## Abstract

In Avikainen (2009, On irregular functionals of SDEs and the Euler scheme. Finance Stoch., 13, 381-401) the author showed that, for any p, q ∈ [1,∞), and any function f of bounded variation in R, it holds that E[|f (X) - f (X)|q] ≤ C(p, q)E[|X -X|p]1/p+1, where X is a one-dimensional random variable with a bounded density, and X is an arbitrary random variable. In this article we will provide multidimensional versions of this estimate for functions of bounded variation in Rd, Orlicz-Sobolev spaces, Sobolev spaces with variable exponents and fractional Sobolev spaces. The main idea of our arguments is to use the Hardy-Littlewood maximal estimates and pointwise characterizations of these function spaces. We apply our main results to analyze the numerical approximation for some irregular functionals of the solution of stochastic differential equations.

Original language | English |
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Pages (from-to) | 840-873 |

Number of pages | 34 |

Journal | IMA Journal of Numerical Analysis |

Volume | 42 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2022 |

## Keywords

- Avikainen's estimates
- Euler-Maruyama scheme
- fractional Sobolev spaces
- functions of bounded variation in R
- Hardy-Littlewood maximal estimates
- multilevel Monte Carlo method
- Orlicz-Sobolev spaces
- Sobolev spaces with variable exponents
- stochastic differential equations

## ASJC Scopus subject areas

- Mathematics(all)
- Computational Mathematics
- Applied Mathematics

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