Total variation distance between a jump-equation and its Gaussian\n approximation

2022/12/14 by Bally, Vlad, Yifeng Qin, Qin, Yifeng +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Probability (math.PR) #Probability and Statistical Research #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2212.07417

openalex created_date 2022/09/30 · openalex publication_date 2022/12/14 · openalex updated_date 2026/07/28

Abstract

We deal with stochastic differential equations with jumps. In order to obtain\nan accurate approximation scheme, it is usual to replace the "small jumps" by a\nBrownian motion. In this paper, we prove that for every fixed time t, the\napproximate random variable X^\ε_t converges to the original random\nvariable X_t in total variation distance and we estimate the error. We also\ngive an estimate of the distance between the densities of the laws of the two\nrandom variables. These are done by using some integration by parts techniques\nin Malliavin calculus.\n

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