Time-inhomogeneous stochastic processes on the {$p$}-adic number field

Abstract

The theory of stochastic differential equation on the field of $p$-adics is initiated by Kochubei. In this article, we will focus on a class of random walks with a certain integrability to develop the theory of stochastic analysis in a way similar to the existing theory of stochastic analysis based on the Brownian motion. In fact, for any random walk in the class, we can introduce the notion of stochastic integral with respect to the random walk and justify the existence of the solution of the stochastic differential equations based on the random walk, where the stochastic differential equations admit not only Lipschitz continuous path dependent coefficients but also continuous coefficients growing at most linearly with respect to $p$-adic norm. Finally, we will see an example of stochastic process which can be covered by Dirichlet space theory and obtained also by solving stochastic differential equation.