Definition of a family of probability measures for Ito diffusions

Question

I have a question concerning the definition of a family of probability measures for the solutions to an Ito diffusion $$X_t^x = x + \int_0^tb(X_s^x)ds + \int_0^t \sigma(X_s^x)dB_s$$ as it's given in Oksendal's Stochastic Differential Equations on p. 111 (A PDF of Oksendal can be found here: http://th.if.uj.edu.pl/~gudowska/dydaktyka/Oksendal.pdf).

Oksendal defines $\mathcal{M}_\infty$ to be the $\sigma$-algebra generated by the random variables $\omega\to X_t(\omega) = X_t^y(\omega)$, where $t\geq 0$, $y\in\mathbb{R}^n$. Already I'm a little confused, because he seems to be saying that for fixed $t\geq 0$, $\sigma(X_t^y) = \sigma(X_t^x)$ for all $x,y\in\mathbb{R}^n$. So my first question is: Is my interpretation correct? If so, why is it true that these random variables all generate the same $\sigma$-algebra?

He goes on to define the probability measures $P^x$ on members of $\mathcal{M}$ (he's now dropped the $\infty$ subscript for some reason) by $$P^x[X_{t_1} \in E_1,...,X_{t_k}\in E_k] = P[X_{t_1}^x\in E_1,...,X_{t_k}^x\in E_k]$$ where $E_1,...,E_k$ is any collection of Borel sets and $t_1,...,t_k\in [0,\infty)$. What he seems to be saying here is that we are just going to replace any initial condition $y\in\mathbb{R}^n$ with $x$ under $P^x$, so that $$P^x[X_{t_1}^y \in E_1,...,X_{t_k}^y\in E_k] = P[X_{t_1}^x\in E_1,...,X_{t_k}^x\in E_k].$$ So my second question is again: Is this interpretation correct? If so, it doesn't seem obvious that this definition would be consistent, i.e., it seems possible that one set could be assigned two different probabilities. For example, if $\{X_t^y \in A\} = \{X_s^z \in B\}$, but $P(X_t^x\in A) \neq P(X_s^x\in B)$.

Edit: I now realize this interpretation makes no sense, as up to null sets we have $$\{X_0^x = x\} = \{X_0^y = y\} = \Omega,$$ but $$P^x[\Omega] = P^x[X_0^y = y] = P[X_0^x = y] = 0$$ and $$P^x[\Omega] = P^x[X_0^x = x] = P[X_0^x = x] = 1.$$

Answer 1

One way to make this precise is the following:

Let $(\Omega,\mathcal{F}, \mathcal{F}_t, \mathbb{P})$ be a complete, filtered probability space. For simplicity, I assume that $b$ and $\sigma$ take values on $\mathbb{R}^n$ and $\mathbb{R}^{n\times n}$ respectively. I will also restrict the time domain to $[0,1]$ (the general case follows similarly). Let $X^x:=\{X^x_t\}_{0\leq t\leq 1}$ be the strong solution of the Ito SDE, with initial value $X_0^x=x\in\mathbb{R}^n$, $\mathbb{P}$-a.s. Define $Q^x$ as the law of $X^x$ on the "path-space" $C([0,1];\mathbb{R}^n)$ (the space of $\mathbb{R}^n$-valued continuous functions defined on [0,1], endowed with the topology of uniform convergence). In other words, consider $X^x$ as a random variable $\Omega\ni\omega\mapsto X^x(\omega)\in C([0,1];\mathbb{R}^n)$ and define $Q^x$ to be $Q^x[A]=\mathbb{P}[X^x\in A]$, for $A\in\mathscr{B}\big(C([0,1];\mathbb{R}^n)\big)$.

For example, let $t_1\in[0,1]$, $B^n$= the unit ball on $\mathbb{R}^n$ and $A=\{\omega\in C([0,1];\mathbb{R}^n)\;| \; \omega(t_1)\in B^n\}\in \mathscr{B}\big(C([0,1];\mathbb{R}^n)\big) $. Then $Q^x[A]=\mathbb{P}\big[X^x_{t_1}\in B^n\big]$.

