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Numerical Approximation of Stochastic Volterra Integral Equation Using Walsh Function

Prit Pritam Paikaray, Sanghamitra Beuria, Nigam Chandra Parida

TL;DR

The paper addresses solving linear stochastic Volterra integral equations of the form $x(t)=f(t)+\int_{0}^{t}k_1(s,t)x(s) ds+\int_{0}^{t}k_2(s,t)x(s)dB(s)$, where $B(t)$ is Brownian motion. It proposes a Walsh-function-based discretization using the operational matrices of integration $\wedge$ and $\wedge_S$ and a relationship to block pulse functions to convert the SVIE into a finite-dimensional algebraic system. The authors provide convergence and error analyses under Lipschitz conditions, showing an $O(h^2)$ mean-square error bound for the approximate solution. Numerical experiments with several SVIEs—both analytically solvable and unsolved in closed form—demonstrate improved accuracy over existing methods and validate the practical utility of the Walsh-based approach.

Abstract

This paper provides a numerical approach for solving the linear stochastic Volterra integral equation using Walsh function approximation and the corresponding operational matrix of integration. A convergence analysis and error analysis of the proposed method for stochastic Volterra integral equations with Lipschitz functions are presented. Numerous examples with available analytical solutions demonstrate that the proposed method solves linear stochastic Volterra integral equations more precisely than existing techniques. In addition, the numerical behaviour of the method for a problem with no known analytical solution is demonstrated.

Numerical Approximation of Stochastic Volterra Integral Equation Using Walsh Function

TL;DR

The paper addresses solving linear stochastic Volterra integral equations of the form , where is Brownian motion. It proposes a Walsh-function-based discretization using the operational matrices of integration and and a relationship to block pulse functions to convert the SVIE into a finite-dimensional algebraic system. The authors provide convergence and error analyses under Lipschitz conditions, showing an mean-square error bound for the approximate solution. Numerical experiments with several SVIEs—both analytically solvable and unsolved in closed form—demonstrate improved accuracy over existing methods and validate the practical utility of the Walsh-based approach.

Abstract

This paper provides a numerical approach for solving the linear stochastic Volterra integral equation using Walsh function approximation and the corresponding operational matrix of integration. A convergence analysis and error analysis of the proposed method for stochastic Volterra integral equations with Lipschitz functions are presented. Numerous examples with available analytical solutions demonstrate that the proposed method solves linear stochastic Volterra integral equations more precisely than existing techniques. In addition, the numerical behaviour of the method for a problem with no known analytical solution is demonstrated.
Paper Structure (7 sections, 6 theorems, 65 equations, 5 figures, 4 tables)

This paper contains 7 sections, 6 theorems, 65 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Walsh Function and its Properties
  3. Relationship between Walsh Function and Block Pulse Functions (BPFs)
  4. Numerical Solution of Stochastic Volterra Integral Equation
  5. Error Analysis
  6. Numerical Examples
  7. Conclusion

Key Result

Theorem 3.1

Let the $m$-set of Walsh function and BPF vectors are $W(t)$ and $\Phi(t)$ respectively. Then the BPF vectors $\Phi(t)$ can be used to approximate $W(t)$ as $W(t)=T_W\Phi(t)$, $m=2^k$, and $k=0,1,\hdots$, where $T_W=[c_{ij}]_{m\times m}$, $c_{ij}=w_i(\eta_j)$, for some $\eta_j=(\frac{j}{m},\frac{j+1

Figures (5)

  • Figure 1: Example \ref{['Ex1']}'s approximate and exact solutions for m=32 and m=64
  • Figure 2: Example \ref{['Ex1']}'s error trend for m=32,n=30, and n=100
  • Figure 3: Example \ref{['Ex3']}'s approximate solution for m=32, m=64 and m=128 with 50 iterations
  • Figure 4: Stock model's approximate and exact solutions for m=32 and m=128 of Example \ref{['Stock']}
  • Figure 5: Example \ref{['Stock']}'s error trend for m=32, m=128 and n=20.

Theorems & Definitions (18)

  • Definition 1: Rademacher Function
  • Definition 2: Walsh Function
  • Definition 3: Block Pulse Functions
  • Theorem 3.1
  • proof
  • Lemma 3.2: Integration of Walsh function
  • proof
  • Lemma 3.3: Stochastic integration of Walsh function
  • proof
  • Theorem 5.1
  • ...and 8 more