Work

Markov property for systems driven by the Brownian sheet

Year: 2026

Type: preprint

Abstract: We study stochastic systems driven by a Brownian sheet and formulate a Markov property adapted to their two-parameter structure. For time-space homogeneous Itô sheets, the future evolution from a point $(t,x)$ is not determined by the single value $Y(t,x)$ alone, but also by the boundary data along the horizontal and vertical edges issuing from $(t,x)$. We therefore show that the enlarged state consisting of the triple $(Y,D_1Y,D_2Y)$ satisfies a natural Markov property with respect to the past sigma-algebra generated by the two coordinate histories of the Brownian sheet. The result is illustrated by examples showing the strong martingale property for the Brownian sheet and for simple Brownian sheet systems. We also recall a Brownian sheet Girsanov theorem and use it to treat drifted sheets.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.37805

Authors: Agram, Nacira, Øksendal, Bernt, Proske, Frank, Tymoshenko, Olena

Institutions: Digital Futures, KTH Royal Institute of Technology, University of Oslo, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”

Language: English

Topic: Stochastic processes and financial applications

Subfield: Finance

Field: Economics, Econometrics and Finance

Domain: Social Sciences

Open Access status: green

Funder: Vetenskapsrådet

Award: Advances in Stochastic Control Theory and Applications

OpenAlex ID: W7214987965

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