4.7 Article

Backward Stochastic Differential Equations Driven by a Jump Markov Process with Continuous and Non-Necessary Continuous Generators

Journal

FRACTAL AND FRACTIONAL
Volume 6, Issue 6, Pages -

Publisher

MDPI
DOI: 10.3390/fractalfract6060331

Keywords

backward stochastic differential equations; jump Markov process; comparison principle; quadratic BSDE with jumps

Funding

  1. Deanship of Scientific Research at King Saud University [RG-1441-339]

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This article deals with backward stochastic differential equations driven by a pure jump Markov process and an independent Brownian motion (BSDEJs). We first prove the existence and uniqueness of solutions for this type of equation and provide a comparison of solutions in the case of Lipschitz conditions in the generator. With these tools, we study the existence of a (minimal) solution for BSDE where the coefficient is continuous and satisfies the linear growth condition. An existence result for BSDE with a left-continuous, increasing, and bounded generator is also discussed. Finally, the general result is applied to solve one kind of quadratic BSDEJ.
We deal with backward stochastic differential equations driven by a pure jump Markov process and an independent Brownian motion (BSDEJs for short). We start by proving the existence and uniqueness of the solutions for this type of equation and present a comparison of the solutions in the case of Lipschitz conditions in the generator. With these tools in hand, we study the existence of a (minimal) solution for BSDE where the coefficient is continuous and satisfies the linear growth condition. An existence result for BSDE with a left-continuous, increasing and bounded generator is also discussed. Finally, the general result is applied to solve one kind of quadratic BSDEJ.

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