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I am trying to compute the transition probabilities of the model given by $X_{n+1} = f_{n+1}(X_n,W_{n+1})$ where $X_n's, W_n's$ are $R^k$ valued random variables for $n \geq 0$, $W_n's$ are independent and $f_n's$ are measurable. Also, define $\mathcal{F}_n=\sigma(X_0,W_1,\cdots,W_n)$. (From this definition we get that $X_n$ is $\mathcal{F}_n$ measurable).

From these notes, the transition probabilites are defined as

$$p_{n+1}(X_n,B) := P[X_{n+1}\in B|\mathcal{F}_n]$$

Is it true that $$ P[X_{n+1}\in B|\mathcal{F}_n] (\omega) = P[f(X_n(w),W_{n+1})\in B]~a.s. $$

If so, what is methodology for the proof and if not, what are the transition probabilites for this model?

Thanks for the help.

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1 Answer 1

up vote 1 down vote accepted

Consider $g_{n+1}(x)=\mathrm P(f_{n+1}(x,W_{n+1})\in B)$, then $\mathrm P(X_{n+1}\in B\mid\mathcal F_n)=g_{n+1}(X_n)$ almost surely.

Notes: (1.) One should add the hypothesis that $(W_n)_{n\geqslant1}$ is independent of $X_0$. (2.) I simply do not understand the RHS of your formula (random variable? real number?). (3.) A congenial reference on these matters is the small blue book by David Williams called $\color{blue}{\textit{Probability with martingales}}$. Get it and read it!

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the equation I wrote was $P[X_{n+1}\in B | \mathcal{F}_n](\omega) = P[\{w_1 | f(X_n(\omega),W_{n+1}(w_1))\in B\}]$. $P(X_{n+1}\in B | \mathcal{F}_n) := E[I_{X_{n+1} \in B} | \mathcal{F}_n]$. However, I was unable to show that the last term is indeed equal to $g_{n+1}(X_n)$. I do have a copy of that nice book by Williams but I was unable to solve this problem. –  jpv Jul 18 '12 at 2:50
    
Thanks! I could find the required topic in that book (beginning of Section 9.10) and understand the proof. –  jpv Jul 18 '12 at 5:30

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