Generalized Directional Gradients, Backward Stochastic Differential Equations and Mild Solutions of Semilinear Parabolic Equations

Abstract

We study a forward-backward system of stochastic differential equations in an infinite dimensional framekork and its relationships with a semilinear parabolic differential equation on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows to construct a unique solution of the parabolic equation in a suitable class of locally Lipschitz real functions. The parabolic equation is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions. The use of the generalized directional gradient allows to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay equations and reaction-diffusion equations), where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic equations of Black-Scholes or Hamilton-Jacobi-Bellman type.

Stochastic reaction-diffusion equations on networks with dynamic time-delayed boundary conditions

“…where, besides having used the notations defined along previous sections, we denote by z the control, while we use the notation X z , to indicate the explicit dependence of the process X ∈ E 2 , from the control z. In what follows we exploit the results contained in [23], where a general characterization of stochastic optimal control problem in infinite dimension is given by means of a forward-backward-SDE approach. Therefore, the control problem defined by equation (41), is to be understood in the weak sense, see also, e.g., [19,22].…”

“…As stated in [23], we first fix t 0 ≥ 0 and X 0 ∈ E 2 , then an Admissible Control System (ACS) is given by U = Ω, F , (F t ) t≥0 , P, (W (t)) t≥0 , z , where…”

Random Perturbation of PDEs and Fluid Dynamic Models

“…The two usual strategies are to prove uniqueness for the Fokker-Planck equation or existence of sufficiently regular solutions to the backward Kolmogorov equation. Kolmogorov equation applies also to other problems, like control theory and averaging; a full list is not appropriate here, let us mention only Gozzi et al [118,119], Fuhrman and Tessitore [112,113], Cerrai and Freidlin [47]. [27], Barbu et al [28], Ambrosio et al [10], Manca [154], Bogachev et al [36][37][38].…”

Weak Solutions of Semilinear PDEs in Sobolev Spaces and Their Probabilistic Interpretation via the FBSDEs

“…Of course, when the solution of the PDE is smooth enough, the meaning of this connection is as usual straightforward. Such a regularity may be obtained under strong assumptions on the coefficients of (1) and the nonlinearity; see, e.g., [1,2]. However, when the coefficients are no more so smooth, the PDE has to be solved in a weak way.…”

“…The improved result on the equivalence of norm and its corollary are shown in Section 3. Finally, Section 4 is devoted to the weak formulation of PDE (1) and FBSDE (2) and provides connection between these solutions.…”