# 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 $t-s$.

2,531 questions
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### Proof: the $\max{(B_t,0)}$ is submartingale without using convex function with Jensen's inequality

Proof: the $\max(B_t,0)$ is submartingale without using convex function with Jensen's inequality To prove it using convex function and jensen's inequality, we know the max funxtion is convex and it ...
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### What is the probability a random walk hits x before it hits y?

This problem was motivated by my bitcoin trading and recalling some of my math education back in the day. I thought I'd ask people who know this much better than I... Suppose there is a continuous, ...
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### Is a vector of independent Brownian motions a multivariate Brownian motion?

Given a filtered probability space $(\Omega, \mathcal{F}, \mathcal{F}_{t\geq 0}, P)$: If $B_1, B_2, \dots, B_m$ are all real $\mathcal{F}_t$ Brownian motions, jointly independent. Is the resulting ...
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### Self similarity of fractional Brownian motion

The fractional Brownian motion with Hurst Paramter $H\in(0,1)$ is a Gaussian Process $\{X(t),t\ge0\}$ with mean $0$ and covariance function $$\gamma(t,s)=1/2(|t|^{2h}+|s|^{2H}-|t-s|^{2H}).$$ I want ...
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### Interpolating process that follows geometric Brownian motion

So I have a set of time series data that follows (at least assumed) GBM but there are missing data. To interpolate, I'm thinking about simulation and create some sample paths. I have two options in ...
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### Density function related to Brownian motion

I am dealing with a question listed below. I am trying to use the running maximum of Brownian motion to deal with the problem, but it does not work out. Let $\tau_{M}=\inf\{t;W(t)=M\},M>0,$ and ...
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### GBM within brownian bridge

I noticed that a brownian bridge between say a and b amy start at B0 and B1 where B1 can be greater than B0. I wonder if this path can be a GBM. If so, what’s the SDE like?
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### scaling invariance of brownian local time

I am studying Brownian local time processes and several references mentioned the scaling invariance of local time. For example, page 10 of this reference (https://hal.archives-ouvertes.fr/hal-00091335/...
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### Reference/suggestion needed - Hitting time of domain equal everywhere -> {domain = sphere, x0 = centre}

So, given a Brownian motion under drift $dX(t) = c dt + dB(t)$ where B(t) is a std Brownian process. Let the process start within some (convex, etc) domain $A$, i.e. $x_0\in A$. Does the following ...
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### If Bt is a Brownian movement. Show that for any t and s, P (Bt> Bs) = 1/2. [closed]

My teacher said it was because of the symmetry but I don't really understand ): somebody can help me?
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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 $\int_0^t X_sdY_s+\int_0^t Y_sdX_s$. ...