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Let

  • $(\Omega,\mathcal{A})$ be a measurable space
  • $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$
  • $I\subseteq\mathbb{R}$
  • $X_t$ be measurable with respect to $\mathcal{A}$-$\mathcal{E}$, for all $t\in I$

Given all these beautiful objects, I would like to call $(X_t,t\in I)$ a stochastic process on $(\Omega,\mathcal{A})$ with time domain $I$ and values in $(E,\mathcal{E})$.

However, most people first introduce a probability measure $\operatorname{P}$ and then say that $(X_t,t\in I)$ is a stochastic process on $(\Omega,\mathcal{A},\operatorname{P})$ with time domain ...

Why do we need $\operatorname{P}$ at this point? The only thing I can imagine is the following: When we talk about stochastic processes, we often talk about properties that hold almost surely (more exactly: $\operatorname{P}$-almost surely!), e.g. continuity properties of the path $$I\to E\;,\;\;\;t\mapsto X_t(\omega)$$ for $\omega\in\Omega$.

So, am I right? Is this the reason why most definitions of stochastic processes include the probability measure $\operatorname{P}$?

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  • $\begingroup$ Wayyy before continuity, already to be able to define the distribution of each X_t one needs P. $\endgroup$ – Did Apr 20 '15 at 17:32
  • $\begingroup$ I agree with you that $P$ is not necessary at this point - however, as long as we do not know the measure $P$, we cannot say anything about the process: neither about distributional properties, nor about almost sure properties. Actually, it's very similar in probability theory: Sure, we can define random variables without knowing the underlying probability measure. But as soon as we want to do something with random variables, we need to know the measure. $\endgroup$ – saz Apr 20 '15 at 18:22
  • $\begingroup$ @Did I need to disagree. Why can't we define a random variable $X$ as being a measurable function between two measure spaces? Then, we could take a probability measure $\operatorname{P}$ and think about $\operatorname{P}[X\in A]$ as being the probability of the event $X\in A$, but we would be free to look at $\mu[X\in A]$ for any other (probability-)measure. When we talk about the expectation or distribution of $X$, then we need a probability measure. But until that point, it does make no sense to me to define a random variable on a probability space. $\endgroup$ – 0xbadf00d Apr 20 '15 at 18:24
  • $\begingroup$ @saz The whole question came into my mind as I read the definition of a Markov process: "$(X_t)_{t\in I}$ is a Markov process with distribution $(\text{P}_x)_{x\in E}$, if $X$ is a stochastic process on a probability space $(\Omega,\mathcal{A},\text{P}_x)$ with $\text{P}_x[X_0=x]=1$, for all $x\in E$ ...". Why can't we say: "$(X_t)_{t\in I}$ is a Markov process with distribution $(\text{P}_x)_{x\in E}$, if $\text{P}_x$ is a probability measure on $(\Omega,\mathcal{A})$ and $\text{P}_x[X_0=x]=1$ ...? Saying $X$ is a stochastic process on $(\ldots,\text{P}_x)$ holds no information, does it? $\endgroup$ – 0xbadf00d Apr 20 '15 at 18:31
  • $\begingroup$ I fail to see where "we disagree". But if it makes you feel better, so be it. $\endgroup$ – Did Apr 20 '15 at 18:40

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