proof of the dΓ©but theorem

Let $(\mathcal{F})_{t\in\mathbb{T}}$ be a right-continuous filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on the measurable space $(\Omega,\mathcal{F})$, It is assumed that $\mathbb{T}$ is a closed subset of $\mathbb{R}$ and that $\mathcal{F}_{t}$ is universally complete for each $t\in\mathbb{T}$.

If $A\subseteq\mathbb{T}\times\Omega$ is a progressively measurable set, then we show that its dΓ©but

 $D(A)=\inf\left\{t\in\mathbb{T}:(t,\omega)\in A\right\}$

is a stopping time.

As $A$ is progressively measurable, the set $A\cap((-\infty,t)\times\Omega)$ is $\mathcal{B}(\mathbb{T})\times\mathcal{F}_{t}$-measurable. By the measurable projection theorem it follows that

 $\left\{D(A)

is in $\mathcal{F}_{t}$. If there exists a sequence $t_{n}\in\mathbb{T}$ with $t_{n}>t$ and $t_{n}\rightarrow t$, then

 $\left\{D(A)\leq t\right\}=\bigcap_{n}\left\{D(A)

On the other hand, if $t$ is not a right limit point of $\mathbb{T}$ then

 $\{D(A)\leq t\}=\{D(A)

In either case, $\{D(A)\leq t\}$ is in $\mathcal{F}_{t}$, so $D(A)$ is a stopping time.

Title proof of the dΓ©but theorem ProofOfTheDebutTheorem 2013-03-22 18:39:15 2013-03-22 18:39:15 gel (22282) gel (22282) 6 gel (22282) Proof msc 60G40 msc 60G05