In this work, we apply the Stochastic Grid Bundling Method (SGBM) to numerically solve backward stochastic differential equations (BSDEs). The SGBM algorithm is based on conditional expectations approximation by means of bundling of Monte Carlo sample paths and a local regress-later regression within each bundle. The basic algorithm for solving the backward stochastic differential equations will be introduced and an upper error bound is established for the local regression. A full error analysis is also conducted for the explicit version of our algorithm and numerical experiments are performed to demonstrate various properties of our algorithm.

h i g h l i g h t s• An efficient approximant for the Gerber-Shiu functions is proposed.• The approximation is based on the Fourier-cosine method.• The approximation is of linear computational complexity. • An explicit error bound is provided for functions satisfying some mild technical conditions. a b s t r a c tIn this article, we provide a systematic study on effectively approximating the Gerber-Shiu functions, which is a hardly touched topic in the current literature, by incorporating the recently popular Fouriercosine method. Fourier-cosine method has been a prevailing numerical method in option pricing theory since the work of Fang and Oosterlee (2009). Our approximant of Gerber-Shiu functions under Lévy subordinator model has O(n) computational complexity in comparison with that of O(n log n) via the fast Fourier transform algorithm. Also, for Gerber-Shiu functions within our proposed refined Sobolev space, we introduce an explicit error bound, which seems to be absent from the literature. In contrast with our previous work (Chau et al., 2015), this error bound is more conservative without making heavy assumptions on the Fourier transform of the Gerber-Shiu function. The effectiveness of our result will be further demonstrated in the numerical studies.

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We propose a numerical algorithm for backward stochastic differential equations based on time discretization and trigonometric wavelets. This method combines the effectiveness of Fourier-based methods and the simplicity of a wavelet-based formula, resulting in an algorithm that is both accurate and easy to implement. Furthermore, we mitigate the problem of errors near the computation boundaries by means of an antireflective boundary technique, giving an improved approximation. We test our algorithm with different numerical experiments.In this article, we shall adopt the quick variant of the SWIFT formula proposed in [13].The reason behind this is left for the error section, but this gives an approximation for E x p [v(t p+1 , X ∆ t p+1 )], which we see in the discrete-time approximation of FBSDE,

In this work, we developed a Python demonstrator for pricing total valuation adjustment (XVA) based on the stochastic grid bundling method (SGBM). XVA is an advanced risk management concept which became relevant after the recent financial crisis. This work is a follow-up work on Chau and Oosterlee in (Int J Comput Math 96(11):2272–2301, 2019), in which we extended SGBM to numerically solving backward stochastic differential equations (BSDEs). The motivation for this work is basically two-fold. On the application side, by focusing on a particular financial application of BSDEs, we can show the potential of using SGBM on a real-world risk management problem. On the implementation side, we explore the potential of developing a simple yet highly efficient code with SGBM by incorporating CUDA Python into our program.