We consider a class of stochastic differential equations driven by a one-dimensional Brownian motion, and we investigate the rate of convergence for Wong–Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the pointwise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider Itô equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise toward the original Brownian motion. We also prove, in analogy with a well-known property for exact solutions, that the solutions of approximated Itô equations solve approximated Stratonovich equations with a certain correction term in the drift.

Rate of Convergence for Wong–Zakai-Type Approximations of Itô Stochastic Differential Equations

Lanconelli A.
Investigation
2019

Abstract

We consider a class of stochastic differential equations driven by a one-dimensional Brownian motion, and we investigate the rate of convergence for Wong–Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the pointwise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider Itô equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise toward the original Brownian motion. We also prove, in analogy with a well-known property for exact solutions, that the solutions of approximated Itô equations solve approximated Stratonovich equations with a certain correction term in the drift.
File in questo prodotto:
File Dimensione Formato  
BenAmmouBK-LanconelliA_JTP_2019_postprint.pdf

embargo fino al 11/06/2019

Tipo: Postprint
Licenza: Licenza per accesso libero gratuito
Dimensione 301.53 kB
Formato Adobe PDF
301.53 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11585/710977
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 2
social impact