# Questions tagged [martingales]

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For the laws of two pure-jump Markov processes $\mu_1$ and $\mu_2$ on $\mathbb R^n$, which generators are $H_1f(x)=\int h(x,dy) (f(y)-f(x))$ and $H_2f(x)=\int e^{-g(x,y)} h(x,dy) (f(y)-f(x))$ (...
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### Stopping times about Brownian motion with draft

Assumet $M(t) = B(t) + \mu t$ where $B(t)$ is a standard Brownian Motion. Denote: $$T_a := \inf \{ t \geq 0, \, M(t) = a\}, \quad T_b := \inf \{ t \geq 0, \, M(t) = b\}$$ The question asks to ...
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### Proof of extended supermartingale convergence theorem

There is a supermartingale convergence theorem which is often cited in texts which use Stochastic Approximation Theory and Reinforcement Learning, in particular the famous book "Neuro-dynamic ...
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### An inequality in harmonic analysis with the BMO flavour

I am asking myself this question (which seems to be a natural generalization of Remark 4.4 of these lecture notes). Question. Let $I_s, s \in \mathcal{S}$ be a collection of intervals included in ...
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### Prove that fractional Brownian motion is not a semimartingale using the p-variation

What follows, up to the horizontal line, is taken from Rogers "Arbitrage with fractional Brownian motion". Consider an interval $[0,T]$ on which is defined the fractional Brownian motion $B$, and ...
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### Martingales and intersection of random walks

Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...
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### Martingale covariation operator in infinite-dimensions

Let $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ be a filtered probability space $U,H$ be separable $\mathbb R$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ and $(f_n)_{n\in\mathbb N}$...
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### Does variants of Bernstein and Freedman concentration inequalities exist with NO uniform bound on the range of RV or martingale differences

A classic formulation of the Bernstein inequality (from Wikipedia) is as follow: Let $X_1, \ldots, X_n$ be independent zero-mean random variables. Suppose that $|X_i|\leq M$ almost surely, for all $i$...
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### CLT for Martingales

I posted this question originally in math stack exchange, but I got no answer. (https://math.stackexchange.com/questions/2604591/clt-for-martingales) In wikipedia, there is a version of a CLT for ...
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### Martingale version of Bernstein-type inequality for (slightly) heavy-tailed distributions?

It is known that for sub-exponentially distributed martingale difference sequence, the following Bernstein-type inequality holds:  ℙ\left(\left| \sum_{i=1}^N a_i X_i \right| \ge t \right) \le 2\...
Let $\varepsilon_1, \dots, \varepsilon_N$ be a martingale difference sequence in $R^d$ with $\|\varepsilon_n\| \le B_n, a.s.$ for each $n=1,\dots,N$. Do we have some Azuma-type concentration ...
### Demonstrations on an $L^1$ martingale [closed]
If $(X_n,\mathcal{F_n})_{n\in \mathbb{N}}$ is a martingale such that $\forall$ n $\in \mathbb{N}, \frac{X_{n+1}}{X_n}\in L^1$ How can be demonstrated that: $\mathbb{E}[\frac{X_{n+1}}{X_n}]=1$ and ...