# Questions tagged [brownian-motion]

Questions related to Brownian motion, a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.

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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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### 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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### 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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### Brownian motion — using limiting distribution to approximate probabilities of random walk

I managed to find these answers just by converting them to normal distributions of large n and using the rules about that I already understand, which I think is similar to using the Brownian motion ...
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### Find $P(W_{t} < a | W_{2t} > a)$

Here are my thoughts: $P (W_{t} < a | W_{2t} > a) = \frac{P(W_{2t} > 2a, W_{t} < a)}{P(W_{2t} > 2a)} = \frac{P(W_{2t} - W_{t} > a,\, W_{t} < a)}{P(W_{2t} > 2a)}$ Then, taking ...
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### Limit to infinity of Lebesgue-Stieltjes integral with Brownian Motion

Prove that for every fixed $x\in \mathbb{R}$ $$\mathbb{E}_x \left[ \exp\big(\int_0^t \frac{1}{1+B_s^2} ds \big) \right]$$ goes to $\infty$ as $t\to \infty$.
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### System of Coupled differential equations with Stochastic frequency parameter

I have a second-order differential equation that has been put in the following form: $$dx = y dt$$ $$dy = -ax dt- bydt+ c \sin{(\omega t)} dt - xdB_t$$ where $dB_t = W_t dt$, $B_t$ being the ...
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### Ornstein–Uhlenbeck stochastic differential equation help

Consider the Ornstein–Uhlenbeck stochastic differential equation: $$dX_t =−aX_tdt+σdW_t, X_0 =x_0 ∈ R$$ where $a$ and $σ$ are constants, and $W = (W_t)$, $t≥0$ is a Brownian motion. i) Using Itô’s ...
Let $W = (W_t)$, $t≥0$ be a Brownian motion. Let $s, t ≥ 0$. Then: i) Show that $E(W_sW_t)=min(s,t)$ ii) Show that $E|W_t−W_s|^{4} =3|t−s|^{2}$ iii) Find the distribution function of the random ...