Mathematics – Probability
Scientific paper
2011-12-25
Mathematics
Probability
35 pages
Scientific paper
The aim of this paper is to study, in the infinite dimensional framework, the existence and uniqueness for the solution of the following multivalued generalized backward stochastic differential equation, considered on a random, possibly infinite, time interval: \[\{{array} [c]{r} -dY_{t}+\partial_{y}\Psi(t,Y_{t}) dQ_{t}\ni\Phi(t,Y_{t},Z_{t}) dQ_{t}-Z_{t}dW_{t},\;t\in[ 0,\tau], \multicolumn{1}{l}{Y_{\tau}=\eta,} {array} \,.\] where $\tau$ is a stopping time, $Q$ is a progresivelly measurable increasing continuous stochastic process and $\partial_{y}\Psi$ is the subdifferential of the convex lower semicontinuous function $y\longmapsto\Psi(t,y)$. Our results generalize those of E. Pardoux and A. R\u{a}\c{s}canu (Stochastics 67, 1999) to the case in which the function $\Phi$ satisfies a local boundeness condition (instead of sublinear growth condition with respect to $y$) and also the results from Ph. Briand et al. (Stochastic Process. Appl. 108, 2003) by considering the multivalued equation. As applications, we obtain from our main result applied for suitable convex functions, the existence for some backward stochastic partial differential equations with Dirichlet or Neumann boundary conditions.
Maticiuc Lucian
Rascanu Aurel
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