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5
votes
0answers
153 views

A generalization of Jensen's Inequality

Jensen's inequality is well known as $$E\big[f(X)\big]\le f\big(E[X]\big)$$ where $X$ is a integrable random variable and $f: R\to R$ is a bounded concave function, see also ...
5
votes
0answers
154 views

A note on Doob's theorem

I have faced the following problem, regarding to the Martingale Theory. Because this area far from my area I don't know whether this problem is in literature or this can be simple question for ...
2
votes
0answers
96 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
0
votes
0answers
41 views

Question about Skorokhod embedding problem

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion on some probability space. Now for every centered probability distribution $\mu$ on $R$, i.e. $\int_{R}|x|d\mu(x)<+\infty$ and ...
3
votes
1answer
116 views

Conditional Form of Rosenthal's Inequality

Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following: If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and ...
1
vote
1answer
145 views

Can we give any upper bound on $E[\max_{n \leq N} X_n]$ in terms of $\max_{n \leq N} E[X_n]$

Consider a sequence $\{X_n\}$ of $N$ random variables. Can we give any upper bound on $E[\max_{n \leq N} X_n]$ in terms of $\max_{n \leq N} E[X_n]$. I think in general it is not possible. If ...
1
vote
1answer
98 views

An identity for the exponential of a martingale

I am trying to understand a Lemma in Olav Kallenberg's book "Foundations of Modern Probability" (Lemma 26.19 in the second edition or 23.19 in the first edition). The part of the lemma that I do not ...
2
votes
0answers
84 views

A result on absolute mean of a stopped supermartingale

The reason of posting the following problem here is that I heard that it is a result from some paper. Let $(X_n, \mathscr{F_n}), n \geq 0$ be a super martingale and $T$ an $\{F_n\}$-stopping time ...
0
votes
0answers
18 views

Zeros of non-lipschitz functions (when noisy estimates are available only)

Given noisy (martingale difference) of a Lipschitz continuous function $f$ it is known how to compute zeros of it. It is the stochastic approximation approach (by Borkar, Kushner and Yin etc.). Is ...
3
votes
1answer
143 views

Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?

If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
1
vote
2answers
47 views

Bounded martingales of infinite path length

Let $(X_t)_{t \in \mathbb{N}}$ be a real-valued martingale that is bounded, i.e., there are $a, b \in \mathbb{R}$ such that $a \leq X_t \leq b$ for all $t$. Define the path length $L$ of $(X_t)_{t ...
4
votes
0answers
78 views

Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
2
votes
1answer
140 views

Examples of a continuous martingale with $E[\sup\limits_{0\leq s\leq t} |M_s|]=\infty$?

A local martingale is a martingale iff it is in the class DL. The condition: for every $t\in[0,\infty)$ $$E[\sup\limits_{0\leq s\leq t} |M_s|]<\infty\tag1$$ guarantees a local martingale $M$ is ...
4
votes
2answers
320 views

Extension of the Azuma-Hoeffding inequality (when the differences are bounded with large probability)

Let $(X_i)$ be a super-martingale and suppose their differences are bounded ''with high probability'', that is $$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$ ...
2
votes
0answers
90 views

Pointwise convergence of ergodic averages of unconventional conditional expectations

Let $(X_i,Y_i)_{i\in\mathbb{Z}}$ be a finite-valued stationary process whose $\sigma$-algebra of tail events is trivial. Let $\mathcal{F}_n^m$ be the $\sigma$-algebra generated by $X_n,\dots,X_m$ ...
1
vote
0answers
56 views

question related to Tanaka Formulae

Supposse $X=(X_t)$ is a cadlag martingale taking values in $\mathbb{R}$. If $f:\mathbb{R}\to\mathbb{R}$ is a convex function, then we have Tanaka Formulae. Now let $g: ...
1
vote
1answer
45 views

question about the optimal decomposition of supermartingale

Given a filtered probability space $(\Omega, \mathbb{F}, \{\mathcal{F}_t\}_{0\le t\le 1}, \mathbb{P})$, let $X$ be a cadlag martingale and $V$ be cadlag supermartingale. Suppose $V$ has the following ...
0
votes
0answers
91 views

a question about Dambis, Dubins-Schwarz Theorem

Let $M=(M_t)_{0\le t\le 1}$ be a continous $\mathbb{F}=\{\mathcal{F}_t\}_{0\le t\le 1}$-martingale s.t. $M_0=0$. Now my question is whether there exists a Brownin motion $B$ s.t. ...
1
vote
0answers
36 views

a question about the modification of a supermartingale

Let $\mathbf{D}\subset\mathbf{D}([0,1],\mathbb{R}_+)$ denote the space of positive cadlag functions $\mathbf{x}$ defined on $[0,1]$ with $\mathbf{x}(0)=1$. Define the canonical process ...
2
votes
1answer
121 views

explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition: $$V_t=V_0+\int_0^tH_sdX_s-K_t$$ where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...
1
vote
0answers
83 views

question about Doob-Meyer decomposition

Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition: ...
7
votes
2answers
343 views

Can every discrete martingale be embedded in a continuous martingale?

Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale ...
1
vote
0answers
114 views

asymptotic variance of sample autocorrelation of two iid random variables

I am trying to prove that the variance of the sample lag-1 autocorrelation $$\hat{\rho}=\frac{\sum_{t=1}^n(x_t-\bar{x})(x_{t-1}-\bar{x})}{\sum_{t=1}^n(x_{t-1}-\bar{x})^2}$$ for an i.i.d. R.V is ...
0
votes
0answers
222 views

Sufficient condition for local martingale property of stochastic integral

Is the following correct and/or a (simple) known result? Let $X$ be a local martingale and $H$ an integrand for $X$, such that the stochastic integral $\int H\cdot dX\ge x$ for some random variable. ...
6
votes
0answers
196 views

Doob's inequality for martingale “convolution”

Let $(X_t, t \in \mathbb{N})$ be a martingale, and let $a \leq b \leq T \in \mathbb{N}$ be constants. Is there something like Doob's inequality for $\mathbb{E} \sup_{a \leq t \leq b} X_t(X_T-X_t)$, ...
0
votes
1answer
145 views

a dominated convergence theorem for martingale (II)

The question is presented in a dominated convergence theorem for martingale Let $\{(X_1^n, X_2^n)\}_n$ be a sequence of martingales defined some probability space. (which means ...
2
votes
1answer
120 views

Can $<.>$ of a martingale determine it only?

Let $\Omega$ be the space of continuous functions defined on $[0,1]$. Define the canonical process $B$ by $$B_t(\omega)=\omega_t,~ \forall\omega\in\Omega$$ Let us equip $\Omega$ with the usual ...
7
votes
3answers
317 views

Rosenthal like inequality for weak $\mathbb L^p$-norms

Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if ...
3
votes
1answer
153 views

a $L^1$ convergence for backward martingale

I have a question which may be naive, but I can not find the related result in the classical reference such as "Foundations of Modern Probability" and "Probability"(Billingsley). So if someone knows ...
0
votes
1answer
176 views

Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...
0
votes
1answer
87 views

construction of a approximate martingale

everyone. Given a probabilistic space $(\Omega, \mathcal{F}_t, \mathbb{P})$ and a martingale $(M_t)_{t\leq 1}$ on it. Suppose $$M_1\stackrel{\mathbb{P}}{\sim}\mu$$ where $\mu$ is a probability ...
4
votes
0answers
118 views

compactness of a probability set

I have a question about the compactness of a set of martingale measures. Let $\Omega=\mathcal{C}[0,1]$ be the space of continuous functions on $[0,1]$ and $\mathcal{M}_{\Omega}$ be the family of ...
2
votes
0answers
74 views

weaker version of the martingale convergence theorem

Let $\mathcal{A}_n$ be a sequence of finite sigma-algebras, let $\mathcal{B}_{q,p}= \sigma(\mathcal{A}_n, q \geq n \geq p )$. Moreover, we suppose $\mathcal{A}_k \subset \mathcal{B}_{\infty,p}$ for ...
4
votes
2answers
208 views

Probability of winding number of 2D Brownian Motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
2
votes
0answers
88 views

a generalization of Monge-Kantorovich Problem

I am thinking about the martingale version of Monge-Kantorovich Problem. Let $\mu(x)$ and $\nu(y)$ denote two density laws on $\mathbb{R}$, and define $M(\mu,\nu)$ the set of densities $f(x,y)$ on ...
1
vote
0answers
80 views

Supermartingale inequality on a particular event

Say, I have a supermartingale $Y_t$ with respect to the filtration $F_t$. Let $T$ and $S$ two stopping times greater than $t>0$ such that on the event $A$, $T>S$, then since $Y_t$ is a ...
11
votes
0answers
145 views

Savings property: A transformation which turns nonnegative martingales into uniformly integrable ones

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 Mises). However, since ...
1
vote
1answer
368 views

Iterated Ito Integral, Gaussian Volterra Process

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 deterministic ...
1
vote
0answers
188 views

What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?

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 $\mathcal{F}_t$ such that $t \mapsto ...
1
vote
1answer
261 views

On martingale representation theorem

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 such that $|f'(x)| ...
1
vote
2answers
289 views

Uniform law of large numbers for martingale difference

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 all fixed $\theta \in ...
-1
votes
1answer
203 views

Maximal inequalities for certain functions of a martingale difference sequence

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),\ldots, ...
3
votes
1answer
136 views

Does martingale convergence hold for arbitrary time?

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 $\mathcal ...
4
votes
0answers
195 views

Some constants in Martingale Stein inequality

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 filtration of ...
8
votes
3answers
741 views

Compactness of the set of densities of equivalent martingale measures

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 set $\mathcal ...
1
vote
0answers
120 views

Second Equality of Wald [closed]

Hi, I'm doing an exercice about the second equality of Wald. Let $(X_i)_{i\ge 1}$ be a sequence of integrable random variables. Let $F = (F_i)$ be a filtration such as $X$ is adapted. We suppose ...
4
votes
1answer
242 views

What is the optimal growth of the constant in BDG?

Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} ...
7
votes
5answers
669 views

Properties preserved under passage to augmented filtration

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 if ...
6
votes
2answers
737 views

De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales

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 $\mathbb{R}^\infty$ that is ...
4
votes
1answer
188 views

Reference request: Martingale decompositions (positive/negative and u.i./singular)

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 martingale which ...