Existence and Uniqueness of Solutions to SDEs with Jumps and Irregular Drifts
We focus on solving stochastic differential equations driven by jump processes (SDEJs) with measurable drifts that may exhibit quadratic growth. Our approach leverages
a space transformation and Itô-Krylov’s formula to effectively eliminate the singular component of the drift, allowing us to obtain a transformed SDEJ that satisfies classical solvability conditions. By applying the inverse transformation proven to be a one-to-one mapping, we retrieve the solution to the original equation. This methodology offers several key advantages. First, it extends the well-known result of Le Gall (1984) from Brownian-driven SDEs to the jump process setting, broadening the range of applicable stochastic models. Second, it provides a robust framework for handling singular drifts, enabling the resolution of equations that would otherwise be intractable. Third, the approach accommodates drifts with quadratic growth, making it particularly relevant for financial modeling, insurance risk assessment, and other applications where such growth behavior is common. Finally, the inclusion of multiple examples illustrates the practical effectiveness of our method, demonstrating its flexibility and applicability to real-world problems.
We focus on solving stochastic differential equations driven by jump processes (SDEJs) with measurable drifts that may exhibit quadratic growth. Our approach leverages
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