[Paper Review] Discrete Itô Formulas and Their Applications to Stochastic Numerics

Jirô Akahori|ArXiv.org|Mar 14, 2006

Stochastic processes and financial applications3 references3 citations

TL;DR

This paper introduces discrete Itô formulas as a discrete-time analogue of Itô's lemma, using orthogonal expansions in conditional Fourier series to derive finite difference schemes for stochastic PDEs. It enables high-dimensional Monte Carlo simulations via quasi-Monte Carlo methods by constructing dimension-free numerical schemes based on orthogonal expansions of increments, with convergence rates tied to moment matching in the discretization.

ABSTRACT

This is a survey note of the author's observations on the discrete-time analogues of Itô formulas.

Motivation & Objective

• To develop a discrete-time analogue of Itô's lemma using orthogonal expansions instead of stochastic integrals.
• To establish a computational framework for simulating high-dimensional PDEs via finite difference schemes derived from stochastic difference equations.
• To analyze the convergence of Euler-Maruyama-type schemes using discrete Itô calculus, particularly in complete market models.
• To propose a dimension-free Monte Carlo simulation method for high-dimensional SDEs using Walsh system-based orthogonal expansions.
• To generalize the correspondence between Markov chain approximations and finite difference schemes to nonlinear and non-Gaussian settings.

Proposed method

• Derives discrete Itô formulas via orthogonal expansions in $L^2(\nu)$, using orthonormal bases $\{H_n\}$ for the increment distribution $\nu$.
• Applies the expansion to random walks and stochastic difference equations, decomposing increments into 0th, 1st, and higher-order chaos terms.
• Constructs finite difference schemes by discretizing the generator of the SDE using $L^N$ operators defined via $N \int f(\mathbf{x} + F(\mathbf{x}, \Delta t, \mathbf{H}(y))) \nu(dy)$.
• Employs the Walsh system as an orthonormal basis for $[0,1)$ to generate quasi-random increments, enabling dimension-free simulation.
• Uses Monte Carlo or quasi-Monte Carlo sampling over $[0,1)^N$ to simulate paths of the discrete process, ensuring stability and convergence.
• Establishes convergence of the scheme by verifying consistency of the discrete operator $\partial^N_t + L^N$ with the continuous generator $\partial_t + L$.

Experimental results

Research questions

• RQ1 How can the Itô formula be reformulated in discrete time without relying on stochastic integrals?
• RQ2 What is the structure of the discrete Itô formula when the increment distribution is non-Gaussian or non-atomic?
• RQ3 Can finite difference schemes for high-dimensional PDEs be constructed via discrete Itô calculus and Monte Carlo methods?
• RQ4 What is the rate of convergence of discrete schemes derived from discrete Itô formulas, especially in complete market models?
• RQ5 How can orthogonal expansions in $L^2(\nu)$ be used to design dimension-free numerical schemes for SDEs?

Key findings

• The discrete Itô formula decomposes the increment of a function of a discrete process into a conditional expectation (0th chaos), a linear term (1st chaos), and higher-order orthogonal corrections.
• For Euler-Maruyama schemes, the discrete Itô formula recovers the standard Itô formula in the limit as $\Delta t \to 0$, with convergence rate at least $\sqrt{\Delta t}$ in complete markets when $n \geq 2$.
• When the increment distribution $\nu$ is Lebesgue on $[0,1]$, and the Walsh system is used as an orthonormal basis, the resulting scheme is effectively dimension-free and supports simulations with up to ~3000 dimensions.
• The finite difference scheme derived from the discrete Itô formula converges to the solution of the corresponding PDE if the discrete operator $\partial^N_t + L^N$ is consistent with the continuous generator $\partial_t + L$.
• The method enables efficient Monte Carlo simulation of high-dimensional PDEs by reducing the problem to path simulation via uniform sampling in $[0,1)^N$ with orthogonal increments.
• The framework generalizes to weak approximation schemes and non-linear SDEs, extending the classical Kushner correspondence between Markov chains and finite difference schemes.