# American Institute of Mathematical Sciences

April  2011, 29(2): 595-613. doi: 10.3934/dcds.2011.29.595

## Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls

 1 Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63, 35121 Padova, Italy 2 Dipartimento di Metodi e Modelli Matematici, per le Scienze Applicate, Via Trieste, 63, 35121 Padova

Received  September 2009 Revised  March 2010 Published  October 2010

We develop a notion of generalized solution to a stochastic differential equation depending in a nonlinear way on a vector--valued stochastic control process $\{U_t\},$ merely of bounded variation, and on its derivative. Our results rely on the concept of Lipschitz continuous graph completion of $\{U_t\}$ and the generalized solution turns out to coincide a.e. with the limit of classical solutions to (1). In the linear case our notion of solution is equivalent to the usual one in distributional sense. We prove that the generalized solution does not depend on the particular graph-completion of the control process $\{U_t\}$ both for vector-valued controls under a suitable commutativity condition and for scalar controls.
Citation: Monica Motta, Caterina Sartori. Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 595-613. doi: 10.3934/dcds.2011.29.595
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