A unified approach to gradient type formulas for BSDEs and some applications
Xiliang Fan, Michael R\"ockner, Shao-Qin Zhang
TL;DR
This paper introduces a unified method for deriving gradient and Bismut formulas for BSDEs, enabling new representations and estimates useful in control problems and PDE analysis.
Contribution
It develops a novel unified approach combining derivative formulas and solution expressions for BSDEs, with applications to McKean-Vlasov BSDEs and PDE gradient estimates.
Findings
Unified gradient formulas for BSDEs
Representation formulas for control solutions
Gradient estimates for PDEs
Abstract
In this paper we present a unified approach to establish gradient type formulas and Bismut type formulas for backward stochastic differential equations (BSDEs). This approach relies on a mix of derivative formulas with respect to the conditional probability of forward SDEs and the expression of the solution of BSDEs. Some concrete examples are given to illustrate the results. As applications, we provide representation formulas for the control solutions to McKean-Vlasov BSDEs and derive gradient estimates for related PDEs.
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Taxonomy
TopicsStochastic processes and financial applications · Mathematical Biology Tumor Growth · Financial Risk and Volatility Modeling