# Iterated stochastic processes : simulation and relationship with high order partial differential equations

## TL;DR

This paper introduces an Euler-based simulation algorithm for iterated stochastic processes, extends the Feynman-Kac formula to higher-order PDEs, and demonstrates their connection to complex stochastic models.

## Contribution

It presents a natural, convergent simulation method for iterated processes and generalizes the Feynman-Kac formula to relate these processes to high-order PDEs.

## Key findings

- The simulation algorithm converges almost surely with rate 1/4.
- Extended Feynman-Kac formula to higher-order PDEs.
- Simulated solutions to fourth order PDEs using iterated processes.

## 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 Feynman-Kac formula, we simulate functionals of the trajectories and solutions to fourth order PDEs that are naturally associated to a general class of iterated processes.