# 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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### PDE on the probability that the brownian motion stays in [a,b]

This was part of an exam question. Let $a<b$, $(X_t)$ a brownian motion and $$\forall x \in \mathbb R, t\ge 0, \quad \pi(x,t):=P(\forall s \in [0,t], x+X_t \in [a,b]).$$ Given that $\pi$ is $C^2$ (...
1 vote
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### Brownian motion construction on the unit sphere in Oksendal text book

In example 8.5.8 of Oksendal text book, the Brownian motion is constructed on the unit sphere. I don't understand how he defines the time change $Z_t(\omega) = Y_{a(t,\omega)}(\omega)$. He defines the ...
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### Intuitive explanation for Feller processes

We know that a Markov process is a stochastic process in which the future evolution of the system depends solely on its current state and is independent of its past states given the present state. ...
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### Why A Lévy process may be viewed as the continuous-time analog of a random walk

It seems that paths of the Brownian motion are always continuous (please correct me if i am wrong). Levy process is the generalization of Brownian motion by allowing for jumps at random times. So why ...
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1 vote
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### Why I can numerically use signatures on rough paths?

In the paper deep signature transforms ({bonnier, kidger, perez, salvi, tlyons}) they use the signatures on Brownian motion and they invert it with the inversion method defined in (The insertion ...
1 vote
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### Infinitesimal generator of brownian motion [closed]

It the literature we see that the infinitesimal generator of Brownian motion is $\frac{1}{2} \Delta$. However, when we search about the generator of the Brownian motion on the (unit) sphere, we see ...
1 vote
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### Expected value $\mathbb{E}\left(\int_0^t e^{B_s}ds\right)$

I am asked to solve this problem. Let $B$ be a BM and $t>0$ a real number. What is the value of : $$\mathbb{E}\left(\int_0^t e^{B_s}ds\right) \text{ ?}$$ I have tried to use Ito formula to have ...
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### From Large Deviations to Finite Time Probability Tails

Let $(B_t)$ be a standard $d$-dimensional Brownian motion. It is well-known that $$\mathbb P(\sup_{s\in[0,t]}|B_s|\ge \alpha) \le 4de^{-\alpha^2/2dt}.$$ One possibility to obtain such a result is ...
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### Prove that Brownian Motions have normal distribution using central limit theorem

In the book Brownian Motion, 3rd edition by Rene Schilling, he defines a $d$-dimensional Brownian motion $B = (B_t)_{t\geq0}$ indexed by $[0,\infty)$ taking values in $\mathbb R^d$ as a process that ...
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### Fractional Brownian Motion is not a semimartingale

In some sources (for example here: Why is a fractional Brownian motion not a semi-martingale?) you can find a proof why the fractional Brownian motion is not a semimartingale via the Bichteler-...