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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.

16128 questions
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Constant mean Markov process must be a martingale?

Is it true that every constant mean ($F_t$ adapted) Markov process must be a martingale? I have not found this statement anywhere but I feel like it must be true. Below I attempt a proof. \begin{align} &E(X_t) = E(X_s) \\ \implies &E(X_t - X_s) = 0…
user438083
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Are holding times independent in a continuous-time Markov chain and in a semi-Markov process

I was wondering if the holding times are independent in a continuous-time Markov chain? Similar question in a semi-Markov process? From what I have read, it is not mentioned that the holding times are independent in both cases, but it is in…
Tim
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Galton-Watson branching process: expectation of a product

So I've been reading about brunching processes and came across the following statement: $Z_n$ is a Galton-Watson process. Let $x$=$E[Z_1]$ and $n>m$ then: $E[Z_nZ_m] = \sum_{r} P(Z_m=r)$$E[rz_n|Z_m=r]$=$\sum_{r} P(Z_m=r)$$r^2x^{n-m}$=…
whatanhonor
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Finding the extinction probability when the offspring has negative-binomial distribution.

We have that $G(s) = \displaystyle(\frac { p } { 1-(1-p)s})^r $ . The extinction probability is the smallest non-negative $\alpha$ such that $\alpha = G(\alpha)$, which is when: $$ \alpha = \displaystyle(\frac { p } { 1-(1-p)\alpha})^r $$ I am…
Mathsguy123
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Generator of a Markov process

For a (index-)homogeneous Markov process $X_t$, its infinitesimal generator A is defined to act on suitable functions $f : \mathbb R^n → \mathbb R$ by $$ A f (x) = \lim_{t \downarrow 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}. $$ In the…
Tim
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Why can all adapted left-continuous stochastic processes be adapted processes?

The definition of an adapted process $X$ is that $X_i$ be $(\mathcal{F_i}, \Sigma)$-measuriable where $\mathcal{F.} = (\mathcal{F_i})_{i \in S}$ is a filtration of the sigma algebra $\mathcal{F}$ (probability space) and $\Sigma$ is part of the…
Zeus
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invariant probability measure of a stochastic differential equation

I have the next exercice: Let $dX_t=-aX_tdt+dW_t$, with $a>0$. Show that $\mathcal{N}(0,\frac{1}{2a})$ is an invariant measure of the process. My idea: We say that a measure $\mu$ is invariant for the process $X_t$ if $$\mu P_t = \mu,$$ where…
Juan Saknussem
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In a Renewal Process, show that sum of convolutions is convergent

I have been doing a course on coursera on stochastic processes. The first process that was introduced was Renewal process, defined as follows; $$S_n = S_{n-1} + \xi_n$$ where $\xi_n$ are IID. We define $F^{n*}$ as convolution of $n$ IID variables,…
Ramesh Kadambi
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How does one read the $1_{[T_n,T_{n+1})} (t)$ in exploding random process?

How does one read the $1_{[T_n,T_{n+1})} (t)$ in exploding random process? I.e. in $X_t = \sum_{n=1}^{\infty} n 1_{[T_n,T_{n+1})} (t)$. I read that $t$ should belong to $[T_n,T_{n+1})$ for it to take value $1$. But since $T_n=S_1+...+S_n$, $S_i \sim…
mavavilj
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example of process s.t. $X_t\sim Y_t$ for all $t$, but $(X_t)$ and $(Y_t)$ has not the same finite dimensional distribution

To prove that two processes $(X_t)$ and $(Y_t)$ has the same distribution, I proved that $X_t\sim Y_t$, but my teacher said that it's not enough, I have to prove that they I the same finite dimensional distribution, i.e. that $(X_{t_1},...,X_{t_n})$…
user657324
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Ito to Stratonovich and Drift Term

I should point out that I have not found myself in the subject of stochastic processes from a Mathematical stand-point. Rather my aim is to appreciate the origin of certain terms in equations describing fluid flow, in particular, simulating…
OP1603
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Alternating distributions of inter-arrival times in a stochastic process

I have some difficulty in deciding if this is a form of {alternating} renewal process or not. The description of the problem is as follows. -> There are 2 sources, which emit 0 and 1 respectively, with rates $\lambda$1 and $\lambda$2. (The 2…
user68468
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Convergence of finite dimensional distributions, poisson random measure

Let $\pi_K$ be Poisson random measure with intensity $\mu = K\lambda$, where $\lambda$ is Lebesgue measure. I have a task to do and the first part was to show that for $f \in L^1({\mathbb{R}^d}) \cap L^3(\mathbb{R}^d$) (shouldn't $3$ be $2$…
Barabara
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Ehrenfest model limiting distribution

In my textbook (Dobrow) p.89. The stationary distribution for the Ehrenfest model with transtition matrix for $\{0,1,...,N\}$ $$P_{ij}=\begin{cases}\frac{i}{N}, &\text{if }j=i+1\\ \frac{N-i}{N}&\text{if }j=i-1\\ 0 &\text{otherwise} \end{cases}…
johnson
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Why Girsanov theorem doesn't allow you to change volatility?

I see Girsanov/Cameron-Martin as a generalization of change of measure from single random variables to stochastic processes (random functions). It is simple to change measure from one non-degenerate normal distribution to another normal distribution…
Shuheng Zheng
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