On any probability space $(\Omega, \mathcal{F} , \mathbb{P})$ with a Brownian motion $W$, we consider the following one-dimensional SDE:

$$dX_t = F(t, X_t) \, dt + dW_t,$$

where $F(t,x) = \mathbb{E} b(x, Z_t)$, where $Z_t$ satisfies law $\mu_t$ for any $t \geq 0$, and $b: \mathbb{R}^2 \to \mathbb{R}$ is a smooth function.

It is natural to define the following approximation by Monte-Carlo for the drift:

$$dY_t = \tilde{F} (t, Y_t) \,dt + dW_t,$$

where $\tilde{F} (t,x) = \frac{1}{N} \sum_{i=1}^N b( x, Z^i_t)$ is a random function, and $Z^i$ are processes (not necessarily independent) such that for any $t \geq 0$, $Z^i_t$ has law $\mu_t$.

Is there any known result about the weak error $|\mathbb{E} \phi(X_t) - \mathbb{E} \phi(Y_t) |$ (for any reasonably smooth $\phi$)? There is not much result in the literature about the estimation of drift by Monte-Carlo...


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.