On Non-Lipschitz Path-Dependent SDEs with Weighted Running Extrema: Existence, Uniqueness, and Strong Convergence Rates
Abstract
Abstract Stochastic differential equations (SDEs) depending on historical running maxima and minima appear naturally in various applied fields, including quantitative finance (e.g., lookback and barrier options), structural reliability engineering (fatigue under peak loads), and dam control theory. In this paper, we investigate a general class of path-dependent SDEs driven by running extrema, where the drift and diffusion coefficients satisfy a non-Lipschitz condition of Bihari-Osgood type. Under mild structural assumptions on the weight functions and pathwise extrema, we establish the existence and uniqueness of strong solutions using a generalized Bihari inequality combined with Picard iterations. Furthermore, we construct a discrete Euler-Maruyama approximation scheme tailored to running extrema and prove its strong convergence rate in $L^2(\Omega)$. The theoretical results provide both a rigorous analytical foundation for non-Lipschitz extremal SDEs and a reliable computational tool for practical simulations in applied probability. 2020 MSC: 60H10, 60G17, 65C30, 91G30.
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