• 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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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).

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

First available in Project Euclid: 17 March 2015

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


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.

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  • [1] Alòs, E., Mazet, O. and Nualart, D. (2001). Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 766–801.
  • [2] Decreusefond, L. and Üstünel, A.S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177–214.
  • [3] Debicki, K. (2002). Ruin probability for Gaussian integrated processes. Stochastic Process. Appl. 98 151–174.
  • [4] Fernique, X. (1975). Regularité des trajectoires des fonctions aléatoires gaussiennes. In École d’Été de Probabilités de Saint-Flour IV – 1974. Lecture Notes in Math. 480 1–96. Berlin: Springer.
  • [5] Frangos, N.E., Vrontos, S.D. and Yannacopoulos, A.N. (2005). Ruin probability at a given time for a model with liabilities of the fractional Brownian motion type: A partial differential equation approach. Scand. Actuar. J. 4 285–308.
  • [6] Frangos, N.E., Vrontos, S.D. and Yannacopoulos, A.N. (2007). Reinsurance control in a model with liabilities of the fractional Brownian motion type. Appl. Stoch. Models Bus. Ind. 23 403–428.
  • [7] Frangos, N.E., Vrontos, S.D. and Yannacopoulos, A.N. (2007). Evidence of long memory in automobile third party liability insurance data. Preprint.
  • [8] Gyöngy, I. and Pardoux, É. (1993). On quasi-linear stochastic partial differential equations. Probab. Theory Related Fields 94 413–425.
  • [9] Hu, Y., Nualart, D. and Song, X. (2008). A singular stochastic differential equation driven by fractional Brownian motion. Statist. Probab. Lett. 78 2075–2085.
  • [10] Hüsler, J. and Piterbarg, V. (2004). On the ruin probability for physical fractional Brownian motion. Stochastic Process. Appl. 113 315–332.
  • [11] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. Amsterdam: North-Holland.
  • [12] Krylov, N.V. (1971). A certain estimate from the theory of stochastic integrals. Teor. Veroyatn. Primen. 16 446–457.
  • [13] Michna, Z. (1998). Self-similar processes in collective risk theory. J. Appl. Math. Stoch. Anal. 11 429–448.
  • [14] Michna, Z. (1999). On tail probabilities and first passage times for fractional Brownian motion. Math. Methods Oper. Res. 49 335–354.
  • [15] Mishura, Yu. and Nualart, D. (2004). Weak solutions for stochastic differential equations with additive fractional noise. Statist. Probab. Lett. 70 253–261.
  • [16] Nualart, D. and Ouknine, Y. (2002). Regularization of differential equations by fractional noise. Stochastic Process. Appl. 102 103–116.
  • [17] Protter, P. (1990). Stochastic Integration and Differential Equations – A New Approach. Applications of Mathematics (New York) 21. Berlin: Springer.
  • [18] Rogers, L.C.G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales. Vol. 2. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.
  • [19] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. Wiley Series in Probability and Statistics. Chichester: Wiley.
  • [20] Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993). Fractional Integrals and Derivatives. Yverdon: Gordon and Breach Science Publishers.
  • [21] Situ, R. (2005). Theory of Stochastic Differential Equations with Jumps and Applications. Mathematical and Analytical Techniques with Applications to Engineering. New York: Springer.