Iterated stochastic processes : simulation and relationship with high order partial differential equations
Abstract
In this paper, we consider the composition of two independent processes : one process corresponds to position and the other one to time. Such processes will be called iterated processes. We first propose an algorithm based on the Euler scheme to simulate the trajectories of the corresponding iterated processes on a fixed time interval. This algorithm is natural and can be implemented easily. We show that it converges almost surely, uniformly in time, with a rate of convergence of order 1/4 and propose an estimation of the error. We then extend the well known Feynman Kac formula which gives a probabilistic representation of partial differential equations (PDEs), to its higher order version using iterated processes. In particular we consider general position processes which are not necessarily Markovian or are indexed by the real line but real valued. We also weaken some assumptions from previous works. We show that intertwining diffusions are related to transformations of high order PDEs. Combining our numerical scheme with the FeynmanKac formula, we simulate functionals of the trajectories and solutions to fourth order PDEs that are naturally associated to a general class of iterated processes.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 DOI:
 10.48550/arXiv.1704.00173
 arXiv:
 arXiv:1704.00173
 Bibcode:
 2017arXiv170400173T
 Keywords:

 Mathematics  Probability
 EPrint:
 Methodology and Computing in Applied Probability, pages 129, 2015