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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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What is a Meyer Process?

Let $X$ be a square-integrable martingale. I am reading the following: Let $\langle X \rangle_t$ be a Meyer process, i.e. the unique predictable process with $\langle X \rangle_0=0$ and right-continuous increasing paths such that $X^2-\langle X…
user126540
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Homogenous Poisson process

Consider a homogeneous Poisson process $N$ with rate $\lambda$. For For $0 < s < t$, I'm trying to show that: $$P(N_t-N_s=0\mid N_t>0)= \frac{e^{\lambda s} - 1}{e^{\lambda t} - 1}$$ I'm mainly thinking of using some sort of conditioning and/or…
mary
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Galton Watson Branching process

Let $X_0$,$X_1$,...be a Galton-Watson branching process. Let us denote $\epsilon$ for the probability when $X_0 = 1$ that the population eventually becomes extinct (that is, that $X_n = 0$ for all suciently large $n$). Then, Set $Z_n =…
mary
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Arrival times, joint density

Let $N(t)$ be a non-homogeneous Poisson process with rate function $\lambda(r),r\geq 0$. I have to find joint density function of arrival times $T_1$ and $T_2-T_1$ and to show that they are dependent. This is not a homework. Firstly I tried to…
Julius
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Interpreting Ventzell Boundary conditions

I am trying to understand the article "On boundary conditions for multidimensional Diffusion processes" of A. D. Ventzell (or Wentzell). I copy the images for greater convenience: In the footnote the author says: I don't understand what are…
Conrado Costa
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Should a stochastic process satisfy this condition?

Let $(X_t)_{t\ge 0}$ be a stochastic process. Let $$M_n(a_1,\dotsc, a_n; t_1, \dotsc, t_n) = \mathbb{E}e^{\sum_{i=1}^n a_i X_{t_i}},\,$$ where $t_i,\, 1\le i \le n$ are distinct. Is it essential that $$\lim_{h_1\to 0,\dotsc, h_n\to 0}M_n(a_1,…
user2808118
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Proof that the Ornstein-Uhlenbeck process is continuous

I am trying to prove the continuity of Ornstein-Uhlenbeck process which is a stationary Gaussian process with covariance kernel $k(x,y) = \exp(-|x-y])$. Let $(X_t)_{t\ge 0}$ be an Ornstein-Uhlenbeck process with $0$ mean. The increment $(X_{t+h} -…
user2808118
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Stochastic delay differential equation

$$dX(t)=F(t, X(t), X(t − τ ))dt + G(t, X(t), X(t − τ ))dW(t)$$ In the stochastic delay differential equation(SDDE) given above, can we assume that the delayed time $τ$ is stochastic process? If so, is there any method of finding $F$, $G$?
user249137
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birth and death processes

Suppose we have a system of N balls, each of which can be in one of two boxes. A ball in box I stays there for a random amount of time with exponential(lambda) distribution and then moves instantaneously to box II. A ball in box II stays there for a…
James R.
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Galton process distribution

Let $1 = Z_0,Z_1,Z_2,\ldots$ be a Galton-Watson branching process with offspring distribution $p_0,p_1,p_2,\ldots$. That is, $p_k$ is the probability that an individual will have $k$ offspring. Suppose that $p_0 = 2/3$ and $p_2 = 1/3$. Let $V = Z_0…
mary
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Does Wiener Process to the power of $n$ have independent increments?

I looked at $cov(W_s^n-W_t^n, W_t^n) = \mathbb{E}(W_s^n-W_t^n)(W_t^n)-\mathbb{E}(W_s^n-W_t^n)\mathbb{E}(W_t^n)$, used Binomial theorem and Moments of Normal Distribution to simplify this, but still can't prove that Covariance is equal to zero (or…
user228494
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What is "starting" static distribution?

I'm not sure if I call everything correctly in English in here, but i have a problem with stochastic processes - Markov chains to be more specific. I'm calculating the "starting" stationary distribution, which in my case equals $PI = [2/7 ,5/7]$.…
Przemysław Robert Wilk
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linear second moment of zero mean stochastic process with independent, stationary increments

I'm working on the following problem: Let $X$ be a zero mean stochastic process with independent and stationary increments. I want to prove that the function $t \mapsto \mathbb{E}X_t^2$ is linear. I have already proven that $X$ and…
Frank
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Compound Poisson Process Problem

I have the following review problem I've been working through and would appreciate any help towards solving it. Customers enter a store according to a Poisson process of rate $\lambda$ = 5 per hour. Independently, each customer buys something with…
Arbit
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Maximize value of exponential

How can i go about maximising the value of the following, $$ exp\left( -\alpha e^{rT}x -\alpha\sigma\lambda e^{rT}\int_0^T e^{-rs}\pi_sds+\frac{1}{2}(\alpha\sigma)^2 e^{2rT}\int_0^Te^{-2rs}\pi_s^2ds \right) $$ with respect to the process…
Danny
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