In the proof of theorem 2.3 of the article diffusion processes with boundary conditions (1971) one reads: enter image description here

where $Q_{s,x}$ is the unique solution to the martingale problem for $a,b$ starting from $x$ at time $s$ (which is a probability measure on $\Omega, \mathcal{F}^s = \sigma(X(u), u \geq s)$

The point I don't understand is the following $Q_{s,x}$ is defined for events that occur after time $s$. So in this sense, $Q_{\tau'\wedge T, x(\tau'\wedge T)}$ is defined only for events after $ \tau'\wedge T$. On the other hand the $r.c.p.d$ (regular conditional probability distribution) for $P^T$ given $\mathcal{M}^{t_0}_{\tau'\wedge T}=\mathcal{F}^{t_0}_{\tau'\wedge T}= \sigma(X(u\wedge \tau'\wedge T), u \geq t_0)$ is a probability measure defined on $\mathcal{F}^{t_0}$.

Lemma 3.6 of [8 -diffusion processes with continuous coefficients I] is the following enter image description here

I suspect that $P^T$ of th 2.3 is $P'$ of lemma 3.6, but in that case $P^T$ should equal $P$ on $\mathcal{M}^{t_0}_{\tau'\wedge T}$ and the $r.c.p.d$ of $P^T$ given $\mathcal{M}^{t_0}_{\tau'\wedge T}$ agrees with $Q_{\tau'\wedge T, x(\tau'\wedge T)}$ on $\mathcal{M}^{\tau'\wedge T}$. Is that what is meant in those lines?


This is just a bad Typo. the question contains the right answer


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.