Backward stochastic variational inequalities on random interval

Abstract

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: \[\cases{\displaystyle -\mathrm{d}Y_t+\partial_yΨ(t,Y_t)\,\mathrm{d}Q_t\niΦ(t,Y_t,Z_t)\,\mathrm{d}Q_t-Z_t\,\mathrm{d}W_t,\qquad 0\leq t<τ,\cr \displaystyle{Y_τ=η,}}\] where $\tau$ is a stopping time, $Q$ is a progressively measurable increasing continuous stochastic process and ${\partial }_y\Psi$ is the subdifferential of the convex lower semicontinuous function $y⟼\Psi (t,y)$. As applications, we obtain from our main results applied for suitable convex functions, the existence for some backward stochastic partial differential equations with Dirichlet or Neumann boundary conditions.

Published at http://dx.doi.org/10.3150/14-BEJ601 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Fractional backward stochastic differential euqations and fractional backward variational inequalities
• Multivalued Backward Stochastic Differential Equations with Time Delayed Generators
• Numerical Schemes for Multivalued Backward Stochastic Differential Systems

Cited by in corpus (4)

• A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance
• Multivalued Backward Stochastic Differential Equations with Time Delayed Generators
• $L^p$-Variational Solutions of Multivalued Backward Stochastic Differential Equations
• Continuity of the Feynman-Kac formula for a generalized parabolic equation