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://doi.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 - <p>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.</p>