0
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0answers
21 views

Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets. For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...
1
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0answers
23 views

Reference: Computing Martin Capacity

For Borel set A the Martin Capacity is defined as: $Cap_{M}(A)=[inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu $probability measure on A $\}]^{-1}$ and Green's function ...
-1
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0answers
36 views

References for Last exit density function for a Borel set A [closed]

I want to find the probability $P_{x}(B(\gamma)\in A,0<\gamma<T))$, where $\gamma=\sup\{t\in [0,T]: B_{t}\in K\}$ be the last exit time of $d$-dimensional Brownian motion $\{B_{t}:0<t<T$ ...
0
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0answers
23 views

Computing equilibrium measure for Borel sets eg. Ball

I am asking for methods to compute equilibrium measures. The more the better. Here is the definition of equilibrium measure in the Brownian motion setting: Let $\gamma=\sup\{t\in [0,T]: B_{t}\in ...
0
votes
1answer
36 views

How big a Brownian bridge can get? Confidence band.

If we know the endpoints of the Brownian path, is there any theorem telling us if it can be contained within a ball a.s. (with probability one)? For example contained in two big enough balls (call it ...
2
votes
0answers
45 views

A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian ...
8
votes
2answers
213 views

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 ...
1
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1answer
56 views

Fake Brownian Motion

Does there exist a martingale which has Marginal distributions same as Brownian Motion marginals but the process itself not being Brownian motion? Any references are highly appreciated. Thanks.
1
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1answer
1k views

Questions and Solutions in Brownian Motion and Stochastic Calculus?

I am currently studying Brownian Motion and Stochastic Calculus. I believe the best way to understand any subject well is to do as many questions as possible. Unfortunately, I haven't been able to ...
3
votes
2answers
279 views

Brownian motion introduction

I didn't get any answers to my previous question; so I am trying a different tack. I am familiar with a first course in probability theory using measure theory, to the extent of proving the Central ...
9
votes
2answers
540 views

Joint moments of Brownian motion

My approach to this SE question uses the following joint moments of Brownian motion. For $n=1,2$ they are obvious and well-known, the others are not terribly hard to work out. Is there a reference ...