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Recently I've been dealing with a problem involving stochastic processes. However, I found myself not so familiar with this topic. I have the following two questions regarding whether there is a canonical way to define smoothness, and whether there is a way to uniquely determine a stochastic process.

  1. By definition, a stochastic process can be considered as a random variable that is evolving along time. Now assume $K\subset{R}$ is compact. Let us consider a stochastic process $X(t, \beta)$, where $t$ is the time; and for any fixed $t$, $X(t, \beta)$ is a random variable subject to a distribution $\mathbb{F}(\beta)$ with $\beta\in K$. What is the canonical way to define the continuity/smoothness of $X(t, \beta)$ with respect to $t$ so that this stochastic process can be considered as a `dynamical system of random variable'?

  2. For a random variable $X$ (with the assumption that its distribution is nice enough), we can uniquely recover its distribution through its moment generating function (MGF). Is there any tool, like the MGF of random variable, to uniquely determine a stochastic process?

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By definition, a stochastic process can be considered as a random variable that is evolving along time.

Your "definition" is not nearly precise enough to allow you to consider the questions you're asking.

Try looking up the actual standard definition and you'll find your questions have standard answers.

A stochastic process is a function $$ X(t, \omega) : [0, \infty) \times \Omega \rightarrow \mathbb{R}, $$ where $(\Omega, \mathcal{F}, P)$ is a probability space, such that $X$ is measurable in $(t, \omega)$. (Same definition in the discrete-time case, after replacing $[0, \infty)$ by $\mathbb{N}$.)

What is the canonical way to define the continuity...of $X(t,\omega)$ with respect to $t$...?

$X(t,\omega)$ is said to have continuous sample paths almost surely, or continuous almost surely, if $X(t, \omega) : [0, \infty) \rightarrow \mathbb{R}$ is continuous for almost all $\omega$, i.e. for all $\omega$ in an event with probability one.

For a random variable X (with the assumption that its distribution is nice enough), we can uniquely recover its distribution through its moment generating function (MGF). Is there any tool, like the MGF of random variable, to uniquely determine a stochastic process?

No, the generalization to stochastic processes of a distribution of a r.v. is the law of the process. There's no functionals that plays the same role that MGF, or the characteristic function, plays for r.v.'s, and characterizes the law of the process.

Indeed, you should not expect such a functional to exist.

The distribution of a r.v. is a probability measure on the real line---it is the push-forward measure given by a random variable. In the case where $X(t, \omega)$ is continuous in $t$ for all $\omega$, the law of $X$ is a probability measure on $C[0, \infty)$, the set of continuous functions on $[0, \infty)$. This is an infinite dimensional set and very different from the real line.

("...with the assumption that its distribution is nice enough" is not quite correct. The characteristic function always characterizes the distribution. It is true that, in order to recover the moments from the derivatives, you need to assume distribution is somewhat regular.)

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  • $\begingroup$ Hi Michael, thanks for the detailed answer here. In this question, I'm more curious about whether we can treat a stochastic process as a `continuous' family of r.v.'s, which is a bit different from what you have described here. For instance, in the definition of continuous sample path, we consider the function $X(t, \omega)$ by fixing $\omega$ and analyze the smoothness along $t$. In my question, I try to fix $t$ first and treat $X(t, \omega)$ as a random variable. I wonder whether there are special types to stochastic process that we can think of them in this way. $\endgroup$
    – mw19930312
    Jun 18, 2020 at 1:35
  • $\begingroup$ @mw19930312 The standard definition already answers that question, again. For a given $t$, $X(t,\omega)$ is a r.v. Any stochastic process is, trivially, a family of r.v.'s indexed by $t$. $\endgroup$
    – Michael
    Jun 18, 2020 at 2:07
  • $\begingroup$ Sorry I still don't feel my question is addressed... When defining the continuous sample path, we fix $\omega$ and look at the change in $t$. In this way, the smoothness is based on the uni-variate function $X(\cdot, \omega)$. But if we treat a stochastic process as a family of r.v.'s, then how do we define the two r.v.'s indexed by $t$ and $t+dt$, denoted as $X(t, \cdot)$ and $X(t+dt, \cdot)$, respectively, to be 'close enough'? We need to follow some topology in the space of probability distributions, do we? $\endgroup$
    – mw19930312
    Jun 19, 2020 at 3:54
  • $\begingroup$ A topology on some space of r.v.'s, e.g. the Banach space of r.v.'s with finite p-th moments. A stochastic process is then a function $X$ from $[0, \infty)$ to that topological space. $\endgroup$
    – Michael
    Jun 19, 2020 at 5:29
  • $\begingroup$ It seems that this sentence 'A topology on some space of r.v.'s, e.g. the Banach space of r.v.'s with finite p-th moments.' does not specify what the topology is. Would you mind explain it a bit further, or referring me to some materials that explain what topology is adopted when defining a stochastic process? $\endgroup$
    – mw19930312
    Jun 19, 2020 at 19:32

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