Now, fix $x\in\mathbb{R}^n$, and observe that since $X^x$ is a strong solution, it is adapted to the natural filtration of $B$. Thus, there exists a measurable map $F_{x}: C([0,1];\mathbb{R}^n)\rightarrow C([0,1];\mathbb{R}^n)$ such that for $\mathbb{P}$-almost all $\omega\in\Omega$ $$ X^x(\omega)=F_x(B(\omega))$$

By letting $\theta$ be the Wiener measure on $C([0,1];\mathbb{R}^n), \mathscr{B}\big(C([0,1];\mathbb{R}^n)\big)$ (i.e. the law of the standard Wiener process $B$, starting at $0$) we can express the law of $\{X^x\}_{x\in\mathbb{R}^n}$ on path-space by the push-forward $$Q^x=\theta_{*F_x}$$

$$Q^x[A]=\theta_{*F_{x}}[A]=\theta[F^{-1}_x(A)]=\theta\big[\{\omega\in C([0,1];\mathbb{R}^n)| F_x(\omega)\in A\}\big]$$ $$=\mathbb{P}\big[\{\omega\in\Omega | F_x(B(\omega))\in A\}\big]=\mathbb{P}\big[X^x\in A\big]$$

Note 1: In the above, $\omega$ has been used to denote both a function on $C([0,1];\mathbb{R}^n)$ and an element of the initial probability space $\Omega$. In each case, I have explicitly stated the space in which $\omega$ lives to clarify this use of notation.

Note 2: If we use the notation of the book you refer to, then we have to use $Q^x$ instead of $P^x$, since $P^x$ is reserved for the law of a Wiener process starting at $x\in\mathbb{R}^n$ (see index). Furthermore, I have used $\theta$ instead of $P^0$. Note that the measures $\{P^x\}_{x}, \{Q^x\}_{x} $ can only be defined on the path space $C([0,1];\mathbb{R}^n)$.

Answer 2

Already I'm a little confused, because he seems to be saying that for fixed $t\geq 0$, $\sigma(X_t^y) = \sigma(X_t^x)$ for all $x,y\in\mathbb{R}^n$.

No. $t$ and $x$ aren't fixed in the definition, i.e. $$ \mathcal M_\infty = \sigma\bigl(X_t^x, t\ge 0,x\in \mathbb R\bigr). $$

He goes on to define the probability measures $P^x$ on members of $\mathcal{M}$

No. $P^x$ is already defined as distribution of the Brownian motion $B$ started from $x$. (And it indeed can be regarded as the push-forward of $P^0$, which is the distribution of $B$ started from $0$, not that of $X$, as gigaster writes, under the shift $\theta_x$; although this is not very relevant to the discussion.)

What Øksendal defines here, is $Q^x$, not $P^x$.

What he seems to be saying here is that we are just going to replace any initial condition $y\in\mathbb{R}^n$ with $x$ under $P^x$, so that [...]

No, again. Actually, the use of $X$ in the definition of $$Q^x[X_{t_1} \in E_1,...,X_{t_k}\in E_k] = P^0[X_{t_1}^x\in E_1,...,X_{t_k}^x\in E_k]$$ may be indeed quite misleading. The measures $Q^x$ are defined on (the Borel subsets of) $C([0,T])$, the space of continuous functions, by $$ Q^x(A) = P^0(X_\cdot^x \in A), A\in \mathcal B\bigl( C([0,T])\bigr). $$

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Unfortunately, this part in Øksendal's book is written very sloppily (despite having undergone six editions). Say, in Theorem 7.1.2, which follows the definition of $Q^x$, $E^x$ is used to denote the expectation w.r.t. $Q^x$; this is admitted in the book immediately, but creates some confusion anyway, as $E^x$ was earlier used to indicate expectation w.r.t. $P^x$.

But there is a much bigger problem: Øksendal somehow omits the very intricate measurability questions. E.g., when he says that $X_r = F(X_t, t,r,\omega), r\ge t$ "by uniqueness", this is not so straightforward as it seems. Indeed, imagine that we want to argue by uniqueness and plug $F(X_t, t,r,\omega)$ as a candidate for solution into the equation, but why this process is adapted? (Adaptedness is crucial for the integration w.r.t. $B$.) Why is it even measurable? To claim this, one first need to show the joint measurability of $F(x,t,r,\omega)$ w.r.t. $x,t,r,\omega$.

This part, I hope, is better explained in our textbook with Yuliya Mishura (I can send you the corresponding chapters on request).