Electronic Communications in Probability

Existence and uniqueness for backward stochastic differential equations driven by a random measure, possibly non quasi-left continuous

Elena Bandini

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We study the following backward stochastic differential equation on finite time horizon driven by an integer-valued random measure $\mu$ on $\mathbb R_+\times E$, where $E$ is a Lusin space, with compensator $\nu(dt,dx)=dA_t\,\phi_t(dx)$:\[Y_t = \xi + \int_{(t,T]} f(s,Y_{s-},Z_s(\cdot))\, d A_s - \int_{(t,T]} \int_E Z_s(x) \, (\mu-\nu)(ds,dx),\qquad 0\leq t\leq T.\]The generator $f$ satisfies, as usual, a uniform Lipschitz condition with respect to its last two arguments. In the literature, the existence and uniqueness for the above equation in the present general setting has only been established when $A$ is continuous or deterministic. The general case, i.e. $A$ is a right-continuous nondecreasing predictable process, is addressed in this paper.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 71, 13 pp.

Accepted: 6 October 2015
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60G57: Random measures

Backward stochastic differential equations random measures

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Bandini, Elena. Existence and uniqueness for backward stochastic differential equations driven by a random measure, possibly non quasi-left continuous. Electron. Commun. Probab. 20 (2015), paper no. 71, 13 pp. doi:10.1214/ECP.v20-4348. https://projecteuclid.org/euclid.ecp/1465320998

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