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PreprintCC BY 4.0arXiv

Markov property for systems driven by the Brownian sheet

  • Nacira Agram
  • Bernt Øksendal
  • Frank Proske
  • Olena Tymoshenko
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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,D1Y,D2Y) 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.

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Cite

BibTeXRIS
BibTeX
@misc{agram2026markov,
  title = {{Markov property for systems driven by the Brownian sheet}},
  author = {Nacira Agram and Bernt Øksendal and Frank Proske and Olena Tymoshenko},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37805},
  eprint = {2609.37805},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37805},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Markov property for systems driven by the Brownian sheet
AU  - Nacira Agram
AU  - Bernt Øksendal
AU  - Frank Proske
AU  - Olena Tymoshenko
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37805
UR  - https://arxiv.org/abs/2609.37805
AB  - 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.
ER  -