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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

1 vote
0 answers
152 views

Convergence of approximate quadratic variation in $L^p$

For a diffusion $X_t$, I can set $$[X]^N_t = \sum_{j=1}^N \bigl(X_{t\frac{j}{N}}-X_{t\frac{j-1}{N}}\bigr)^2$$ Then it is well-known that the process $[X]^N_t$ tends to the quadratic variation $[X]_t$ …
Matthias Ludewig's user avatar
2 votes
Accepted

How to evaluate the wiener measure of sets?

First let $t$ be fixed and look at the set $\{ w \mid w(t) \in U\}$. Take a sequence of continuous functions $f_k$ that are uniformly bounded and converge to the indicator function of a set $U$ in the …
Matthias Ludewig's user avatar
3 votes
1 answer
105 views

Density for Translated Process

Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito proces …
Matthias Ludewig's user avatar
15 votes
4 answers
2k views

Positivity of certain Fourier transform

Is the Fourier transform of the function $$ f(\xi) = e^{-t|\xi|^{2m}}$$ positive for $t>0$ and $m \in \mathbb{N}_0$?
Matthias Ludewig's user avatar