On the wavelet-based SWIFT method for backward stochastic differential equations

Abstract

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,

Exploration of a Cosine Expansion Lattice Scheme

“…First, the cosine coefficient f cos may be difficult to calculate, especially in a recurring situation. In our previous work [3], we aimed to remedy this shortcoming by adopting a wavelet basis such that we can approximate Equation (1.1) as a weighted sum of local values, in the form of…”

“…We place our work in the context of the construction of wavelets in [3] to see the similarities and differences between the two approaches. Instead of using the full Fourier series for the wavelet construction, as in [3], we choose a "cosine function only" basis set as our starting point, corresponding to the half-period cosine space we used in Section 2.…”

“…Readers are referred to [12] for further properties of such functions and for a condensed proof of the above properties one may follow the derivation of Theorem 3.1 in [3].…”

“…Further, we restrict our lattice to the box [−4.5, 4.5] s and we study the cases of dimension s ∈ [3]. We applied the full scheme in Equation (2.8) here with K = 2 7 .…”

An SGBM-XVA demonstrator: a scalable Python tool for pricing XVA

“…In terms of numerical analysis, there is also significant research on the efficient calculation or approximation for BSDEs. This includes Monte Carlo-based research, like [4], chaos decomposition method [5], cubature methods [6] or Fourier and wavelet-based methods, like in [7] and [8]. However, there are relatively few studies on the practical application of BSDEs.…”