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Strong convergence of path sensitivities

Michael B. Giles

Abstract

It is well known that the Euler-Maruyama discretisation of an autonomous SDE using a uniform timestep $h$ has a strong convergence error which is $O(h^{1/2})$ when the drift and diffusion are both globally Lipschitz. This note proves that the same is true for the approximation of the path sensitivity to changes in a parameter affecting the drift and diffusion, assuming the appropriate number of derivatives exist and are bounded. This seems to fill a gap in the existing stochastic numerical analysis literature.

Strong convergence of path sensitivities

Abstract

It is well known that the Euler-Maruyama discretisation of an autonomous SDE using a uniform timestep has a strong convergence error which is when the drift and diffusion are both globally Lipschitz. This note proves that the same is true for the approximation of the path sensitivity to changes in a parameter affecting the drift and diffusion, assuming the appropriate number of derivatives exist and are bounded. This seems to fill a gap in the existing stochastic numerical analysis literature.

Paper Structure

This paper contains 7 sections, 6 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. SDE sensitivity analysis
  3. Strong convergence analysis
  4. Extensions
  5. Vector SDEs and vector parameters
  6. Higher order sensitivities
  7. Conclusions

Key Result

Lemma 1

For a given time interval $[0,T]$, and any $p\geq 2$, there exists a constant $c_p^{(1)}$ such that

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Lemma 4