I am trying to read Oksendal's book "Stochastic Differential Equations". In chapter 7, he consider processes like $$dX_t=b(X_t)dt+\sigma(X_t)dB_t$$ with the usual conditions ensuring unicity. Denote by $X^x_t$ the unique solution of the above equation such that $X_0=x$. Let $Q^x$ denote the probability law of $X^x_t$ when $X_0=x$ and $E^x(\cdot)$ be the expectation with respect to $Q^x$. Now this is claimed
[...]Hence we have $$E^x[f_1(X_{t_1})\ldots f_k(X_{t_k})]=E[f_1(X^x_{t_1})\ldots f_k(X^x_{t_k})]$$ for all Borel functions $f_1,\ldots,f_k$ and all times $t_1,\ldots,t_k$ where $E=E_P$ denotes the expectation with respect to the probability law $P=P^0$ for $\{B_t\}_{t\ge0}$ when $B_0=0$.
First I don't understand what are the processes on the left hand side : Is $X_{t_1}=X^0_{t_1}$? Or $X_{t_1}=X^x_{t_1}$? Or something else? This is not clear to me. Second, how do you show the identity properly?
Following Øksendal's notation, the LHS means: $$ \text{LHS}= \int_{\left(\mathbb{R}^n\right)^{[0,+\infty)}}\prod_{i=1}^k f_i\left(Y_{t_i}\right) dQ^x(Y)=\int_{\mathbb{R}^{n\cdot k}}\prod_{i=1}^k f_i\left(y_{i}\right) dQ^x_{t_1,\ldots,t_k}(y), $$ where the second equality follows from Kolmogorov's Extension Theorem, see, for instance, Chapter 2 of Øksendal's book. The RHS means: $$ \text{RHS}= \int_{\Omega}\prod_{i=1}^k f_i\left(X_{t_i}^{x}(\omega)\right) dP(\omega). $$ Then, the equality follows from a standard result called "the law of the unconscious statistician" applied to the random vector $(X^x_{t_i})_{i=1}^k$ in $\mathbb{R}^{n\cdot k}$. See, for instance, Theorem 5.11 (figure below) in these notes of A. Grigoryan or Proposition 10.1 of Folland "Real Analysis".