Uniform large deviations for multivalued stochastic differential equations with Poisson jumps
The paper deals with multivalued stochastic differential equations driven by countably many Wiener processes and a Poisson random measure of the form \[ \begin{multlined} dX^\epsilon(t)\in -A(X^\epsilon(t))dt+b(X^\epsilon(t))dt+\sqrt{\epsilon}\sigma(X^\epsilon(t))dW(t)\\ +\int_\mathbb{Y}\gamma(X^\epsilon(t-),y)\bigl(\epsilon N^{\epsilon^{-1}}(dy,dt)-\nu(dy)dt\bigr)\end{multlined} \] with initial condition \(X(0)=x_0\in \overline{D(A)}\subset\mathbb{R}^d\). Here, \(\epsilon\) is a small parameter; \(A:\mathbb{R}^d\to 2^{\mathbb{R}^d}\) is a multivalued maximal monotone operator with domain \(D(A)\); the coefficients \(b,\,\sigma,\,\gamma\) are Lipschitz continuous vector fields; \(W\) is an sequence of independent Brownian motions; and \(N\) is a Poisson random measure with compensator \(\nu(dy)dt\), independent of \(W\) and a locally compact Polish space \(\mathbb{Y}\) for the marks. In the first part of the paper, necessary technical aspects and notations regarding multivalued stochastic differential equations and the Laplace principle are presented. Then, in the main part of the paper the uniform large deviation principle is proved for perturbed equations of the above form employing a variational representation of bounded measurable functionals of a Poisson random measure and an infinite-dimensional Brownian motion. The example of a reflected stochastic differential equation with Poisson jumps concludes the paper.
multivalued stochastic differential equations
stochastic differential equations with reflection
scientific article; zbMATH DE number 5960763
scientific article