Questions tagged [martingales]

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2
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1answer
138 views

Calculate Radon-Nikodym derivative

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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0answers
30 views

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 ...
2
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1answer
68 views

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 ...
5
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1answer
115 views

Is there an i.i.d sequence in the unit cube $[-1,1]^d$ with $\mathbb E \left[ \Big \| \sum_{i=1}^N X_N \Big \|_\infty\right] = \sqrt {dN}$?

There are loads of concentration results for sums of scalar-valued independent sums $X_1,X_2,\ldots, X_N$ with $\mathbb E[X_n]=0$. For example Hoeffding's Inequality says if all $|X_1|\le 1$ then $\...
0
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2answers
62 views

Martingale optional stopping before a stopping time

Here’s an easy one, I hope: Suppose $\tau$ is a stopping time and $(M_t)$ is a martingale which together satisfy the hypotheses of the optional stopping theorem so that $\mathbb{E}[M_\tau]= \mathbb{E}...
4
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0answers
202 views

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 ...
0
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1answer
57 views

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 ...
0
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0answers
39 views

Generator of Markov process applied to the expectation of exponential moments of randomized stopping time

For a weak Feller process $(X_t)_{t \geq 0}$ with values in a topological set $X$ and for a measurable (let say closed) set $C$, we define the randomized stopping time $$ \tilde{\tau}_C = \inf \{t &...
-2
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1answer
50 views

Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?

I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find ...
1
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1answer
63 views

Non-relativized, Computable and Schnor randomness w.r.t a measure

Riemann and Slaman have some great work classifying what reals are 1-random with respect to a measure $\mu$ relative to $\mu$. In that paper they cite Levin and Kautz (but not to refs I can find) for ...
3
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0answers
59 views

Martingale polynomial functions

If $B_t$ is a Brownian motion then using Hermite polynomials one can find that $$1, B_t, B_t^2-t, B_t^3 - 3tB_t,...$$ are martingales. If $X_t$ is a diffusion $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)...
8
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2answers
254 views

Show that this process is not a martingale

I am cross-posting this question from MSE since I did not received any answer, furthermore I tried asking some professors in my university but still we could not find an answer. The most surprising ...
0
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0answers
41 views

Construction of equivalent local martingale measures

Assuming No Free Lunch with Vanishing Risk (NFLVR), the market $(\Omega, \mathcal F, \mathbb P, S)$ admits a measure $\mathbb Q$ equivalent to $\mathbb P$ such that $S$ is a $\mathbb Q$-local ...
6
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0answers
111 views

Delayed Pólya's urn process

The standard Pólya's urn process can be stated as follows: You have an urn with red and green balls. At any time unit you choose one ball at random, note the colour, and give the ball back. At the ...
1
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1answer
134 views

Length of longest subsequence as a martingale

Consider a sequence of continuous random variables $(X_n)_{n \geq 1}$. Let $Y_n$ denote the longest increasing subsequence in the tuple $(X_1,\dots,X_n)$. Does $Y_n$ form a martingale? If not, can I ...
1
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0answers
86 views

Expected number of games for three-player gambler's ruin

Three gamblers each start with $a$, $b$ and $c$ chips, respectively. In each round of the game, a gambler is selected uniformly at random to give up one chip, and one of the remaining two gamblers is ...
4
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1answer
97 views

Expected supremum of normalised random walk

Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$. Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix. Define $S^k=...
3
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0answers
56 views

Embedding a continuous-time martingale in Brownian motion

Using the Skorohod embedding, we can embed any square-integrable discrete time martingale $(M_n)$ into a Brownian motion, obtaining times $(T_n)$ such that $(B(T_n))_{n\ge 0}$ is a version of $(M_n)$. ...
2
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0answers
96 views

Wiener isometry for semimartingales

Suppose that $Y_t$ is a special square-integrable $\mathbb{R}$-valued semi-martingale and let $\mathcal{L}^2(Y)$ denote the set of $Y$-predictable processes satisfying $$ \mathbb{E}\left[ \int_0^{\...
3
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1answer
266 views

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 ...
2
votes
1answer
144 views

Bernstein Inequality for continous local martingale

I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time. Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then : $$P\left(\sup_{t\in [0,...
4
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1answer
159 views

The Dirichlet space of martingales

The Hardy space of martingales can be defined in terms of martingale differences. I'll stick to the simplest case of dyadic martingales. Notation The underlying probability space can be taken to be $[...
4
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0answers
112 views

For a martingale $f_0,f_1,\ldots $ how can we bound $P(\frac{1}{n} \|f_n\| \le 1$ for all $ n \ge N)$?

Suppose $f_0,f_1, \ldots$ is a martingale (or i.i.d sequence) in $\mathbb R^d$ with $f_0=0$ and all $\|f_n - f_{n-1}\| \le L$ say. There are many concentration results for the initial segment of the ...
0
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1answer
274 views

Ito integral and true martingale

Consider a twice diferentiable function $F$ on $R$ with bounded first derivative $F'$ and a Brownian motion $W$. Show that $F(W_t)-\frac{1}{2} \int_{0}^{t} F'' (W_s)ds$ is a true martingale. I tried ...
7
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1answer
246 views

Do i.i.d. sums concentrate any faster than martingales?

