# Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

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### Probability of Seeing "X" % of Balls in "Y" Turns?

Here is a math problem I thought of: Set up: Suppose we have integers 1,2,3...99, 100 Each integer has an equal probability of being selected Game: In round=1, we pick 5 numbers randomly without ...
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### Tightness under quenched and annealed law

If a random process in random environment $X_n$ is tight under the quenched measure, does it mean it is also tight under the annealed measure?
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### On the equivalence in measurability of two different definitions of a playing strategy for a multi-armed bandit

Consider a $d$-armed bandit. At each time $t\in\mathbb{N}^+$ one of the arms is pulled and some reward is gained. For each arm $i=1,\ldots,d$ let $\pi_{i,t}$ be the number of times that arm $i$ has ...
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### Meaning of asymptotic in the context of stochastic process

So for a fractional Brownian motion, I define $k^{th}$ variation as $$S_n = \frac{1}{n}\sum_{i = 1}^{n}{|B_{H}(i\times\frac{T}{n}) - B_{H}((i - 1)\times\frac{T}{n})|^k}$$ where T is fixed and $B_H$ ...
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### Markov chain question : Calculating lim[n→∞] P(X[n+1] = X[1])

i have the following matrix, and $P(X_0=i)$ i=[1,3] is also given: the question is to calculate $lim_{n\to \infty}$ $P(X_{n+1} = X_1)$ . what i did is: I've found the stationary distribution $\pi$...
1 vote
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### Is a Gaussian process with covariance $C''(0)=0$ allowed?

Let $y(t)$ be a zero mean, homogeneous isotropic Gaussian process: $$\left<y\right>=0 \qquad,\qquad \left<y(t_1)y(t_2)\right>=C(t_2-t_1)$$ where $\left< \dots \right>$ denotes ...
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### For $X_t$ a continuous-time Markov chain, show $M_t := \frac{f(X_t)}{f(X_0)}\exp\left(-\int_0^t \frac{Gf(X_s)}{f(X_s)} \; ds\right)$ is a martingale

Let $(X_t)_{t \geq 0}$ be a continuous-time Markov chain on finite state space $E$ with generator $G$ and let $f: E \to \mathbb{R}_{> 0}$ be a function. Then I would like to show that \begin{...
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### Survival probability Amoeba population

I have a question regarding the already covered Amoeba problem. Here is the set up: Every minute, an amoeba may die, stay the same, split into two or split into three with equal probability. All its ...
1 vote
42 views

### Finding the law of a Brownian motion triple

Let $B=(B_t)_{t\geq 0}$ be a standard Brownian motion and let $S_{s,t} = \sup_{s \leq r \leq t} B_r$. It is well known from a classic result of Lévy that $S_{0,t} - B_t$ realises a Brownian motion ...
1 vote
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### Stochastic Simulation - Simulation from the Marginal Distributions

I am reviewing some material on MCMC / simulation and I realised I never quite understood this point. Given a joint distribution $f(x_1, ..., x_n) = f(x_1) f(x_2 | x_1)...f(x_n | x_{n-1}, ..., x_1)$ ...
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### How to show Geometric Brownian motion is not a Gaussian process?

Let's consider Geometric Brownian motion: $$X_t = e^{\mu t + \sigma B_t}$$ where $B_t$ is Brownian motion. Question: How to prove that this process is not Gaussian? I understand that $B_t$ itself is ...
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1 vote
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### Does weak convergence of a sequence of stochastic processes impliy convergence of their supremum?

Consider a sequence of stochastic processes $X_n$ on $L^\infty([0,1]^d)$ and assume that $\mathscr{L}(X_n)$ converges weakly to $\mathscr{L}(W)$, where $W$ is a Brownian bridge and $\mathscr{L}(\cdot)$...
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### Levy Process is a Feller Processs

I want to show that every Levy Process is a Feller Process. Let $X=(X_t)_{t\geq 0}$ be a Levy Process and $\mu_t(dy):=P\circ X^{-1}(dy)$ . I found a proof where it is shown that the transition ...
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1 vote
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### Is there a notion of approximation of continuous-time Markov processes by finite-valued Markov processes?

Recall that in practice, to simulate a Brownian motion on $[0,1]$, we usually use the interpolated process $X^n=(X^n_t)_{t\in[0,1]}$ between the jumps of a random walk $(S_k)_{k=1,...,n}$ with $n$-...
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### Is Feller semigroup in this definition strongly continuous?

Let $C_0=C_0(\mathbb R^d)$ be the space of continuous functions vanishing at infinity, equipped with the usual supremum norm $\| \cdot \|$. Let $T=(T_t)_{t\geq 0}$ be a semigroup of operators ...
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### A cadlag Feller process for $\mathcal F$ is Markov w.r.t $\mathcal F_+$ (Th. 46, Chap. 1, Stochastic Integration - Protter)

In page 35 of the book Stochastic Integration by P. Protter, he defines a Feller process as follows: Then he states the following theorem. In the proof, he used the following strategy: Next, he ...
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### Generating MA Process in matlab using gaussian nosie

I have an MA process that is: $x[n] = a * w[n] + b * w[n - 1] + c * w[n - 2]$ Where $a, b, c \in \mathbb{R}$ and $w \sim \mathcal{N}(0, \sigma^{2})$ How can I generate $x[n]$ in Matlab ? I know that ...
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### Random walk with indipendent but not identically distributed increments

Suppose $\{Z_i\}_{i=1,2, \ldots}$ are normally distributed (independent but not identically distributed) random variables with mean $\mu(y)>0$ and positive variance $\sigma^2(y)$. Therefore we ...
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### Definition of a Markov process

I found 2 Definitions for a Markov process and I am trying to understand how they are connected. Let $X=\left(X_t\right)_{t\geq 0}$ be a $\mathcal{F}_t$ adapted Process. We say $X$ is a Markov ...
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### Bounded random walk joint distribution

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. normally distributed random variable with zero mean and variance $\sigma^2$. Consider the bounded random walk $(S_n)_{n\in\mathbb{N}}$, defined ...
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### Stochastic Integral and Ito's Isometry: Sharp-bracket process [M] or Angle-bracket process <M>?

I am learning stochastic integral, and I have noticed that the Ito's Isometry is sometimes stated using the quadratic variation process $[M]$ (e.g., pg. 47, Eq. (27.3), vol. 2 of Rogers & William),...
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### Why is it harder to prove lower bounds on expected maxima (than upper bounds)?

I am currently learning upper and lower bounds for the expected maxima of random variables, e.g. bounds for the quantity: $$\mathbb{E}\left[\max_{1 \leq i \leq n} X_i\right]$$ Most of the ...
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### Show this truncated stochastic process is a martingale

Consider a filtered space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq 0}, P)$. Let $(M_t)$ be a uniformly integrable (UI) martinagle w.r.t. $\{\mathcal{F}_t\}_{t\geq 0}$. Let $\tau, \nu$ be two ...
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