Total variation distance between two diffusions in small time with unbounded drift: application to the Euler-Maruyama scheme. (English) Zbl 1517.65006
Summary: We give bounds for the total variation distance between the solutions to two stochastic differential equations starting at the same point and with close coefficients, which applies in particular to the distance between an exact solution and its Euler-Maruyama scheme in small time. We show that for small \(t\), the total variation distance is of order \(t^{r/(2r+1)}\) if the noise coefficient \(\sigma\) of the SDE is elliptic and \(\mathcal{C}_b^{2r}\), \(r\in \mathbb{N}\) and if the drift is \(C^1\) with bounded derivatives, using multi-step Richardson-Romberg extrapolation. We do not require the drift to be bounded. Then we prove with a counterexample that we cannot achieve a bound better than \(t^{1/ 2}\) in general.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
Keywords:
Aronson’s bounds; Euler-Maruyama scheme; Richardson-Romberg extrapolation; stochastic differential equation; total variationReferences:
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