Point Process Differentials with Evolving Intensities

Abstract

The theory of stochastic differential equations customarily assumes one is a priori given known coefficients and known driving terms (e.g., white noise, Poisson white noise, etc.). Motivated by simple models in microeconomics ([3], [5]), in this article we investigate stochastic differential equations where one of the driving terms is not a priori given but evolves in a fashion depending on the paths of the solution. For example, a driving term may be a point process N = N[λ(X)] with an instantaneous stochastic intensity λ_S = λ(X)_S, where X is a solution of the equation:

$${X_t} = {K_t} + \int\limits_0^t F {\left( X \right)_s}d{Y_s} + \int\limits_0^t {G{{\left( X \right)}_S}} dN{\left[ {\lambda \left( X \right)} \right]_s}.$$

((1.1))

These results were obtained jointly with Jean Jacod while the author was visiting the Université de Rennes.