Tn the book "Stochastic Differential Equation" from Oksendal one can find the following theorem(6th edition, Theorem 8.6.8):
Let $X(t)=X^x(t)$ and $Y(t)=Y^x(t)$ be an Itô diffusion and an Itô process, respectively, of the forms
$$\begin{align*}dX(t) &= b(X(t))dt + \sigma(X(t))dB(t) \\ dY(t) &= [\gamma(t,\omega) + b(Y(t))]dt + \sigma(Y(t))dB(t), \end{align*} $$ then there exists a stochastic process $M_t$ and a probability measure $Q$ $$dQ:= M_t dP$$ such that the $Q$-law of $Y^x_t$ is the same as the $P$-law of $X^x_t$
Questions
- What is $P$ in the above theorem? Is it the probability law $P^0$ of the brownian motion starting at $0$? Or is $P$ any measure? Could I choose for $P$ the probability law of $Y_t$?
- According to the definition of a law of a process the theorem claims $$ P(X_t^x \in A) = Q(Y_t^x \in A). $$ Later it is claimed that also $$ \mathbb{E}_P[f(X_t^x)] = \mathbb{E}_Q[f(Y_t^x)]$$ holds for $f \in C_0(\mathbb{R}^n)$, how does this follow?
- I have read a different version of the theorem, claiming that if $P$ denotes the path measure of $X_t$ and $Q$ the path measure of $Y_t$, then $$ dQ = M_t dP$$ holds. How does this fit and what is meant with path measure? Is it $$ Q(A):=P^0 (Y_t \in A ) \quad P(A):=P^0(X_t \in A) ? $$
The two processes are defined on some probability space $(\Omega, \mathcal{F}, P)$: $P$ is that probability measure.
Riesz representation theorem says that those two statements (either define a measure explicitly on Borel sets of $\mathbb{R}^n$ or define a linear functional on $C_0(\mathbb{R}^n)$) are equivalent.
Those two statement are also equivalent. A stochastic process, with continuous sample paths in your case, is just a probability measure on $C([0,\infty), \mathbb{R})$.