Approximation of conditional expectation of unknown function

Question

I am given a multidimensional markovian stochastic process $X_1,X_2,...X_n$ with continuous state space and unknown to me function $V$. I want to approximate expectation $E(V(X_k)|X_{k-1} = x)$ which is a function of $x$.

Suppose that I can simulate the entire sequence $X_1,X_2,...X_n$ using Monte Carlo. And for each $X_{k-1}^i$ from the black box I am given the value of $V(X_k^i)$ where $i$ denotes simulation index.

I understand that my question is too general, I hope that you will give me some references for this topic.

Answer 1

This type of problem can be approached by "curve fitting" or "least squares Monte Carlo". Both are basically just two names for the same thing: least squares regression. The approach is straightforward and just requires a basic understanding of regression. So no SDEs, functional analysis or measure theoretic conditional expectation is required.

The setting

To recognize how and why regression can be applied set $x_i = X_{k-1}^i$ and $y_i=V(X_k^i)$. With this terminology your simulation gives you iid pairs $(x_i,y_i)$. You know that $E[V|X_{k-1}=x_i]$ is a function $f(x_i)$ but you cannot observe this function you only observe a (crude) estimate $y_i$. Call $e_i = y_i -f(x_i)$ the error of the estimate. Notice that

1. $y_i = f(x_i) + e_i$ by definition.
2. The $e_i$ are independent, since you sample each path independent.
3. $E[e_i|X_{i-1}] = E[y_i - f(x_i)|X_{i-1}] = E[y_i | X_{i-1}] - f(x_i) = 0.$

Now make the assumption that the error $e_i$ has a normal distribution and you are in the regression setting.

Will it work? It depends ...

How successful this will be depends. You need to "guess" the structure of $f$. A standard (initial) guess is assuming $f(x)=\sum \beta_j p_j(x)$ where $p_j(x)$ are polynomials. You choose the $\beta_j$ such that they minimise the quadratic error with your observations, i.e. $\hat\beta = \text{argmin} \sum_i(y_i - \sum \beta_j p_j(x_i))^2$.

Depending on the structure of $f$ other low dimensional families might be better. For example if you know that $f$ is piecewise linear, you would choose piecewise linear functions for the approximation. If $f$ is periodic, use periodic ones and so on.

Another important issue is how fine you can sample $X_{k-1}$. If $f$ is very irregular (i.e. wiggles a lot) and you have only few samples, you are in trouble. Then the variance of $e_i$ will be large and samples with adjacent $x_i$ will be far apart and provide only very little information. If $f$ is more or less linear you only need few samples and linear maybe quadratic polynomials.

There is a rich literature and many tools available to do regression and to analyse the results. Specifically for the pricing of American options have a look at this review

Finally you should try to incorporate as much prior information as possible into your approximation. I mentioned already the structure of $f$ and explained the problem for a single location $k$. But if the $V$ functions at different $k$ have a known relation, there are ways to incorporate this as well. But to have a reasonable discussion about this, you need to provide a more detailed specification of your problem.

Answer 2

Few complements to g g's proposition:

So your function $V$ is unknown, but despite this lack of information, you are able to simulate $V(X_{k}^{i})$ for $X_{k-1}=X_{k-1}^{i}$. In this case, for a given value of $X_{k-1}^{i}$ why not generating $S$ values $\left\{ V(X_{k}^{s})\right\} _{s=1}^{S}$ and then applying the law of large numbers to find $$ S^{-1}\sum_{s=1}^{S}V(X_{k}^{s})\overset{p}{\rightarrow }E\left[ V(X_{k})|X_{k-1}=X_{k-1}^{i}\right] . $$ Then iterating over $i$ and $k$, this should identify the whole function $E(V(X_k)|X_{k-1} = x).$ There is a literature on simulated least squares that could be helpful.