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Referencing a question I asked previously, but asking more about methodology.

Suppose we have two Ito processes

$$dX_t = b_1(X_t)dt + \sqrt{2} dW_t \,,$$

$$dY_t = b_2(Y_t) dt + \sqrt{2} dW_t \,.$$

If we have a good reason to believe that the two processes are close or converge to the same stationary distribution, for example if $b_1$ and $b_2$ are close, then how can we show that $X_t$ and $Y_t$ are close? For example, by bounding $\mathbb{E}\|X_t - Y_t\|^2_2$? How is Ito's formula used in doing this? Say we define $e(t) = X_t - Y_t$ and examine the convergence of $e(t)$?

Thank you.

Edit: per the comment, let's say we want to bound $W^2_2(\text{law}(X_t), \text{law}(Y_t))$.

blue_egg
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  • You should use Wasserstein, not $L^2$, since talking about the invariant distributions being "nearby" is a question about convergence in distribution as $t$ becomes large. – Theo Diamantakis Dec 07 '23 at 19:54
  • @TheoDiamantakis so we'd want to bound something like $W^2_2(\text{law}(X_t), \text{law}(Y_t))$? Sure, I'm fine with that – blue_egg Dec 07 '23 at 22:21

1 Answers1

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Under the assumption that $b_1$ is Lipschitz, we have \begin{align} \mathbb{E}[|X_t-Y_t|^2]&\lesssim \mathbb{E}[\int_0^t |b_1(X_s)-b_1(Y_s)|^2\,d s]+ \mathbb{E}[\int_0^t |b_1(Y_s)-b_2(Y_s)|^2\,d s]\\ &\lesssim \int_0^t \mathbb{E}[|X_s-Y_s|^2]\,ds + \int_0^t \mathbb{E}[|b_1(Y_s)-b_2(Y_s)|^2]\,ds. \end{align} If we additionally assume that $\sup_{x\in\mathbb{R}}|b_1(x)-b_2(x)|<\varepsilon$, Gronwall's inequality implies $$ \mathbb{E}[|X_t-Y_t|^2]\lesssim \varepsilon,\quad t\in[0,T]. $$

I hope my answer can help you even though I did not use Ito's formula.

El_mago
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  • Thanks for the answer, but I'm looking for a reference/explanation of the use of Ito's rule to prove convergence. – blue_egg Dec 11 '23 at 16:10