Backward Stochastic Differential Equations

Data-driven Feynman-Kac Discovery with Applications to Prediction and Data Generation ArXiv ID: 2511.08606 “View on arXiv” Authors: Qi Feng, Guang Lin, Purav Matlia, Denny Serdarevic Abstract In this paper, we propose a novel data-driven framework for discovering probabilistic laws underlying the Feynman-Kac formula. Specifically, we introduce the first stochastic SINDy method formulated under the risk-neutral probability measure to recover the backward stochastic differential equation (BSDE) from a single pair of stock and option trajectories. Unlike existing approaches to identifying stochastic differential equations-which typically require ergodicity-our framework leverages the risk-neutral measure, thereby eliminating the ergodicity assumption and enabling BSDE recovery from limited financial time series data. Using this algorithm, we are able not only to make forward-looking predictions but also to generate new synthetic data paths consistent with the underlying probabilistic law. ...

Constrained mean-variance investment-reinsurance under the Cramér-Lundberg model with random coefficients ArXiv ID: 2406.10465 “View on arXiv” Authors: Unknown Abstract In this paper, we study an optimal mean-variance investment-reinsurance problem for an insurer (she) under a Cramér-Lundberg model with random coefficients. At any time, the insurer can purchase reinsurance or acquire new business and invest her surplus in a security market consisting of a risk-free asset and multiple risky assets, subject to a general convex cone investment constraint. We reduce the problem to a constrained stochastic linear-quadratic control problem with jumps whose solution is related to a system of partially coupled stochastic Riccati equations (SREs). Then we devote ourselves to establishing the existence and uniqueness of solutions to the SREs by pure backward stochastic differential equation (BSDE) techniques. We achieve this with the help of approximation procedure, comparison theorems for BSDEs with jumps, log transformation and BMO martingales. The efficient investment-reinsurance strategy and efficient mean-variance frontier are explicitly given through the solutions of the SREs, which are shown to be a linear feedback form of the wealth process and a half-line, respectively. ...

Deep multi-step mixed algorithm for high dimensional non-linear PDEs and associated BSDEs ArXiv ID: 2308.14487 “View on arXiv” Authors: Unknown Abstract We propose a new multistep deep learning-based algorithm for the resolution of moderate to high dimensional nonlinear backward stochastic differential equations (BSDEs) and their corresponding parabolic partial differential equations (PDE). Our algorithm relies on the iterated time discretisation of the BSDE and approximates its solution and gradient using deep neural networks and automatic differentiation at each time step. The approximations are obtained by sequential minimisation of local quadratic loss functions at each time step through stochastic gradient descent. We provide an analysis of approximation error in the case of a network architecture with weight constraints requiring only low regularity conditions on the generator of the BSDE. The algorithm increases accuracy from its single step parent model and has reduced complexity when compared to similar models in the literature. ...