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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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Limiting property of a sequence of stochastic processes - How to interpret it?

What is the interpretation of $$ \lim_{a \to \infty} \sup_n P \Big( \sup_t |X_t^n| \geq a \Big) = 0$$ where $\{X^n_t\}_n$ is a sequence of stochastic processes? Could we say that the paths $X^n$ almost never "explode"?
harisf
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state space model

i would like to understand basic idea of state space model,generally definition of state space model says that State space model (SSM) refers to a class of probabilistic graphical model (Koller and Friedman, 2009) that describes the…
dato datuashvili
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Sum of two autocovariances

I need to prove, that sum of two autocovariance functions is an autocovariance function. I take two random processes $X,Y$ for which $X_1(t),X_2(t)$ random variables are independent. Their autocovariance functions are respectfully $\Gamma_1$ and…
atomoutside
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Example 1.4 Karatzas & Shreve

This example shows that two processes $X,Y$ can be a modification of one another, but they don't necessarily need to be indistinguishable. Let $T$ be a continuous random variable with continuous distribution. Let $X_t\equiv 0$ and $Y_t =…
user311475
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Notation around processes: Is Yt the process or a single observation?

I have troubles interpreting this paragraph in an introduction to Stationarity: A time series {Yt} is said to be stationary if for every integer m, the set of variables Yt1, Yt2, ..., Ytm depends only on the distance between the times t1, t2, ...,…
Arnaud Le Blanc
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Autocovariance of moving average process

Let $\epsilon_t\text{ ~ i.i.d.}(0,1)$, and $X_t=\epsilon_{t}+0.5\epsilon_{t-1}$. I need to find its autocovariance function. I know that $E(X_t)=0$, $E(\epsilon_{t})=0$. Let's say, that…
atomoutside
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Understanding theorem involving a supermartingale being $0$ a.s. on $[T, \infty[$.

I'm having some trouble understanding the following theorem: How do you interpret the sentence after the "then", involving the a.s. property for $X_s$ when $s \in [T,\infty[$? I interpret it as $P(X_s=0 \ \forall_{S \in [T,\infty[})=1$. However,…
An old man in the sea.
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Find process correlation function

I have following task: There are two normal centered random processes $X(t)$ and $Y(t)$ given. They have correlation functions $K_{x}(t_{1}, t_{2}), K_{y}(t_{1}, t_{2})$ and cross-correlation function $R_{xy}(t_{1}, t_{2})$. Get correlation function…
Angelika
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What are the units of the variance/rate parameter in a 1D continuous Brownian diffusion process?

In a 1-dimensional continuous Brownian diffusion process following the SDE: $$ dx(t) = \sigma dW(t) $$ where $W(t)$ represents the Wiener process, what are the units of $\sigma$? Given that the probability density of, say, $x(s) = 0$ at time $s>0$,…
iq447
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Stochastic Process problem

I recently started learning about stochastic processes and i found a problem in order to familiarize myself somehow better. Since this is something new for me, the thing is i don't even know how to approach such a problem. Suppose you have 2…
Notoriousphd
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Does there exist a smooth stochastic process with nonzero quadratic variation?

Pick a sequence $b_i\in C^\infty_c(\Bbb R)$ and define $S_t=\sum_{i=1}^N X_ib_i(t)$ where $N$ is a discrete r.v. and the $X_i$'s are i.i.d. standard normals. It is clear that $S_t$ is a.s. smooth regardless of the distribution of $N$. However, when…
GuPe
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Stochastic Process | Doubt with an example

I'm was reading an pdf about introduction to stochastic processes and the example say: A coin it's tossed a few times. By each head the player win $1$ unit and by each tail the player loss $1$ unit. So then in the example we have $n=6$ and $\omega =…
Caeta
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Probability further tests does not provide different results

Could you please suggest me a distribution or stochastic process which would fit the following problem? I have a device under test, and there is a buffer within it, of which I would like to find out the maximum and minimum fill level occupancy when…
Christian
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How can I make my figure resemple more the example?

Currently I'm learning about SDE's. In my course notes the following example was given for the following SDE: This results in the following image: \begin{array}{l} d B_{t}=\left(-K_{1} B_{t}+s_{t}+\frac{1}{2} B_{t} \sigma^{2}\right) d t-B_{t} \sigma…
Tim
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Backward and forward Kolmogorov equation for pure birth

I am dealing with a pure birth process and trying to solve the forward and backward kolmogorov equations. I am stuck solving the backward equation. $\begin{bmatrix}P'_{0,0}(t)&P'_{0,1}(t)\\P'_{1,0}(t)&P'_{1,1}(t)\end{bmatrix}= \begin{bmatrix}-…
Jimjamlorde
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