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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 ItĂ´’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
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