Suppose $X_1,X_2, \ldots, X_N \in \mathbb R^d$ are random variables with each $\|X_n\|_2 \le 1/2$ (this choice of the constant simplifies later formulae). The simplest concentration inequality I ...
0
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1answer
401 views

On the martingale representation theorem

I seen several sources claim that any martingale in a Brownian filtration is continuous. However while working with processes of the form $\mathbb{E}(X\mid \mathcal{F}_{s})$ for some random variable $...
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0answers
65 views

Prior state dependent transition probability ABRACADABRA problem

The power of the martingale trick for computing the expected stopping time is amply demonstrated in this question and this answer as an advanced version of the ABRACADABRA problem. However, it seems ...
2
votes
1answer
101 views

Concavity, martingales and stopping time

Suppose $(x_t)_t$ is a bounded $\mathbb F_t$ martingale and $f(t,x)$ is continuous, bounded, and concave in $x$. So, for any $s \ge t$, $$\mathbb E_t f(s,x_s) \le f(s,\mathbb E(x_s)) = f(s,x).$$ Does ...
0
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1answer
81 views

Conditioning on an irrelevant variable in a martingale control problem

Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued $\...
10
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2answers
667 views

Can we do better than Azuma-Hoeffding when the variance is small?

The Azuma-Hoeffding Inequality says that if $X_1,X_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X_i - X_{i-1}\| \le 1$ say, then we should not expect the difference $\|...
1
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0answers
50 views

Martingales limit theorems (reference)

I have a sequence of processes $\{X^N(t)\}_{t\in [0,T]}$, $N\in\mathbb N$ such that $X^N(t)=x+M^N(t)$, where $M^N(t)$ is a martingale with expectation $0$ and with quadratic variation $<M^N>(t)$ ...
2
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0answers
49 views

p-Variation distance defines semi-martingales

Question When, does the process $\tilde{X}_t$, defined path-wise by $$ \tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right), $$ define a ...
3
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2answers
161 views

Expectation of the exitpoint distance for the symmetric random walk

Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$. Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...
1
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1answer
119 views

Martingale representation theorem for symmetric random walk

Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that $$ X(t) = \int_0^t ...
3
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1answer
199 views

Concentration of a modified random walk

Let $\varepsilon$ be a number in $(0, 1)$, consider the following random walk on the real line $X^{(0)}, X^{(1)}, \dots$, where $X^{(0)}=0$ If $X^{(t)} > 0$, then with probability $.5$, $X^{(t+1)...
1
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1answer
68 views

n-factor martingale representation theorem

Baxter & Rennie at pag. 162 state the following theorem. Let $W$ be an $n$-dimensional $\mathbb Q$-Brownian motion and let $M_t=(M_1(t),...,M_n(t))$ be an $n$-dimensional $\mathbb Q$-martingale ...
3
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0answers
73 views

Convergence of SDEs

Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...
3
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1answer
92 views

Lindeberg implies convergence of max of conditional variances in L1

The following is taken from Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES, Equation 4.6. $$\{X_{n,k}\}_{n=0,1,...;k=0,1...,k_n}$$ is a (triangular) array of r.v.'s /w ...
3
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1answer
199 views

Inequality for exponential sum in Dvoretzky 1972

I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has ...
7
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1answer
469 views

Is this a martingale sequence?

I have a sequence of random variables $X_1, X_2, \ldots X_N$ such that $|X_i| \leq R \ \forall \ i $, satisfying $$|E[X_n|X_1,X_2,\ldots X_{n-1}]| \leq |X_{n-1}|, $$ Can I construct a sub/super-...
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0answers
96 views

Is martingale solution equivalent to weak solution for SDE driven by stable process

Consider the following SDE $$ d X_t=b(X_t)d t+d L_t, $$ where $L_t$ is the symmetric $\alpha$-stable process. The corresponding generator is given by $$ L=\Delta^{\alpha/2}+b\cdot\nabla. $$ Is the ...
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0answers
133 views

Does Novikov condition imply BMO martingale?

Let $(\Omega,\mathbb{F},P)$ be a complete probability space, equipped with a filtration $\mathcal{F}_t, 0 \le t < \infty$. Consider a continuous local martingale $(X_t, \mathcal{F}_t)$ such that $...
1
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0answers
49 views

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 $(f_n)_{n\in\mathbb N}$...
4
votes
1answer
173 views

Zero-one law for an independence-like structure

I am a number theorist by profession, so apologies if the answer to this question is "trivially true" or "trivially false". Let $(\Omega, \mathcal{A}, P)$ be a (non-atomic) probability space. Let $(\...
2
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1answer
175 views

Existence of solution to a martingale optimal transport type problem

I encounter the following problem during the course of my research: Given a random variable $Y=(Y_1,Y_2)$ with values in $\mathbb R^2$ and the cost function $c(x,y)=(x_1-y_1)(x_2-y_2)$ where $x=(x_1,...
4
votes
1answer
460 views

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 $i$...
2
votes
1answer
315 views

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 ...
7
votes
1answer
208 views

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\...
5
votes
0answers
267 views

Vector martingale concentration

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 ...
-2
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1answer
104 views

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 ...