# 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 $t-s$.

2,496 questions
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### What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have ...
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### Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
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### Integral of Brownian motion is Gaussian?

Let $(W_t)$ be a standard Brownian motion, so that $W_t \sim N(0,t)$. I'm trying to show that the random variable defined by $Z_t = \int_0^t W_s \ ds$ is a Gaussian random variable, but have not ...
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### A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
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### Stochastic Integrals are confusing me; Please explain how to compute $\int W_sdW_s$ for example

I have been trying hard to understand this topic, but only failing.Reading through my lecture notes and online videos about stochastic integration but I just can't wrap my head around it. The main ...
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### What is “white noise” and how is it related to the Brownian motion?

In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering ...
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### How to compute $\mathbb{E}(\exp(\int_0^t W_s ds)|W_t)$?

I am trying to compute the conditional expectation $$\mathbb{E}\left[\exp\left(\int_0^t W_s ds\right)\middle|\, W_t\right]$$ where $W$ is a standard Wiener process and where $s\le t$. To initially ...
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### Sobolev meets Wiener

Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point. Does it at least have weak derivatives?
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### Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$dV_t = - \beta V_t dt + \sigma dW_t$$ with $V_0 = v$, where $W_t$ is a Wiener ...
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### Hitting time of Brownian Motion with a drift

Let $X_t =x+bt+\sqrt{2}W_t$, where $W_t$ is a standard Brownian motion. Let $T=\inf\{t: |X_t|=1\}$. I am trying to find $\mathbb{E}[T]$ for the case $b\neq0$. Firstly, I am going to apply Girsanov to ...
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### Collection: Results on stopping times for Brownian motion (with drift)

The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties: distributions of stopping times (Laplace ...
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### Dominated convergence problems with Wald's identity for the Brownian Motion

In the course of proving Wald's second identity $E(B^2_T)=E(T)$, where $(B_t)_{t\geq0}$ is the Brownian motion and $T$ is a stopping time with $E(T)<\infty$, I got stuck with the following problem. ...
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### Show that Brownian motion on the unit circle is exponentially ergodic and has the uniform measure as its invariant distribution.

My search results keep bring up planar Brownian motion on the unit disk. However, I am specifically referring to $e^{jW_{t}} = [\cos(W_t),\sin(W_t)]^{T}$ where $W_t$ is Brownian motion. I am at a ...
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### Initial Distribution of Stochastic Differential Equations

consider the SDE \begin{align} \begin{cases} X_t= \mu (t,x_t)dt + \sigma(t,X_t) d W_t \quad \forall t\in [0,T] \ (\text{or } t\geq 0),\\ X_0 \sim \xi. \end{cases} \end{align} Suppose that, ...
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### How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
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Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ? A geometric Brownian motion $X=(X_t)_{t \geq 0}$ satisifies $dX_t = \sigma X_t \, dW_t$ where $W=(W_t)_{... 1answer 519 views ### Can anyone solve a stochastic differential equation - related to neuroscience research? I'm a neuroscience grad student, and I'm hoping one of ya'll could help me solve this problem regarding particle diffusion. It relates to my research on molecular-level neural plasticity, but I've ... 1answer 2k views ### Hölder Continuity of Fractional Brownian Motion I would like to prove the following theorem: Let$H\in (0,1)$. The fractional Brownian motion$B_H$admits a version whose sample paths are$a.s.$Hölder continuous of order strict less than$H$. ... 3answers 3k views ### Expected value of average of Brownian motion For a standard one-dimensional Brownian motion$W(t)$, calculate: $$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$ Note: I am not able to figure out how to approach this problem. All ... 1answer 435 views ### Difference between weak ( or martingale ) and strong solutions to SDEs Hi Im fairly new to SDE theory and am struggling with the difference between a weak ( or martingale ) solution and a strong solution to an SDE : $$d(X_{t})=b(t,X_{t})dt + \sigma(t,X_{t})dW_{t}$$ ... 1answer 905 views ### Exponentials of stochastic processes and Brownian motions This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given$X_t=\int_0^ur_sds$for a ... 2answers 4k views ### 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 ... 1answer 348 views ### Jump Process - Random Walk A 1-D random walker strarting at time$t=0$and location$x=0$, moves to the right ($x+1$) or the left ($x-1$) according to independent random variables$R_1,R_2,\ldots$and$L_1,L_2,\ldots$, such ... 0answers 189 views ### Brownian Motion in Confined space, any results? I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ... 1answer 2k views ### Expectation of an integral w.r.t. Brownian Motion I know the following statement: if$f$is a deterministic function and continuous, i.e.$f\in C^0([0,T],\mathbb{R})$, then$\int f(s)dW_s$is normally distributed with mean zero and variance$\int f^...
I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
How do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is: $$\tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\}$$ I understand the optional stopping ...