Fast sampling from time-integrated bridges using deep learning

Abstract

We propose a methodology for sampling from time-integrated stochastic bridges, i.e., random variables defined as ${\int }_{t_1}^{t_2}f\left(Y\left(t\right)\right)dt$ conditional on $Y\left(t_1\right)=a$ and $Y\left(t_2\right)=b$, with $a,b\in R$. The techniques developed in Grzelak et al. (2019) – the Stochastic Collocation Monte Carlo sampler – and in Liu et al. (2020) – the Seven-League scheme – are applied for this purpose. Notably, the time-integrated bridge distribution is approximated using a polynomial chaos expansion constructed over an appropriate set of stochastic collocation points. In addition, artificial neural networks are employed to learn the collocation points. The result is a robust, data-driven procedure for Monte Carlo sampling from time-integrated conditional processes, which guarantees high accuracy and generates thousands of samples in milliseconds. Applications are also presented, with a focus on finance.