28] established the well-posedness of quadratic BSVIEs (which means the generator g(t, s, y, z) of (1.3) has a quadratic growth in z) and explored the applications of quadratic BSVIEs to equilibrium dynamic risk measure and equilibrium recursive utility process. BSVIE of the more general form Y (t) = ψ(t) + T t g(t, r, Y (r), Z(t, r), Z(r, t))dr − T t Z(t, r)dW (r) (1.4)was firstly introduced by Yong [36] in his research on optimal control of forward stochastic Volterra integral equations (FSVIEs, for short). The BSVIE (1.4) has a remarkable feature that its solution might not be unique due to lack of restriction on the term Z(r, t); 0 ≤ t ≤ r ≤ T . Suggested by the nature of the equation from the adjoint equation in the Pontryagin type maximum principle, Yong [36] introduced the notion of adapted M-solution: A pair (Y (·), Z(· , ·)) is called an adapted M-solution to (1.4), if in addition to (i)-(iii) stated above, the following condition is also satisfied:a.e. t ∈ [0, T ], a.s. (1.5) Under usual Lipschitz conditions, well-posedness was established in [36] for the adapted M-solutions to BSVIEs of form (1.4). This important development has triggered extensive research on BSVIEs and their applications. For instance, Anh, Grecksch and Yong [4] investigated BSVIEs in Hilbert spaces; Shi, Wang and Yong [25] studied well-posedness of BSVIEs containing mean-fields (of the unknowns); Ren [24], Wang and Zhang [33] discussed BSVIEs with ...

Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems

“…A similar technique can be seen in the literature of BSDEs (see for example [6]), and in the literature of BSVIEs (see for example [17,18]). We remark that the estimates ( 10) and ( 11) are more detailed than [21]. Indeed, by letting t = t and then taking the supremum over t ∈ [S, T ], we get the estimates in Theorem 3.1 of [21].…”

“…For the sake of the well-posedness of the adjoint equations, in Section 3, we prove the well-posedness of the general EB-SVIE (1) under weaker assumptions than the literature. We provide a direct proof which is different from the original method of [21]. Moreover, we show a new type of regularity property of the solution (Y (•, •), Z(•, •)) = {(Y (t, s), Z(t, s))} (t,s)∈[S,T ] 2 to an EBSVIE with respect to the t-variable.…”

“…Remark 3.1. Compared with [21] and other previous researches on BSVIEs, the last assumption ( 7) on the continuity of the generator g with respect to t ∈ [S, T ] is new and weaker. In the literature, the continuity of g with respect to t ∈ [S, T ] is assumed to be pointwise, that is,…”

“…Shi-Wang-Yong [18] and Wang-Zhang [27] investigated optimal control problems of BSVIEs. Recently, as a generalization of BSVIEs, Wang [21] introduced EBSVIEs, and investigated the Feynman-Kac formula for a non-local quasilinear parabolic partial differential equation (PDE, for short). A similar equation was considered by Hamaguchi [7].…”

“…Remark 3.5. Wang [21] showed the well-posedness of EBSVIE (1) under a stronger assumption. His method is to show the existence and uniqueness of the solution of EBSVIE (1) (defined on S ≤ t ≤ s ≤ T ) firstly when T − S is small, and then, by considering an associated family of BSDEs (or, stochastic Fredholm equations), connect them inductively.…”