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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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Stationary Distribution of a Geometric Brownian Motion with a resetting barrier

Let $X_t$ be a geometric brownian motion with drift $\mu x$ and volatility $\sigma x$. There is a barrier at $x^*$ such that when $X_t$ reaches $x^*$, it is reset to some initial level $x_0 > x^*$. There is no symmetric upper barrier. What is the…
Shffl
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Ornstein-Uhlenbeck like process for non-normal data distributions

I have a set of data which histogram looks like this: The values are not normally distributed since the distribution is not simetric. For another set of data that is normally distributed, I have used a Ornstein-Uhlenbeck stochastic process to…
Santi Peñate-Vera
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Continuous Time Stochastic Process

I am trying to build a stochastic model where two processes happen randomly with different rates that depend on the status of the system. Imagine you have a grid NxN made of 0 or 1. The 1 elements turn into 0 with a constant rate $\lambda_1$. The 0…
lucacerone
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Approximate average answer to time spent working at the current job

Suppose that the distribution of time at a single job is a gamma distribution with a mean of $5$ years and a standard deviation of $2$ years, and suppose that the times at successive jobs are i.i.d. random variables. Every year the person fills out…
ghjk
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Consequence of Doob-Meyer decomposition

Let $\mathcal{H}^2_0$ be the space of all $L^2$ bounded RCLL martingales null at zero. As a consequence of Doob-Meyer we know: For every $M\in \mathcal{H}^2_0$ there exist a unique adapted, increasing RCLL process $[M]$ null at zero with $\Delta…
user20869
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Expectation of product of forward and backward recurrence times of a renewal process?

Let $\{S_n=Y_0+\cdots+Y_n\}_{n\ge 0}$ be a renewal sequence, where $\{Y_n\}_{n\ge 1}$ is a sequence of i.i.d. random variables taking on only non-negative values and assume $Y_0=0$. Let $N(t)=\sum_{n=0}^{\infty}1_{[0,t]}(S_n),$, where $1_{[0,t]}$ is…
Connor
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Property of local martingales

I am currently reading the book 'High Frequency Financial Econometrics' by Jacod and Ait-Sahalia and have a problem with the following statement in chapter 2: 'The usual rationale for first-differencing a time series is to ensure that when T [the…
ArkhamKnight
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distribution function of random variable - negative infinity

This doubt is from the book Stochastic Processes by Ross..Chapter 1, subsection Random variables. The distribution function F of the random variable X is defined for any real number x by F(x) = P{X$\leq$x} = P{X $\epsilon$ (-$\infty$,x]} I have not…
SAK
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Stopping time for Bernoulli trials

Consider an infinite sequence $X_1,\cdots,X_t$ of i.i.d. Bernoulli random variables with parameter $p \in (0,1)$. Consider also an infinite sequence $\alpha(t)$ of integers such that $\alpha(t) \leq t$ for all $t$ and \begin{equation*} \lim_{t…
Oliv
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Finding a transition probability

My transition matrix is $$\left( \begin{array}{ccc} .4 & .3 & .3 \\ .3 & .2 & .5 \\ .7 & 0 & .3 \end{array}\right) $$ So I'm asked for $P[X_3=1,\,X_2=1\,\lvert\,X_1=2]$…
Erick
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Stochastic Differential Equation with noisy drift

I wonder if this type of equation fits in the formalism of SDE $$dX_{t}=(\mu_0+\eta_t)dt+\sigma dW_{t}$$ where $\eta_t$ is uncorrelated Gaussian noise. First I wrote this, but I don't know if it makes sense. $$dX_{t}=(\mu_0+dW_{1t})dt+\sigma…
Pablo Riera
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conditional distribution of $W(t/2)$ given $W(t)=x$

How to get the conditional distribution of $W(t/2)$ given $W(t)=x$ where $W(t)$ represents wiener process. This was a problem in my exam and i couldn't think how to start :( Any help!! Thanks in advance.
Aang
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Finding the probability that the population becomes extinct for the first time in the third generation.

I am trying to solve the following problem and am wondering firstly if my solution is correct and alternatively if there is a shorter way to compute this. In a branching process the number offspring per individual has a binomial distribution with…
Comic Book Guy
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Variation and jumps of spec. stochastic processeses

just wondering, why $\Delta[Var(X)]_t=|\Delta X_t|$ holds for cadlag, adapted and increasing real valued processes X with finite variation over all finite intervals $[0,t]$, which are in addition uniform integrable martingales? $Var(X)_t$ denotes…
peer
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First Passage Problem, Looking for General Method

I'm trying to find a general method for solving problems like the following: Flip a fair coin repeatedly, subtracting 1 if heads and multiplying by 2 if tails. If you currently have X, what is the probability that you reach, say, 1000 before…
Charles
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