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0 votes
1 answer
57 views

Lower bounding an alternating series with signs from a martingale difference sequence

Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that $$M_n := \sum_{i = 0}^n \epsilon_i$$ is a martingale. We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
1 vote
0 answers
182 views

Hardy's inequality proof using Doob's inequalities

Consider a probability space $([0,1],\mathcal{B}([0,1],\lambda),p>1$ and $f \in L^p(]0,\infty[).$ We want to prove Hardy's inequality using martingale theory and Doob's maximal inequalities. Let $\...
-1 votes
1 answer
519 views

Poisson kernel is the Cauchy distribution, reference?

Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
2 votes
1 answer
571 views

Extension of Dynkin's formula, conclude that process is a martingale

This question was asked here, but it did not get enough attention, so I'm crossposting it to MO. Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial ...
1 vote
1 answer
237 views

Poisson kernel, expectation, an absolute value comes in

See here. Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...
8 votes
1 answer
694 views

A generalization of Jensen's Inequality

Jensen's inequality is well known as $$E\big[f(X)\big]\le f\big(E[X]\big)$$ where $X$ is a integrable random variable and $f: R\to R$ is a bounded concave function, see also http://en.wikipedia.org/...
1 vote
1 answer
159 views

$M_t = f(B_{t \wedge \tau}) + (t \wedge \tau)$ local martingale, $\textbf{E}^x[\tau] = f(x)?$

Suppose $D \subset \mathbb{R}^d$ is a domain and $f: \overline{D} \to \mathbb{R}$ is a continuous function, $C^2$ in $D$, satisfying$$f(x) = 0\text{ for }x\in \partial D,$$$${1\over2} \Delta f(x) = -1 ...