Forward–Backward SDEs in Algorithmic Trading: BSDEs, Deep BSDE Solvers, and Risk-Sensitive Control With Python (Computational Mathematics Library)

Harness modern stochastic control and deep learning to design, price, and manage algorithmic trading strategies in continuous time. This book takes you from first principles of stochastic price dynamics through fully coupled forward backward stochastic differential equations and on to state of the art deep BSDE solvers, always in the concrete setting of execution, market making, and portfolio problems. Every chapter is accompanied by complete Python code that reproduces all numerical examples, so you can move directly from theory to implementation. Starting from trading PnL modeled as stochastic differential equations, you will see how execution, inventory, and risk constraints naturally lead to BSDE and FBSDE formulations. You will learn how risk neutral, risk averse, and risk sensitive objectives are embedded in generators, how they connect to nonlinear expectations and entropic risk, and how they change optimal trading behavior in realistic market impact models. On the computational side, the book shows how to discretize and solve BSDEs with Monte Carlo and regression, then scales up to high dimensional settings with deep BSDE solvers implemented in Python. You will build full workflows that simulate forward dynamics, propagate backward value and costate processes, and train neural networks to approximate optimal controls in problems that are out of reach for classical PDE methods. You will learn how to: Model prices, inventory, and execution costs with controlled SDEs tailored to high frequency and institutional trading Formulate optimal execution, market making, and portfolio problems as BSDE or FBSDE systems Incorporate permanent, transient, and nonlinear market impact into continuous time trading models Encode risk neutral, mean variance, and risk sensitive objectives using quadratic and nonlinear BSDE generators Apply regression based Monte Carlo methods to solve discretized BSDEs in Python Use deep BSDE solvers for high dimensional control, including multi asset execution and multi venue trading Design risk sensitive strategies with dynamic risk constraints and robust formulations that account for model uncertainty Calibrate SDE and BSDE parameters to real market data and adapt models online in changing conditions Each chapter includes: A carefully chosen trading or risk management problem Step by step derivation of the associated BSDE or FBSDE Discussion of existence, uniqueness, and qualitative behavior of solutions A complete, well commented Python implementation with simulation and numerical results The result is a self contained path from mathematical foundations to production ready prototypes. Whether you are a quant researcher, a practitioner responsible for algorithmic execution and risk, or an advanced student in financial engineering, this book gives you a rigorous framework and executable code to develop and test sophisticated continuous time strategies driven by forward backward SDEs.