[Math] Why are Optional Stochastic Processes Important

soft-questionstochastic-analysisstochastic-processesterminology

I understand to some degree why adapted processes, progressive processes, and predictable processes are important. EDIT: I am referring only to the continuous time case, NOT discrete time.

But why do we care about optional processes?

What is significant about the optional sigma-algebra?

Are optional times and optional processes ever as important as stopping times and predictable processes?

(Also I know that given the "usual assumptions", specifically the second regarding a right-continuous filtration, that optional and stopping times are equivalent — are optional and predictable processes equivalent?)

Finally, does the concept of optional processes have something to do with the "Optional Sampling Theorem"? The name always seemed somewhat arbitrary to me.

Best Answer

Let's assume we are in the context of a filtered probability space $(\Omega,(\mathcal F_t),\Bbb P)$ satisfying the usual conditions.

Here are a few features of optionality that make it important.

  1. If $T$ is a stopping time and $A\in \mathcal F_T$ then there is a bounded optional process $Z$ such that $Z_T=1_A$ on $\{T<\infty\}$.

  2. If $Z$ is an optional process such that $Z_T=0$, $\Bbb P$-a.s. on $\{T<\infty\}$, for each stopping time $T$, then $\Bbb P[\sup_t|Z_t|=0]=1$. (In this event $Z$ is said to be evanescent.)

  3. If $D\subset\Omega\times[0,\infty)$ is an optional set, and if $0<\delta<\Bbb P[\pi_\Omega(D)]$, where $\pi_\Omega(D)$ is the projection of $D$ unto $\Omega$, then there is a stopping time $T$ such that $\Bbb P[T<\infty]\ge\delta$ and $\{(\omega,T(\omega)): T(\omega)<\infty\}\subset D$.

  4. If $X$ is a bounded measurable process, there is an optional process $Z$ (unique up to evanescence) such that $\Bbb E[X_T\,|\,\mathcal F_T]=Z_T$, $\Bbb P$-a.s. on $\{T<\infty\}$, for each stopping time $T$. ($Z$ is the optional projection of $X$.)

The point is that for an optional process $Z$, properties of the random variables $Z_T$, as $T$ varies over the class of stopping times, lead to conclusions about the entire process $Z$.

An example showing that "progressive" is a weaker property. Consider the natural filtration of a standard Brownian motion $B$. Let $G$ denote the set of times $t\ge 0$ such that $B(t)=0$ but $B(t+s)\not=0$ for all sufficiently small $s$. It can be shown that $G$ is progressive. But the strong Markov property of $B$ implies that if $T$ is a stopping time, then $\Bbb P[T\in G]=0$. If $G$ were optional then we could conclude from point 2. above that $G$ was evanescent. But this is clearly not the case, so $G$ is not optional.

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