Bernoulli

  • Bernoulli
  • Volume 21, Number 1 (2015), 303-334.

Stochastic differential equations driven by fractional Brownian motion and Poisson point process

Lihua Bai and Jin Ma

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Abstract

In this paper, we study a class of stochastic differential equations with additive noise that contains a fractional Brownian motion (fBM) 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 the long term dependence between the claim payment and the past history of liability becomes the main focus. We establish some new fractional calculus on the fractional Wiener–Poisson space, from which we define the weak solution of the SDE and prove its existence and uniqueness. Using an extended form of Krylov-type estimate for the combined noise of fBM and compound Poisson, we prove the existence of the strong solution, along the lines of Gyöngy and Pardoux (Probab. Theory Related Fields 94 (1993) 413–425). Our result in particular extends the one by Mishura and Nualart (Statist. Probab. Lett. 70 (2004) 253–261).

Article information

Source
Bernoulli, Volume 21, Number 1 (2015), 303-334.

Dates
First available in Project Euclid: 17 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1426597071

Digital Object Identifier
doi:10.3150/13-BEJ568

Mathematical Reviews number (MathSciNet)
MR3322320

Zentralblatt MATH identifier
1319.60123

Keywords
discontinuous fractional calculus fractional Brownian motion fractional Wiener–Poisson space Krylov estimates Poisson point process stochastic differential equations

Citation

Bai, Lihua; Ma, Jin. Stochastic differential equations driven by fractional Brownian motion and Poisson point process. Bernoulli 21 (2015), no. 1, 303--334. doi:10.3150/13-BEJ568. https://projecteuclid.org/euclid.bj/1426597071


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