Backward stochastic differential equations associated to a symmetric Markov process
The authors consider a semi-linear symmetric second order differential operator \(L\) (in divergence form, without non degeneracy assumption but with some closability condition) having measurable coefficients, and they study the system of PDE's \[ (\partial_{t} +L)u(t,x)+f(t,x,u,\nabla u\sigma)=0,\;(t,x)\in[0,T]\times R^{d},\quad u(T,x)=\Phi(x),\tag{1} \] to which they associate backward stochastic differential equations (BSDE). If \(f\) and the coefficients of \(L\) are Lipschitz continuous, (1) has a viscosity solution \(u\); a diffusion process \(X_{s}^{t,x}\), \(s\in[ t,T],(t,x)\in[0,T]\times R^{d}\), with infinitesimal operator \(L\) and driven by some Brownian motion \(B\) can be defined, and a stochastic interpretation of \(u\) can be given with the help of the solution \(Y_{s}^{t,x}=u(s,X_{s}^{t,x})\), \(Z_{s}^{t,x}=(\nabla u\sigma)(s,X_{s}^{t,x})\) of the associated BSDE [\textit{E. Pardoux} and \textit{S. Peng}, Syst. Control Lett. 14, No.~1, 55--61 (1990; Zbl 0692.93064) and in: Stochastic partial differential equations and their applications. Lect. Notes Control Inf. Sci. 176, 200--217 (1992; Zbl 0766.60079)]. Further studies about the connection between BSDE and PDE were done by \textit{G. Barles} and \textit{E. Lesigne} [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 47--80 (1997; Zbl 0886.60049)] and \textit{V. Bally} and \textit{A. Matoussi} [J. Theor. Probab. 14, No.~1, 125--164 (2001; Zbl 0982.60057)] (for solutions \(u\) in Sobolev spaces and associated BSDE). For measurable coefficients of \(L\), \textit{A. Lejay} [Stochastic Processes Appl. 97, No.~1, 1--39 (2002; Zbl 1058.60045)] studied BSDE's associated to PDE's, which are driven by a Dirichlet process. In the case of only measurable coefficients, the symmetric Markov process \(X\) generated by \(L\) does no more satisfy an SDE. The authors replace the usual stochastic calculus corresponding to SDE's by Fukushima's stochastic calculus (1994). By using functional analytical methods they first prove that, for measurable \(f:[0,T]\times R^{d}\times R^{\ell}\times R^{\ell\times k}\rightarrow R^{\ell}\), Lipschitz continuous in the \(z\)-variable, continuous and satisfying some monotonicity assumption with respect to the \(y\)-variable, there is a unique weak solution \(u\) of (1) in some functional space. They show that a BSDE with respect to the ``coordinate martingale'' of \(X\) (introduced by Fukushima and playing the role of \(\sigma(X_{r}^{t,x})dB_{r}\) from the Lipschitz case) can be solved on each time interval \([0,T-t]\) under the law \(P^{x}\) of \(X\), for any \(t\in[0,T]\) and any \(x\) outside some exceptional set. The solution of the BSDE gives a stochastic interpretation of \(u\). To solve such BSDE's, as an important technical ingredient the authors prove a representation theorem for arbitrary martingales; this generalizes Fukushima's result for martingale additive functionals.
divergence form semilinear partial differential equations
scientific article; zbMATH DE number 2168219