## Backward stochastic differential equations and quasilinear parabolic partial differential equations.(English)Zbl 0766.60079

Stochastic partial differential equations and their applications, Proc. IFIP Int. Conf., Charlotte/NC (USA) 1991, Lect. Notes Control Inf. Sci. 176, 200-217 (1992).
[For the entire collection see Zbl 0754.00020.]
The first aim of this paper is to study the regularity properties of the solution of a new type of backward stochastic differential equation (BSDE) defined on $$[t,T]$$: $-dY_ s^{t,x}=f(X_ s^{t,x},Y_ s^{t,x},Z_ s^{t,x})ds-Z_ s^{t,x}dW_ s,\quad Y_ T^{t,x}=g(X_ T^{t,x}),$ where $$W$$ is a Brownian motion. The solution of this BSDE consists of a pair of adapted processes $$(Y,Z)$$ valued in $$\mathbb{R}^ k\times\mathbb{R}^{k\times d}$$. Here, for any given parameter $$(t,x)\in[0,T)\times\mathbb{R}^ n$$, $$X^{t,x}$$ is considered as a given process solved by a classical SDE of Itô’s type: $dX_ s^{t,x}=b(X_ s^{t,x})ds+\sigma(X_ s^{t,x})dW_ s,\quad t\leq s\leq T,$ with initial datum $$X_ t^{t,x}=x$$. This type of BSDE has been studied by the authors [Syst. Control. Lett. 14, No. 1, 56-61 (1990; Zbl 0692.93064)], and it has been used by the second author [Stochastics Stochastics Rep. 37, No. 1/2, 61-74 (1991; Zbl 0739.60060)] to represent the given solution of the following $$(k$$-dimensional) system of quasilinear parabolic PDE $-{\partial u\over\partial t}={\mathcal L}u+f(x,u,(\nabla u\sigma)),\quad u(T,x)=g(x),$ as $$u(t,x)=Y_ t^{t,x}$$ (a generalized Feynman-Kac formula), where $${\mathcal L}$$ is the infinitesimal operator generated by the Markovian $$X^{t,x}$$.
The second aim of this paper is then to apply the obtained regularity results of $$Y^{t,x}$$ with respect to $$(t,x)$$ to deduce a converse of the above result, namely to show that a function given by $$u(t,x)=Y_ t^{t,x}$$ solves the above mentioned parabolic PDE. It gives an existence, uniqueness and regularity $$(C^{1,2})$$ result for this (possibly degenerate) PDE.
Reviewer: S.Peng (Jinan)

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60J60 Diffusion processes

### Citations:

Zbl 0754.00020; Zbl 0692.93064; Zbl 0739.60060