Abstract

We present a general stochastic representation for a general class of processeswith resetting. It allows to describe any stochastic process intermittently terminated and restartedfrom a predefined random or non-random point. Our approach is based on stochastic differentialequations called jump-diffusion models. It allows to analyze processes with resetting both, analytically and using Monte Carlo simulation methods. To depict the strength of our approach, wederive a number of fundamental properties of Brownian motion with Poissonian resetting, such as:the Ito lemma, the moment-generating function, the characteristic function, the explicit form ofthe probability density function, moments of all orders, various forms of the Fokker-Planck equation, infinitesimal generator of the process and its adjoint operator. This way we build a generalframework for the analysis of any stochastic process with intermittent random resetting.