Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right) \mid \mathcal F_t^X\right)$

I have found a theorem (see below) in two papers an I try to figure how it could be proved. The result seems to be intuitive, but I'm not able to prove it in a rigorous way. Assumptions: Consider a ...
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1answer
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Optional stopping/sampling for right-continuous supermartingales

Let $\mathbb{F}$ be a filtration $(X_t)_{t\ge 0}$ be a right-continuous $\mathbb{F}$-supermartingale $\sigma,\tau$ be bounded $\mathbb{F}$-stopping times with $\sigma\le \tau$ and ...
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Show $\int_{-\infty}^{\infty}\,f(u,t)dG(u)$ is a ch.f. where $G$ is a d.f. ; $f(u,\cdot)$ is a ch.f. and $f(\cdot,t)$ is continuous.

Show $$\int_{-\infty}^{\infty}\,f(u,t)dG(u)$$ is a ch.f. where $G$ is a d.f. ; and $f(u,\cdot)$ is a ch.f. for each $u$ and $f(\cdot,t)$is coutinuous for each $t$. Note that ch.f. means ...
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how do I parametrise a stochastic matrix

I have a matrix $\mathbf{t}$ that maps one $d$ dimensional probability distribution to another $\mathbf{t}^T x = q$, i.e. with $\sum\limits_i t_{ij} x_i = q_j$ and $\sum\limits_j t_{ij} = 1$ $\forall$ ...