# Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ be some function. The task is to use Monte-Carlo to compute the integral $$\mathsf E[g(\xi)] = \int_\Bbb R g(x)f(x)\mathrm dx \approx \frac1N\sum_{i=1}^N g(\xi^i)$$ where $\xi^i$ are iid random variable distributed as $\xi$. Since the approximation error may have a big variance, the idea of the importance sampling is to change a density of $\xi$ to some $\hat f$ and use that $$\int_\Bbb R g(x)f(x)\mathrm dx = \int_\Bbb R g(x)w(x)\hat f(x)\mathrm dx$$ where the weighting function is given by the fraction $w = f/\hat f$. As a result, $$\mathsf E[g(\xi)] = \mathsf E[g(\hat \xi)w(\hat \xi)] \approx \frac1N\sum_{i=1}^N g(\hat\xi^i)w(\hat\xi^i)$$ where $\hat\xi^i$ are iid with densities $\hat f$. Then one runs the optimization problem to find the best choice of $\hat f$, that is the one which minimizes the variance.

In my case I have a similar problem. Let us consider a discrete-time stochastic process $X$ with a state space $E$, given by the stochastic difference equation of the form $$X_{k+1} = r(X_k,\eta_k), \quad X(0) = x\in E, \tag{1}$$ where $\eta_k$ is a sequence of iid real-valued random variables with some density $h$, and $r$ is a jointly measurable function. Let $\mathsf P$ be the induced probability measure on $E^{n+1}$ and let $A$ be measurable subset of $E^{n+1}$. I am interested in using importance sampling to evaluate $\mathsf P(A)$.

The method of importance sampling described above easily extends to the case of random elements with a range $\Bbb R^m$, when their density is precisely known. In my case $E$ is a subset of $\Bbb R^m$, and I know the function $h$, but it would be almost impossible to get an expression of the density of $X = (X_0,X_1,\dots,X_n)$ since the function $r$ may have an extremely complicated shape. Due to this reason, I restrict myself just to change of $\mathsf P$ which is induced by the change of the distribution of $\eta$: \begin{align} \mathsf P(A) &= \int_{E^{n+1}} 1_A(x_0,\dots,x_n)\mathsf P(\mathrm dx_0\times\dots\times\mathrm dx_n) \\ & = \int_{\Bbb R^n}1_A(R(y_0,\dots,y_n))h(y_0)\dots h(y_{n-1})\mathrm dy_0\times\dots\times \mathrm dy_{n-1} \\ & = \int_{\Bbb R^n}1_A(R(y_0,\dots,y_n))w(y_0,\dots,y_{n-1})\hat h(y_0)\dots \hat h(y_{n-1})\mathrm dy_0\times\dots\times \mathrm dy_{n-1} \\ & = \mathsf E[1_A(\hat X_0,\dots,\hat X_n)w(\hat \eta_0,\dots,\hat \eta_{n-1})] \end{align} where $R$ is a function which for every noise realization $(\eta_0,\dots,\eta_{n-1})$ gives the path of the process $(X_0,\dots,X_n)$ according to $(1)$, and further $\hat h$ is a new density, $\hat \eta_k$ are distributed according to $\hat h$, $$(\hat X_0,\dots,\hat X_n) \sim R(\hat \eta_0,\dots,\hat\eta_{n-1})$$ is distributed according to the new distribution of the noise.

My question is the following: in case I need to do importance sampling of $(1)$, I need to perform the change of measure from $\mathsf P$ to some $\hat{\mathsf P}$ in such a way, that I get an explicit shape of the Radon-Nikodym derivative $w = \mathrm d\mathsf P/\mathrm d\hat{\mathsf P}$.

Is the way I described the only one which guarantees the knowledge of explicit shape of $w$? I am pretty sure, there shall be some literature on the importance sampling of stochastic difference equations that deals with such problems, but I didn't find anything particularly appropriate yet.

Was asked on MSE but didn't get much attention.