Time discretization of BSDEs with singular terminal condition using asymptotic expansion

Abstract

We consider a class of backward stochastic differential equations (BSDEs) with singular terminal condition and develop a numerical scheme to approximate their solution. To this end, we extend an asymptotic development of the BSDE solution known from the power case, which arises from optimal liquidation problems, to more general generators. This expansion allows to obtain a suitable approximation of the BSDE solution close to the terminal time. Using this as a terminal condition, we analyze the error of a backward Euler implicit scheme and detail its dependence on the terminal condition.

Summary

Develops an asymptotic expansion for BSDEs with singular (blow-up) terminal conditions near $t=T$. The expansion $Y_t=\varphi ((T-t)/{\eta }_t)-{\varphi }^{\prime }((T-t)/{\eta }_t)H_t$ decomposes the solution into a leading singular profile plus a bounded correction term $H$. This is an expansion in the temporal singularity variable $(T-t)$, NOT a perturbative $\epsilon$-expansion. Previously known only for the power case (optimal liquidation); extended here to general generators satisfying concavity and monotonicity conditions.

Key Findings

• Convergence proved via contraction mapping on a weighted Banach space (Theorem 2)
• Error bound for backward Euler discretization: $|Y_{t_i}-{\bar{Y}}_i|\le C[{\Delta }^{\alpha }+h^{\beta }+{\Psi }_1(\Delta )h^{\beta }+{\Psi }_2(\Delta )h]$ with explicit trade-off between approximation distance $\Delta$ from $T$ and step size $h$

Critical Notes

Limited relevance to XVA HJB paper

This paper addresses BSDEs with blow-up terminal conditions (e.g., from optimal liquidation), not the perturbative $u^{\epsilon }=U^{(0)}+\epsilon U^{(1)}$ structure of the XVA paper. The asymptotic methodology is fundamentally different: expansion in $(T-t)$ near a singularity, not expansion in a small parameter $\epsilon$.

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