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

Monica Motta; Caterina Sartori

doi:10.3934/dcds.2011.29.595

Article Contents

2011, Volume 29, Issue 2: 595-613. Doi: 10.3934/dcds.2011.29.595

This issue Previous Article Euler-Lagrange equations for composition functionals in calculus of variations on time scales Next Article Generalized exterior sphere conditions and $\varphi$-convexity

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

Monica Motta^1, and
Caterina Sartori^2,

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

Received: August 31, 2009

Revised: February 28, 2010

Published: May 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 49E05, 49N25; Secondary: 93E03.

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