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Take a time interval $[0,T]$, and a filtered probability space $(\Omega,P,\mathcal{F},\mathcal{F}_t)$. If $X \in L^1(\mathcal{F}_T)$, then $M_t = E [X \ | \ \mathcal{F}_t]$ is a martingale. If I wan …

The Arzela-Ascoli function basically says that a set of real-valued continuous functions on a compact domain is precompact under the uniform norm if and only if the family is pointwise bounded and equ …

Let $(\Omega,P,\mathcal{F})$ be a probability space with filtration $\mathbb{F} = (\mathcal{F}_t), t \in [0,T]$, where $T$ can be finite or infinite. Let $M$ be a cadlag (local) martingale with respe …

Let $W$ be a standard Brownian Motion. Let $\epsilon>0$ be given. Let $X^\epsilon$ be the process which diffuses like $W$ on $(-\epsilon,\infty)$, but when $X^\epsilon$ reaches the level $-\epsilon$ …

Let $(X_t)_{t \geq 0}$ be a standard Poisson process with intensity $\mu$. Let $\tau_b = \inf ( t>0 : X_t= at + b)$, where $a>0$ and $b<0$, and let $\sigma = \inf (t>0 : X_t \geq at)$. Is there any …

Let $X$ be a continuous 1-d diffusion: $$ dX_t = a(X_t)dt + b(X_t)dW_t, X_0 = x. $$ W is a standard Brownian Motion and $a(\cdot)$ and $b(\cdot)$ can have nice regularity properties. Let $Z^n_1,Z^n_ …

It seems to me that in the statement of the Neyman-Pearson Lemma, equivalence isn't assumed, just absolute continuity. I think in general, it is these sets of absolutely equivalent local martingale m …

Let $H^p$ be the Hardy space of analytic functions on the open unit disk $\mathbb{D}$: $f \in H^p$ if $f$ is analytic on $\mathbb{D}$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta < \inf …