# 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 $t-s$.

2,576 questions
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### Convex functions of a martingale.

Let $X_1(t)=e^{B(t)}$ and $X_2(t)=e^{-B(t)}$ where $B(t)$ is the standard brownian motion and $\{G_t\}$ is the filtration generated by the brownian motion. Determine what kind of martingale $X_1(t)$ ...
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### Significance of the term $\int_0^t X_sdY_s+\int_0^t Y_sdX_s$.

Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s-\int_0^t Y_sdX_s$. Is there any significance of the term $\int_0^t X_sdY_s+\int_0^t Y_sdX_s$. ...
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### Why $\mathbb P(B_t\geq L)=\mathbb P(B_t\geq \ell, \tau\leq t)$?

Let $(B_t)$ a Brownian motion. I want to prove that for all $L\geq 0$, $$\mathbb P(\sup_{0\leq s\leq t}B_s\geq L)=2\mathbb P(B_t\geq L).$$ The proof start by : let $\tau=\inf\{t\geq 0\mid B_t= L\}$. ...
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### Is this stochastic Picard iterator well-defined?

Preliminaries Let $x_0 \in \mathbb{R}^d$. Let $T \in (0, \infty)$. Let $$\sigma: \mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}$$ and $$\mu: \mathbb{R}^d \rightarrow \mathbb{R}^{d}$$ be affine ...
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### Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the ...
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### Minimal stopping time of brownian motion

Suppose $W$ is a Brownian motion, let $H_B$ be the hitting of $B \in \mathbb{R}$ and let $\tau$ be another stopping time that is taken to be minimal, i.e $(W_{t\wedge \tau})_{t \geq 0}$ is uniformly ...
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### Why is L an interesting random time for a Brownian motion?

Let $B$ be a Brownian motion and define $L=\sup \{ t \leq 1 : B_t = 0 \}$. My question is: Why is $L$ an interesting random time? Durrett's probability book proves something about the ...
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### Compute moments of Brownian motion stopped at exit time of $[a,b]$

Given $B_t$ a standard brownian motion and $a < 0 < b$ Set $T = \inf\{ t : B_t = a \vee B_t = b\}$ For any $\alpha \in \mathbb{Z}^+$, find $EB_T^\alpha$. I know I can use optional ...
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### Proof that $p$-th total variation of a brownian motion is $0$ while $p>2$

The p-th total variation is defined as $$|f|_{p,TV}=\sup_{\Pi_n}\lim_{||\Pi_n||\to n}\sum^{n-1}_{i=0}|f(x_{i+1}-f(x_{i})|^p$$ And I know how to calculate the first total variation of the standard ...
Consider the following SDE: $$\mathrm{d} X_t = - \lambda X_t + \mathrm{d} B_t$$ with initial condition $X_0 = x$, and where $B_t$ is a standard Brownian motion. An application of Ito's formula gives ...
Given a (standard) Brownian Motion $W_t$ if we do some sort of scaling, inversion or reversal then we also get a Brownian motion. I have seen proofs but the proofs only seem to rely on showing ...