Time-dependent weak rate of convergence for functions of generalized bounded variation
| Autor: | Antti Luoto |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Approximation using simple random walk weak rate of convergence 01 natural sciences Stochastic solution 41A25 65M15 (Primary) 35K05 60G50 (Secondary) 010104 statistics & probability Exponential growth FOS: Mathematics 0101 mathematics Brownian motion stokastiset prosessit Mathematics osittaisdifferentiaaliyhtälöt Applied Mathematics Probability (math.PR) 010102 general mathematics Mathematical analysis finite difference approximation of the heat equation Function (mathematics) Rate of convergence Bounded function Bounded variation numeerinen analyysi approksimointi Statistics Probability and Uncertainty Mathematics - Probability |
| Zdroj: | Stochastic Analysis and Applications. 39:494-524 |
| ISSN: | 1532-9356 0736-2994 |
| Popis: | Let $W$ denote the Brownian motion. For any exponentially bounded Borel function $g$ the function $u$ defined by $u(t,x)= \mathbb{E}[g(x{+}\sigma W_{T-t})]$ is the stochastic solution of the backward heat equation with terminal condition $g$. Let $u^n(t,x)$ denote the corresponding approximation generated by a simple symmetric random walk with time steps $2T/n$ and space steps $\pm \sigma \sqrt{T/n}$ where $\sigma > 0$. For quite irregular terminal conditions $g$ (bounded variation on compact intervals, locally H\"older continuous) the rate of convergence of $u^n(t,x)$ to $u(t,x)$ is considered, and also the behavior of the error $u^n(t,x)-u(t,x)$ as $t$ tends to $T$ Comment: 41 pages. Addition of Remark 2.4 and Proposition 3.1(i) (which improves Theorem 2.6(A)(i)), minor improvements in the presentation |
| Databáze: | OpenAIRE |
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