Asymptotic expansion for forward-backward SDEs with jumps

ABSTRACT

This work provides a semi-analytic approximation method for decoupled forward-backward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with σ-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications.

KEYWORDS:

• BSDE
• jumps
• random measure
• asymptotic expansion
• Lévy process

Acknowledgments

All the contents expressed in this research are solely those of the authors and do not represent any views or opinions of any institutions. The authors are not responsible or liable in any manner for any losses and/or damages caused by the use of any contents in this research.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 In [10], the authors successfully applied the asymptotic expansion method proposed in [22, 23] to a collateralized debt obligation with 120 underlying names to evaluate credit/funding valuation adjustments.

2 See Mania and Tevzadze [40], Pham [46] and Fujii [21] for concrete examples.

3 See, for example, Chapter XIII in [29]. If one assumes the predictable representation property, this construction is irrelevant.

4 The additional factor $+2$ (instead of $+1$) arises basically from the need to bound the approximation error for the control variables $(Z,\psi )$.

5 In p=2, one can see more directly $\parallel \tilde{X}-X^n{\parallel }_{{\mathbb{S}}^2}^2\to 0$ since the integral of $\delta L^n$ can be replaced by that of $\delta {\tilde{\gamma }}_n$ (See a remark below Lemma A.3.). Taking an appropriate subsequence if necessary, one can also show that $(X_s^n,s\in [t,T]{)}_{n\ge 1}$ is almost surely uniformly convergent to $({\tilde{X}}_s,s\in [t,T\left]\right)$ by the Borel-Cantelli lemma.

Additional information

Funding

The research is partially supported by Center for Advanced Research in Finance (CARF) and JSPS KAKENHI (Grant Number 25380389).