A discretized paths-based sequential integration method involving the self-similarity of the fractional Brownian motion

Abstract

The discretized paths-based sequential integration method (SIM) is a quite versatile approach for solving various problems, including barrier problems, first passage problems, reflecting barrier problems and so on. This method builds upon the Chapman–Kolmogorov equation and is not applicable to non-Markovian problems, as in the case of fractional Brownian motion (FBM). In this paper, it is shown that the loss of the Markovian property can be overcome by utilizing the self-similarity of the FBM. In order to apply the discretized paths-based SIM, we have to solve a specific stochastic boundary value problem, also called stochastic “bridge” problem, which involves selecting only the trajectories of the FBM that ends at an assigned value, say $\bar{x}$ at $t_k$, at the beginning of the time interval $t_k-t_{k+1}$. It is shown that, due to self-similarity, the stochastic “bridge” problem may be solved only once, regardless of the value $\bar{x}$ at $t_k$. It is also shown that the trajectories of the stochastic “bridge” problem exhibit self-similarity, which circumvents the loss of Markovian property in FBM, thus allowing the discretized paths-based SIM to be employed without invoking the classical Chapman–Kolmogorov equation. Further, an application involving the classical first passage problem is presented.