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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Question on Ito's Formula with some piecewise defined process.

I have another question on Ito's formula, but maybe a rather strange one. In the book of Protter (Theorem 33, pg. 81) we can see that for a general $n$ dimensional semimartingale $(X_t)_{t \geq 0}$ and a $C^2$ function $f$ we can write: Suppose I…
Barreto
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WSS implies IID?

If I have two WSS (wide-sense stationary) processes, $x(t)$ and $y(t)$, can I state that they are IID (independent and identically distributed) sequences?
Rubem Pacelli
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Interchanging two essential suprema of a random variable

Setting Let's start with the definition of the essential supremum of a random variable: A $\mathcal G$-measurable random variable (rv) $Y$ is called essential suprema of a family of $\mathcal G$-measurable rv $(X_i)_{i\in I}$ (with values in…
user681025
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Is an decreasing and convex function over super-martingale also super-martingale?

I learned from class that if $\phi(x)$ is an increasing and convex function and $X$ is a sub-martingale, thus $\phi(X_{t})$ is sub-martingale. Is there any similar theorem about the super-martingale? Like decreasing function on super-martingale also…
nonlinear
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Construct martingale with given quadratic variation

This question is from Protter's Stochastic calculus Problem 4 chapter 2. Let $f:\mathbb{R}_+ \rightarrow \mathbb{R}_+$ be non-decreasing and continuous. Show that there exists continuous martingale $M$ such that $[M,M]_t=f(t)$ I only know how to…
Zorualyh
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Reflected in zero, or Reflected at zero

When considering a stochastic process such as a geometric Brownian motion. Is it correct to say that it is reflected in zero? Or would it be reflected at zero? Purely interested in what the correct wording is.
Sven
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why Gauss-Markov process expected value is zero?

I know that a stationary Gaussian process $X(t)$ that has an exponential autocorrelation is called a Gauss-Markov process. $$R(t)=\sigma^2 e^{-t/τ}$$ In a book, it says that the autocorrelation function approaches zero as the time constant τ is…
Hengwei
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Shift operator of a stopping time, what does it mean exactly?

I'm trying to figure out this question: Let $X$ be a canonical, right-continuous Markov process with values in a Polish state space $E$, equipped with Borel $\sigma$-algebra $\mathcal{E}$. Assume $t \mapsto \mathbb{E}_{X_t}f(X_s)$ right-continuous…
BallzofFury
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Details of the Realization of a stochastic process

A well known example of a strict-sense stationary random process is along the lines of $X_t = \sin(2\cdot \pi\cdot f\cdot t + \theta)$ where $\theta$ is some random variable, usually $\theta\sim \text{Uniform}(0,2\pi)$. f is just a fixed frequency…
StrictlyStationaryPoster
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how can an irreducible markov chain be transient?

If it's irreudible, then you can "get to anywhere from anywhere." a transient state is one that eventually leaves and never returns. but doesn't this contradict the irreducibility premise that you can get to anywhere from anywhere?
beginner
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Some questions on random sum of i.i.d random variables

Recently, I have started studying stochastic processes based on Ross. However, at the start of the book, I have struggled with some difficulties. Actually, I can't understand the following example: Indeed, I can't understand why in the highlighted…
Optimized Life
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Difference between $X_t = Y_t$ a.s. and $X_{\tau} = Y_{\tau}$ a.s. ($\tau$ a stopping time)?

We work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F})_{t\in[0,T]},P)$. Let $X,Y$ be two làdlàg adapted stochastic processes. What is the difference between the following two conditions: $X_t= Y_t \enspace P$-a.s. $\forall t\in…
user711386
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Why are independent identically distributed sequences on a discrete probability space necessarily constant sequences, and what does this mean?

proof of only iid sequences on a discrete probability space are constant sequences I am having trouble with the proof linked in the image above. In particular, my questions are: Is $\Omega$ (the discrete probability space mentioned in the theorem…
curiousconfused
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Question on stochastic matrix with stirctly positive entries

Let $M$ be a stochastic matrix and assume all its entries are striclty positive. Does this imply that the limit as $n \rightarrow \infty$ of $M^n \vec{x}$ is a vector with all entries equal? I know that if we assume the entries are just positive…
wwinters57
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change of variable in SDE

Here is the original question, Suppose you have a call option on the square of a log-normal asset $V_t$. what equation does the price satisfy? my question is how this corresponds to a change of variable stated in the solution to this question. in…
junyaozheng98
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