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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...

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