Discritization Error Diffusion Process

Question

Let $X_t$ be an $\mathbb{R}^d$-valued diffusion process with initial condition $x\in\mathbb{R}^d$ and $X_t^{\Delta}$ be it discretization along an evenly spaced grid on $[0,1]$ with spacing $1>\Delta>0$. Is there a large deviations principle for $$ \mathbb{P}\left( \left\|X_t - X_t^{\Delta} \right\|> \delta \right) \leq \exp(I(\delta,\Delta)) + \omicron(?) , $$ where $I$ is a rate function depending on $\delta$ and on $\Delta$.

Answer 1

Probably not, and if it does, it is unlikely to be meaningful, because it will depend on your discretization.

The issue is that both $X_t$ and $X_t^\Delta$ are random variables, whereas a Large Deviations Principle is concerned with deviations from a deterministic quantity that is identified from a Law of Large Numbers. The typical large deviations result for a process discretized as $$X_{t+\Delta}^\Delta = X_t^\Delta + \Delta \xi_t(X^\Delta_t)$$ for some i.i.d. state-dependent random variables $\xi_t$ (e.g. $\xi_t(x) = x + Z_t$ for $Z_t$ i.i.d. $N(0, 1)$) is $$P(\|X^\Delta - \phi\|_\infty \geq \delta) \leq \exp(-I(\phi(\cdot))/\delta),$$ where $I(\phi(\cdot)) = \int_0^t L(\phi(s),\dot\phi(s)) ds$ and $L$ depends on the distribution of $\xi_t(\cdot)$. Here $\phi$ is a deterministic absolutely continuous function.

In your question $X_t$ is a diffusion process at some fixed $t \geq 0$. Say for simplicity $t=1$ and that $X_t$ is a standard Brownian motion. One can discretize $X_1$ as $$X_1^\Delta = \sum_{i=1}^{\lfloor 1/\Delta\rfloor} \sqrt{\Delta}\xi_i,\quad \xi_i \stackrel{\text{i.i.d.}}{\sim} N(0, 1).$$ Then $X_1^\Delta \Rightarrow X_1 \sim N(0, 1)$ in distribution, but you cannot upgrade the result to almost sure convergence. You can certainly rig the discretization so that the convergence is almost sure, but then you would probably just have $X_1^\Delta = X_1$.

Your question may have a better answer if you specify the discretization and/or a LLN.