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Time discretization of BSDEs with singular terminal condition using asymptotic expansion

Thomas Kruse, Julia Ackermann, Alexandre Popier

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.

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.
Paper Structure (16 sections, 15 theorems, 132 equations)

This paper contains 16 sections, 15 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Breakdown of the paper.
  3. Notations.
  4. Time discretization of singular BSDEs using the implicit Euler scheme
  5. Error analysis for the implicit Euler scheme for \ref{['eq:sing_BSDE']} with bounded terminal condition
  6. Error analysis for the implicit Euler scheme for \ref{['eq:sing_BSDE']} with singular terminal condition
  7. Expansion of $Y$ near $T$
  8. Main result
  9. Proof of Theorem \ref{['thm:gene_case']}
  10. Examples
  11. Power case
  12. Exponential case
  13. Comparing the generator with the exponential case
  14. Conclusion and perspectives
  15. Proofs of some technical results
  16. ...and 1 more sections

Key Result

Lemma 1

Assume that Conditions A1 and A3 are satisfied. Then there exists a unique solution $(Y,Z)\in \mathbb S^2(0,T)$ of eq:reg_BSDE. The solution satisfies for all $t\in [0,T]$ a.s. that

Theorems & Definitions (20)

  • Example 1
  • Remark 1: Comments on \ref{['A1']} and \ref{['A2']}
  • Remark 2: Comments on \ref{['A3']} and \ref{['A4']}
  • Remark 3
  • Lemma 1
  • Proposition 1
  • Corollary 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • ...and 10 more