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 $t-s$.

2,574 questions
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Brownian motion and Running Maximum

Take B$_t$ as a standard Brownian motion such that B$_0$ = 0. And M$_t$ is the corresponding running maximum. i.e. M$_t$ = max$_{0\leq s \leq t}$ B$_s$. My goal is to compute: (i) Quadratic ...
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Deriving Ito process with a drift from geometric Brownian motion.

Please help me solve this question. Thank you. Let the Geometric Brownian motion be: $$\frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t}$$ $\Delta S$ = change in stock price (s) ...
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Expectation of the exponent of a constant times exiting time of a Brownian motion (i.e. $\mathbb{E}_x[e^{n\sigma}]$)

Suppose $a, n>0$ and $B_t$ is a Brownian Motion, define $$\sigma=\inf\{t:B_t\in\{-a,a\}\}.$$ I want to find $\mathbb{E}_x[e^{n\sigma}]$. (Notice since $n>0$ it is $\textbf{not}$ Laplace ...
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Is there anything known about $\sup_{s\leq t} \vert B_s \vert - f(s)$, where $B$ is a Brownian motion and $f$ measurable.

If we consider the process $$Y^f_t := \sup_{s\leq t} \vert B_s \vert - f(s)$$ is there anything known about the distribution or at least the probability $\Bbb P (Y^f_t \leq 0 )$ ? Of particular ...
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Book-recommendation: Numerical method for stochastic differential equations

Speaking of numerical stochastic differential equations, the book of Peter Kloeden 1992 Numerical Solution of Stochastic Differential Equations is a quite famous and standard reference. But when I ...
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HJB when optimal control responds partially to Brownian motion

I cannot formulate the HJB $V$ in a simple problem where the state $a$ follows a Brownian motion, and the control $\ell$ is (for some states) such that the derivative $V_\ell$ stays constant. In ...
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Brownian Motion and Complex Analysis

I was recently taught in a lecture on complex analysis based off of this paper that much of complex analysis can be rephrased in the language of Brownian Motion. The paper gives some simple proofs of ...
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Supremum of Brownian motion almost surely not at endpoint

Let $(B_t)$ be a standard $\mathbb R$-valued Brownian motion starting from 0. Let $$T = \inf\{t \in [0, 1] : B_t = \sup_{s \in [0, 1]} B_s\}.$$ I wish to show that $T < 1$ almost surely. ...
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Proving that $\int^\infty_0 e^{2B_s} ds=\infty$ [duplicate]

I am struggling to prove that $$\int^\infty_0 e^{2B_s} ds=\infty$$ where $B_t$ is a regular Brownian motion. We know that $\limsup_{t\rightarrow\infty}B_t=\infty$, so I figured this would be a quick ...
If $B_t$ is a standard n-dimensional standard Brownian motion, with is then $B_t\sim \sqrt{t}B_1$, i.e. why are they equivalent in distribution?