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Consider a discrete random variable $N\in\mathbb N$ with $\mathbb P(N=0) = p$, $\mathbb P(N=n) = (1-p)(1-q)q^n$ for $n\neq 0$. Then the probability generating function of $N$ $$\mathbb E(z^N) = \fra …

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form $$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$ for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb R …

There's no general upper bound. Suppose $p_t<1$ for every $t$ and $\sum_{t=1}^\infty p_t = \infty$. For every $N$ there's a positive probability that $l_N = 0$, then $l_t$ will be larger than $NB …

This should probably be a comment but I'm 9 points short. The answer's on wikepedia. It's $e^{\tfrac{it}2} \prod_{i=1}^\infty cos\left(\frac t{3i}\right)$. http://en.wikipedia.org/wiki/Cantor_distr …

Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified dr …

I don't think this is true. Consider one dimensional Brownian motion with $X_0 = 1$ and let $B_i$ be the indicator of the event that the $k$th decimal place is a $0$ (so all of our $B_i$'s are the s …