Volume 32, Issue 4
Stochastic Differential Equations Driven by Multifractional Brownian Motion and Poisson Point Process

Hailing Liu, Liping Xu & Zhi Li

J. Part. Diff. Eq., 32 (2019), pp. 352-368.

Published online: 2020-01

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  • Abstract

In this paper, we study a class of stochastic differential equations with additive noise that contains a non-stationary multifractional Brownian motion (mBm) with a Hurst parameter as a function of time and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which between the claim payment and the past history of liability present the long term dependence. By using the variable order fractional calculus on the fractional Wiener-Poisson space and a multifractional derivative operator, and employing Girsanov theorem for multifractional Brownian motion, we prove the existence of weak solutions to the SDEs under consideration, As a consequence, we deduce the uniqueness in law and the pathwise uniqueness.

  • Keywords

Stochastic differential equations multifractional Brownian motion fractional Wiener-Poisson space Poisson point process Girsanov theorem.

  • AMS Subject Headings

60H15, 60G15, 60H05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

liuhailing@126.com (Hailing Liu)

xlp211@126.com (Liping Xu)

lizhi_csu@126.com (Zhi Li)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-32-352, author = {Liu , Hailing and Xu , Liping and Li , Zhi }, title = {Stochastic Differential Equations Driven by Multifractional Brownian Motion and Poisson Point Process}, journal = {Journal of Partial Differential Equations}, year = {2020}, volume = {32}, number = {4}, pages = {352--368}, abstract = {

In this paper, we study a class of stochastic differential equations with additive noise that contains a non-stationary multifractional Brownian motion (mBm) with a Hurst parameter as a function of time and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which between the claim payment and the past history of liability present the long term dependence. By using the variable order fractional calculus on the fractional Wiener-Poisson space and a multifractional derivative operator, and employing Girsanov theorem for multifractional Brownian motion, we prove the existence of weak solutions to the SDEs under consideration, As a consequence, we deduce the uniqueness in law and the pathwise uniqueness.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v32.n4.5}, url = {http://global-sci.org/intro/article_detail/jpde/13614.html} }
TY - JOUR T1 - Stochastic Differential Equations Driven by Multifractional Brownian Motion and Poisson Point Process AU - Liu , Hailing AU - Xu , Liping AU - Li , Zhi JO - Journal of Partial Differential Equations VL - 4 SP - 352 EP - 368 PY - 2020 DA - 2020/01 SN - 32 DO - http://dor.org/10.4208/jpde.v32.n4.5 UR - https://global-sci.org/intro/article_detail/jpde/13614.html KW - Stochastic differential equations KW - multifractional Brownian motion KW - fractional Wiener-Poisson space KW - Poisson point process KW - Girsanov theorem. AB -

In this paper, we study a class of stochastic differential equations with additive noise that contains a non-stationary multifractional Brownian motion (mBm) with a Hurst parameter as a function of time and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which between the claim payment and the past history of liability present the long term dependence. By using the variable order fractional calculus on the fractional Wiener-Poisson space and a multifractional derivative operator, and employing Girsanov theorem for multifractional Brownian motion, we prove the existence of weak solutions to the SDEs under consideration, As a consequence, we deduce the uniqueness in law and the pathwise uniqueness.

Hailing Liu , Liping Xu & Zhi Li . (2020). Stochastic Differential Equations Driven by Multifractional Brownian Motion and Poisson Point Process. Journal of Partial Differential Equations. 32 (4). 352-368. doi:10.4208/jpde.v32.n4.5
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