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

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).As applications of our results, we first analyse the regularity of densities of solution to coupled FBSDEs. In the second example, we consider a regime switching term structure interest rate models (see for e.g., Ma et al. (2017)) for which the corresponding FBSDE has discontinuous drift. Our results enables us to: firstly study classical and Malliavin differentiability of the solutions for such models, secondly the existence of density of such solutions. Lastly we consider a pricing and hedging problem of contingent claims on non-tradable underlying, when the dynamic of the latter is given by a regime switching SDE (i.e., the drift coefficient is allowed to be discontinuous). We obtain a representation of the derivative hedge as the weak derivative of the indifference price function, thus extending the result in Ankirchner et al. (2010).