# Questions tagged [stochastic-processes]

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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### Stochastic process, stochastic differential equation

A stochastic process $\{X_t , t ≥ 0\}$ satisfies stochastic differential equation $$\frac{dX_t}{X_t} = 3\mu\ dt + 2\sigma dB_t.$$ where $-\infty<\mu<\infty$ and $\sigma>0$ are given ...
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### Decomposition of Gaussian spaces with respect to covariance function

Let $K(t,s):T^2 \to \mathbb{R}$ be a kernel symmetric and type positive (for every $n$ $\sum^n_{i,j}u_iu_jK(t_i,t_j) \geq 0$ and $(u_1,\dots,u_n) \in \mathbb{R}^n$) where $T$ is any set. Thus, it is ...
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Let $\xi_{i}$ - i.i.d., $\mathbb{P}(\xi_{i} = 1) = \mathbb{P}(\xi_{i} = -2) = \frac{1}{2}$; $X_{i} = \sum_{j=1}^{i} \xi_{j}$. How to find the $\mathbb{P}(\exists n \geq 0: X_{i} = 1)$? Give me some ...
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### Convergence of a stochastic process

Suppose that $\{X_n(t),t\in\mathbb{R}\}_{n=0,1,...}$ is a collection of stochastic processes, i.e., for any fixed $n$, we have and stochastic process $\{X_n(t),t\in\mathbb{R}\}$. Assume that for any ...
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### The quadratic variation of the Brownian motion almost certainly tends to T

On the segment $[0, T]$, choose $n$ independent points $t_{n,k}$ (each distributed evenly). Prove that the quadratic variation of the Brownian motion on the sequence of random partitions of the ...
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### Help with Poisson Stochastic Process

Cars pass along the road in accordance with the Poisson process of intensity $\lambda$ . A pedestrian crosses the road at time $W$ as soon as he sees that there will be no cars during time $T$ (...
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### The probability of getting $y$ new coupons from a batch of $k$

In the coupon collector process, the goal is to assemble a collection of $n$ distinct coupons, while we get a random coupon at each time. I am looking at a generalization of this problem, where at ...
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### Can one change the dimension of a Bessel process by a Girsanov change of measure?

Recall that a (squared) Bessel process $X_t$ with the dimension $\delta_0\geq0$ is a solution of the SDE $$d X_t = 2\,\sqrt{X_t}\,d W_t + \delta_0\,d t.$$ A naive application of the Girsanov Theorem ...
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### Finding the expectation of a Wiener Process

My question is on how to find $\mathbb E[W_t^n]$ where $n= 0,1,2,...$ and $W_t$ is a standard normal Wiener process. Would we need to use a moment generating function. Thanks.
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### Applyin Itô's formula in function of quadratic variation

I am learning some basic stochastic calculus and came across the following exercise: Consider a local martingale $M$ with continuous trajectories. Let $Z_t = \exp(M_t −0.5[M]_t)$. Show that Z ...
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### Conditional expectation of poisson procces problem

Let $N_{t}^{i}$ - be three independent Poisson processes of intensity $1$. $\tau$ = $\inf\{t: \,N_{t}^{3} = 1\}$, $X^{i}$ = $N_{\tau}^{i}$ (means that $X^{i}$ - the values of the first two processes ...
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### Need a help with Markov Chain

We have Markov Chain with continuous time, three conditiions and generator: \begin{equation*} Q = \begin{bmatrix} -3 & 1 & 2\\ 1 & -1 & 0\\ 1 & 0 & -1 \end{bmatrix} \end{...
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### Question about Ito Process. Stochastic Processes

How to prove, that $W_{t/(1-t)}$ at $[0,1)$ is Ito Process ? (Have stohastic differential)
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### how to deal with multiple-level conditional probability

Consider a Markov Chain, when I want to prove that $P(X_{n+1} = i_{n+1}|X_{n-1} = i_{n-1}, X_{n-2} = i_{n-2}, \cdots, X_0 = i_0) = P(X_{n+1} = i_{n+1}|X_{n-1} = i_{n-1})$, I tried to use \begin{...
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### Is stopped integral of predictable process predictable?

Assume that $H$ is a predictable process that is locally bounded with localizing sequence $(\tau_n)_n$. And assume that $\langle M, M \rangle$ is a increasing, right-continuous, predictable process. ...
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### Relation between Stopping times

I am having a hard time trying to understand the following relation: Consider that a stopping time is defined by $\{\tau \leq n \} \in \mathcal{F}_{n}$. Now take two stopping times $\phi$ and $\tau$ ...