Abstract. In Rajeev (Indian J. Pure Appl. Math. 44 (2013), no. 2, 231-258.), 'Translation invariant diffusion in the space of tempered distributions', it was shown that there is an one-to-one correspondence between solutions of a class of finite dimensional stochastic differential equations (SDEs) and solutions of a class of stochastic parial differential equations (SPDEs) in the space of tempered distributions, which are driven by the same Brownian motion. Coefficients of the SDEs were related to coefficients of the SPDEs through convolution with the initial value of the SPDEs.In this paper, we consider the situation where solutions of the SDEs are stationary and ask whether solutions of the corresponding SPDEs are also stationary. We provide an affirmative answer, when the initial random variable takes value in a certain set C, which ensures that coefficients of the SDEs are related to coefficients of the SPDEs in the above manner.
We extend the Itō formula [14, Theorem 2.3] for semimartingales with rcll paths. We also comment on Local time process of such semimartingales. We apply the Itō formula to Lévy processes to obtain existence of solutions to certain classes of stochastic differential equations in the Hermite-Sobolev spaces.
In this article we show that a finite-dimensional stochastic differential equation driven by a Lévy noise can be formulated as a stochastic partial differential equation (SPDE) driven by the same Lévy noise. We prove the existence result for such an SPDE by Itô’s formula for translation operators, and the uniqueness by an adapted form of “Monotonicity inequality”, proved earlier in the diffusion case. As a
consequence, the solutions that we construct have the “translation invariance” property.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.