Mathematics Stack Exchange
Mathematics Stack Exchange

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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adapted and right-continuous or left-continuous sample path imply progressively measurable

I am a beginner in stochastic process. Here is a standard proof of a proposition from brownian motion and stochastic calculus. 1.13 Proposition. If the stochastic process $X$ is adapted to the filtration $\left\{\mathscr{F}_{t}\right\}$ and every…
XXX
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Pure Birth Process

I encountered this problem while trying out various practice problems to study for my stochastic processes test. (It's not homework, it's just a practice problem). Consider a pure birth process on the states 0,1,...,n for which $\lambda_k =…
acwang123
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Question regarding Poisson process

Let $N\left(\cdot\right)$ be a Poisson process with rate $1$ and let $\Lambda\left(t\right)$ be a non-decreasing right-continuous function. Define $N_{\Lambda}\left(t\right)=N\left(\Lambda\left(t\right)\right)$ , I need to show that given a…
Serpahimz
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Use of Martingale Representation Theorem

I am working on the following problem, and struggling with it. Can anyone help? Let $$H=e^{\int_0^T B_s\,ds}$$ where $T>0$. Show first $E[H^2]<\infty$. Then find an adapted process ($\epsilon_t$)…
Stupid
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Splitting a Poisson process according to time-dependent probabilities

Let $X_t$ be a homogeneous Poisson process of rate $\lambda$. Suppose we define functions $p_1(t)$, ..., $p_k(t)$, such that for all $i$ and $t$, $p_i(t)\in [0,1]$ and $\sum\limits_{i=1}^kp_i(t) = 1$. Let $Y_1,\dots, Y_k$ be processes. Now, for each…
Ben Derrett
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Characterization of two-step 2x2 stochastic matrices

Show that: A 2 x 2 stochastic matrix is two-step transition matrix of a Markov chain if and only if the sum of its principal diagonal terms is greater than or equal to $1$.
kira
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covariance function for Brownian motion

What would the covariance function be of $V(t) = (1-t) B[t/(1-t)]$ if $B(t)$ is standard Brownian motion. Also $t$ is between $0$ and $1$. Thanks for the help! EDIT: Here is where I am stuck: I believe that $Cov(V(t),V(s)) =…
icobes
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Progressive measurability implies adaptedness

I've read that every progressively measurable process is also adapted, but I can't prove it using the definition of measurability. Can anyone give me a proof of this result ?
Ali Jouahri
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A generic question on stochastic integrals

What is the right approach to take and find the moments of the following: $$\mathcal{Z}_t=\int\mathcal{W}_t^k\,d\mathcal{W}_t=?$$ $$\mathcal{W}_t \sim \mathcal{N}(0, t),\…
Viktor
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Stochastic process and unit variance

What does it mean when in stochastic process, we say that the process has unit variance? What is its exact definition?
Common Knowledge
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Finding expectation of waiting time

Can someone explain this solution to me? The question was find E[W1 | X(t) = 2] where W1 is the time until the first event occurs and X(t) is a Poisson process. V1 represents the smallest of the uniform variables that are a part of this theorem. I…
icobes
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Quadratic Covariation

I am not sure about the answer to this question. For a Brownian motion $B_t$ and a process $M$ defined by $M_t=B_{t-s}$ if $t>s$ and 0 else, what is the Quadratic Covariation $[B,M]_t$ ? I find $(t-s)$ if $t>s$ and $0$ else, but the right answer…
TheBridge
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Backward PDE for a mean-reverting stochastic process

A mean-reverting geometric Brownian motion is defined by a system of the equations: $$dX_t = \mu(X_t, \overline{X}_t) X_t dt + σ X_t dW_t$$ and $$d\overline{X}_t = λ(\overline{X}_t − X_t)dt \, .$$ Suppose we want to calculate…
wsw
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Infinite line of people

Let us assume that we have an infinite line of people, and each person can either move forwards or remain at the same place. They move only one step at a time. (They are jumping from one position to the next if that position is empty). All people…
picakhu
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Stopping times of Poisson process

Given a Poisson process $N$, and let $S_n$ be the $n-$th jump time, i.e. $$S_n = \inf\{t\mid N_t = n\}$$ Question: is there a way to characterize all stopping times? especially, can all (or at least, all the bounded) stopping times be described in…
Jay.H
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