Density analysis for coupled forward–backward SDEs with non-Lipschitz drifts and applications

Rhoss Likibi Pellat and Olivier Menoukeu Pamen

Stochastic Processes and their Applications, 2024, vol. 173, issue C

Abstract: We explore the existence of a continuous marginal law with respect to the Lebesgue measure for each component (X,Y,Z) of the solution to coupled quadratic forward–backward stochastic differential equations (QFBSDEs) for which the drift coefficient of the forward component is either bounded and measurable or Hölder continuous. Our approach relies on a combination of the existence of a weak decoupling field (see Delarue and Guatteri, 2006), the integration with respect to space time local time (see Eisenbaum, 2006), the analysis of the backward Kolmogorov equation associated to the forward component along with an Itô-Tanaka trick (see Flandoli et al., 2009). The framework of this paper is beyond all existing papers on density analysis for Markovian BSDEs and constitutes a major refinement of the existing results. We also derive a comonotonicity theorem for the control variable in this frame and thus extending the works (Chen et al., 2005; Dos Rei and Dos Rei 2013).

Keywords: Forward–backward SDEs; Quasi-linear PDE; Non-smooth drifts; Regularity of density; Quadratic drivers; Integration with local time; Malliavin calculus (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spa.2024.104359

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