Langevin equations with multiplicative noise: uniqueness, self-consistency and new solution methods by a time-discrete approach

TL;DR

This paper introduces a time-discrete approach for Langevin equations with multiplicative noise, ensuring consistency with Fokker-Planck equations and providing accurate numerical path computations.

Contribution

It develops a novel time-discrete method that avoids the 'integration sense' assumption, yielding consistent solutions and explicit FPE solutions for multiplicative noise.

Findings

Path increments are complete in the order of the time step.

The approach yields the 'anti-Ito' integral in the continuous limit.

Explicit solutions for the FPE are obtained for short times.

Abstract

A time-discrete approach avoids the assumption of an 'integration sense'. New path increments (in a short time step) are complete in the order of that step, and not Gaussian distributed when the noise is multiplicative; this eliminates an existing mismatch with the Fokker-Planck equations. By the Markov property these increments can be accumulated in consecutive intervals, to yield the solution for any times. In one dimension, more generally also under a certain condition, it is shown that the limit of continuous time exists and results in the 'anti-Ito' intrgral for the paths; the time step can therefore be diminished arbitrarily. The numerical computation of the paths is particularly accurate, due to increments that agree with the FPE by the mode, in addition to the mean. Under the above condition the FPE takes a simple form and can explicitly be solved for short times; this allows the computation of the density function for any times, by use of the Markov property.