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A stochastic process is a collection of random variables usually indexed by a totally ordered set.

**3**

votes

For Brownian bridge, this is more or less a standard exercise. It is simpler to do it first for Brownian motion and then move to Brownian bridge. There are two steps. I assume $b<0$, otherwise the ans …

answered May 10 '17 by ofer zeitouni

**2**

votes

It all depends on the shape of $P_2$ and on the assumptions you put on $X_i$. In what follows I'll assume that $\Lambda(\lambda)=\log E_1 e^{\lambda X_1}$ is finite
for all $\lambda$. I will also ass …

answered Dec 19 '17 by ofer zeitouni

**2**

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Your question is equivalent to asking about the expectation of the max, and asking for an upper bound on the expectation of the max. You have a bunch of Gaussians $\{Y_\alpha\}_{\alpha\in I}$ indexed …

answered May 28 by ofer zeitouni

**3**

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Since your state space is finite, you will have that $\|p_n-p\|\to 0$ and $\|p_n'-p'\|\to 0$ at exponential rate of decay of probability (simply from finite alphabet large deviations - for example, us …

answered Aug 28 '13 by ofer zeitouni

**3**

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A weak solution (is it good enough for you?) can be obtained by a Girsanov transformation that gets rid of the $b$, so the main issue is the diffusion coefficient, and for that the martingale problem …

answered Jul 1 '13 by ofer zeitouni

**3**

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The following is an example: take a Markov chain with states $0,\pm 1,\pm 1/2,\pm 1/3,..$ with jump rates $\lambda_{0,1}=\lambda_{0,-1}=\lambda_{1/k,1/(k+1)}=\lambda_{-1/k,-1/(k+1)}=1$ and rates $0$ …

answered Dec 11 '13 by ofer zeitouni

**1**

vote

Welcome to MO! This is not quite a research level question, but here is the answer anyway.
Represent the Orenstein-Uhlenbeck process as white noise passing through a low-pass (this is really the repr …

answered May 8 '18 by ofer zeitouni

**3**

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No. the total variation distance is always one, since the quadratic variation of the processes is different, and so the measures are mutually absolutely singular.

answered Sep 18 '13 by ofer zeitouni

**6**

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Solutions do exist locally. Globally they MAY blow up as you already know.
The blowup will be dominated by the deterministic system. You did not write what is your initial condition - note that $0$ is …

answered Jul 12 '13 by ofer zeitouni

**1**

vote

I assumed $\gamma=\sigma$.
If $\sigma$ vanishes somewhere and is only measurable,
then your conditions are clearly not enough to guarantee anything, even if $\alpha=0$.
Indeed, take $\alpha=0$, $X_ …

answered Jun 4 '15 by ofer zeitouni

**1**

vote

Certainly, if $\phi\in C_b$ then you only care about the values of $f$ at in a neighborhood of points in the range of $\phi$. In particular, the behavior of $f$ at infinity is irrelevant.

answered Dec 9 '14 by ofer zeitouni

**7**

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There are links, for example in the theory of singular integrals, see section 6.2
in Stroock's book on probability theory. Also, googling "BMO and martingales" will give you information in the directi …

answered Sep 22 '18 by ofer zeitouni

**4**

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It is not in the uniform topology but with a topology tapered off at infinity it is correct. It is done that way in the book of Deuschel and Stroock on large deviations.

answered Oct 2 '15 by ofer zeitouni

**9**

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A good way to measure the set of maxima is the Hausdorff dimension of the set of records, which for BM is a.s. 1/2. Because of time/scale invariance, the dimension is the same for $\alpha B_\cdot$, an …

answered May 24 by ofer zeitouni

**2**

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I learnt the following from Govind Menon.
Consider the Burgers equation
$$\partial_t g+g \partial_x g=0$$
with initial condition $g(0,x)=1/x$. A self-similar solution is $g(t,x)=t^{-1/2}g_*(x\sqrt{t} …

answered Jun 23 '15 by ofer zeitouni