Stochastic Equations and Equations for Probabilistic Characteristics of Processes with Damped Jumps

Abstract

Processes such as shot noise are an adequate tool for modeling discontinuous random processes with a damping effect. Such processes arise in various fields of physics, technology and human activity, including the field of finance. They allow to simulate not only abrupt changes in the values of processes with jumps, but also the subsequent return of values to their original or near the original position. For this reason, shot noise is a suitable tool for simulating price hikes of various market assets. This paper presents a general formulation of the shot noise type processes, a stochastic differential equation corresponding to the process, and a connection with the integral-differential equation for the transition probability density, which is formulated in terms of generalized functions. In addition, the paper considers a stochastic equation that allows, along with an abrupt change described by a process of the shot noise type, a continuous change in the process. An equation is obtained for the transition probability density of a process containing jump-like and continuous types of evolution, which should also be considered in spaces of generalized functions. Thus, main results of the paper are obtained on the basis of the application of stochastic analysis and the theory of generalized functions.

Keywords

• wiener process
• shot noise
• stochastic differential equation
• probabilistic characteristics
• generalized functions