[Paper Review] Solving Backward Stochastic Differential Equations with quadratic-growth drivers by Connecting the Short-term Expansions

Masaaki Fujii, Akihiko Takahashi|arXiv (Cornell University)|Jun 14, 2016

Stochastic processes and financial applications3 citations

TL;DR

This paper proposes a novel numerical scheme for solving Markovian backward stochastic differential equations (BSDEs) with quadratic-growth drivers by decomposing the time horizon into short intervals and applying semi-analytic asymptotic expansions on each. The method achieves $\mathcal{O}(h^{3p/2})$ error bounds in $L^p$-norm for both the value and control processes, enabling high-accuracy solutions without Monte Carlo simulation or regression-based conditional expectation estimation.

ABSTRACT

This article proposes a new approximation scheme for quadratic-growth BSDEs in a Markovian setting by connecting a series of semi-analytic asymptotic expansions applied to short-time intervals. Although there remains a condition which needs to be checked a posteriori, one can avoid altogether time-consuming Monte Carlo simulation and other numerical integrations for estimating conditional expectations at each space-time node. Numerical examples of quadratic-growth as well as Lipschitz BSDEs suggest that the scheme works well even for large quadratic coefficients, and a fortiori for large Lipschitz constants.

Motivation & Objective

• To develop a computationally efficient numerical scheme for Markovian BSDEs with quadratic-growth drivers, which are prevalent in financial risk management and utility optimization.
• To overcome the limitations of standard Monte Carlo and regression-based methods by avoiding time-consuming conditional expectation estimates at each time node.
• To extend semi-analytic asymptotic methods—previously effective only for small non-linearities—to handle large quadratic coefficients through time decomposition.
• To provide a unified framework applicable to both quadratic-growth and standard Lipschitz BSDEs, enhancing generality and computational scalability.

Proposed method

• Decompose the terminal time interval into $n$ short subintervals $I_i = [t_{i-1}, t_i]$ to localize the non-linear dynamics.
• Apply a two-order asymptotic expansion to each subinterval: a zero-th order solution $\widehat{Y}^{i,[0]}, \widehat{Z}^{i,[0]}$ based on the linearized driver, and a first-order correction $\widehat{Y}^{i,[1]}$ for the residual error.
• Use the solution from the previous interval as the initial condition for the next, ensuring temporal consistency through recursive connection.
• Establish error propagation bounds via a perturbed terminal condition framework, analyzing the difference between true and approximate solutions in $L^p$-norm.
• Derive a total error estimate by substituting the asymptotic expansion error into the perturbation analysis, leading to an $\mathcal{O}(h^{3p/2})$ convergence rate.
• Leverage known regularity results for Lipschitz BSDEs and SDEs to control the growth of increments in state and control processes over short intervals.

Experimental results

Research questions

• RQ1 Can asymptotic expansion techniques be extended to handle quadratic-growth drivers in BSDEs, where standard perturbation methods break down?
• RQ2 What is the error propagation behavior when connecting short-time asymptotic approximations across multiple intervals in a Markovian BSDE?
• RQ3 Can the proposed scheme achieve high accuracy without relying on Monte Carlo simulations or regression-based conditional expectation estimation?
• RQ4 How does the scheme perform for large quadratic coefficients or high volatility regimes, where non-linear effects dominate?
• RQ5 Is the method applicable to both quadratic-growth and standard Lipschitz BSDEs, enabling a unified computational framework?

Key findings

• The proposed scheme achieves an $\mathcal{O}(h^{3p/2})$ error bound in $L^p$-norm for both the solution $Y$ and control process $Z$ over each short interval, with $p \geq 2$.
• The error estimate is derived by analyzing the difference between the true solution and the asymptotic approximation, with the residual error treated as a perturbation in a Lipschitz BSDE.
• The method avoids Monte Carlo simulation and regression-based conditional expectation estimation, significantly reducing computational cost compared to standard schemes.
• Numerical examples demonstrate robust performance even for large quadratic coefficients and high volatility, indicating the scheme's stability beyond the perturbative regime.
• The scheme is applicable to both quadratic-growth and standard Lipschitz BSDEs, suggesting potential for a unified numerical framework.
• Theoretical justification relies on the boundedness of the zero-th order solution and the regularity of the underlying SDE, ensuring the error remains controllable over short intervals.