Background I work in a subfield of computability theory called algorithmic randomness. We have been using martingales as long as probability theory (going back to work of von Mis …

I'm trying to proving the stable limit of a martingale M_n(t). When I calculate the limit in probability of its quadratic variation, I find that it is always null except for a poin …

Let me define $$ J^f_{n}(t) = \, \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} f(t, t_1, \ldots, t_n) \; dB_{t_n} ...dB_{t_1} $$ where $f:[0,1]^{n+1} \to \mathbb{R}$ is a nice de …

There is a theorem as follows: Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ suc …

Let $\xi_{tn}(\theta),t=1,\dots,n$ be a real-valued martingale difference array indexed by a parameter $\theta \in \Theta \subset R$, where the set $\Theta$ is compact. Now, for al …

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(\mathcal{F_{t}})_{0\le t\le T}$ a filtration generated by standard Brownian motion $W_t$. Let $f(x)$ be $C^1$ function su …

Suppose $\xi_1,\ldots \xi_T$ is a martingale difference sequence. Then, 1) For any $a\in \mathbb{R}^{+}$, can we say something about the sequence $\xi_1^2\mathbb{1}(\xi_1\geq a),\ …

If T, X_n are r.v. and A = {T >=n} is specified by X_n then is T a stopping time? How can this be proved?

Let $\{\mathcal B_i:i\in I\}$ be a family of $\sigma$-algebras (over the same set $\Omega$) which are totally ordered by inclusion, in the sense that for any $i,j\in I$ either $\ma …

Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the …

Dear all, generally speaking, my question is about which properties of a stochastic process are preserved when I skip from the original to the augmented filtration. Recall that …

Dear all, the following is a special case of Stein inequalities for martingales. $\textbf{Theorem}$ Let $(\Omega, \mathbb{P})$ be a (standard) probability space equipped with a f …

De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\in …

For a paper I am writing, I need these two facts. The proofs are fairly short, but I would rather just cite them. This is for martingales index by natural numbers. Also, I call a …

I'm not sure if this stuff all falls under what most would just term "probability", but I'm researching applied macroeconomics and need to get a handle on the following concepts AS …