Questions tagged [martingales]
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274
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Martingale diffusions falling in $\{-1,1\}$ at finite maturity
This is a continuation of Characterization of martingale diffusions ending in $\{-1,1\}$
$X=(X_t)_{0\le t\le T}$ is said to be a martingle diffusion if $X_0=0$, $X_T\in\{-1,1\}$ and
$$X_t=\int_0^t a(u,...
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1
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Characterization of martingale diffusions ending in $\{-1,1\}$
Let $\mathcal M$ be the collection of martingle diffusions starting at zero and ending in $\{-1,1\}$. Equivalently, $X\in \mathcal M$ iff there exists a measurable function $a$ s.t. it holds almost ...
2
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Reverse martingale convergence theorem in Banach spaces
In section 1.5 of a course given by Gilles Pisier, the author is claiming that in the excerpt below $\operatorname E[\varphi_i\mid\mathcal A_{-n}]\to\operatorname E[\varphi_i\mid\mathcal A_{-\infty}]$ ...
2
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1
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Inequality for increments of $r$th absolute moments of martingales, $1<r<2$
If $Y_n=\sum_{i=1}^n X_i$ is a martingale, where $X_i$ is a martingale difference sequence, $\mathbb{E}[X_n\mid \mathcal{F}_{n-1}]=0$ for all $n$, we know that
$$ \mathbb{E}\big[Y_n^2-Y_{n-1}^2\big]=\...
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How to quantify the randomness of martingales?
For a real valued random variable (or probability distribution), the (relative) entropy is used to quantify how random it is. Provided a stochastic process, how can we determine whether it is ''very ...
2
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1
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589
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If a continuous function of a Markov martingale is a martingale, does the function have to be affine linear?
Let $M$ be an almost surely continuous martingale that is not almost surely constant in time - that is, it is not the case that almost surely, $M_t = M_0$ for all $t$.
Assume further that $M$ is a ...
4
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2
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233
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Bounded density for diffusions with diffusion coefficients bounded away from $0$
Consider a diffusion given by
$$X_t=\int_0^t a(s,X_s)\,dW_s$$
for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
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2
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A comparison of diffusions
Consider two diffusions given by
$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$
for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
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1
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First hitting time for non-homogeneous diffusion martingale
This question can be seen as a continuation of Lipschitz continuity of $\mathbb P[\tau>t]$ with respect to $t$
Consider the martingale given as
$$X_t=1+\int_0^t a(s,X_s)dW_s,\quad \forall t\ge 0.$$
...
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220
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Martingale representation theorem for almost adapted martingales
Given a filtration $\mathcal F_t$ on a probability space, we say a stochastic process $X$ is almost $\mathcal F_t$-adapted if there exists some $\mathcal F_t$-adapted process $Y$ such that $\underset{...
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An unnatural martingale
What is an example of a real valued stochastic process $X$, and a filtration $\mathcal F_t$ such that $X$ is a martingale with respect to $\mathcal F_t$ but not it’s natural filtration?
Either ...
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273
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When is every Levy martingale of a process a continuous martingale?
Let $X_t$ be a real valued stochastic process, and $\mathcal H_t$ the the natural filtration of $X_t$.
Under what conditions on $X$ does the following statement hold?
For every $\mathcal H_\infty$-...
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Probability of filling a small ball before exiting a big one for $d=2$
Let $S_n$ be the simple random walk in dimension $d=2$. Let $0<r<R$ and $\alpha \in (0,1)$. Let $B_r$ denote the $\{x \in \mathbb Z^2: \|x\|\le r\}$ where $\|\cdot\|$ is the Euclidean norm. ...
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Does this sequence of martingales converge?
Consider a sequence of martingales that are right-continuous with left limits, denoted by $(X^n_t)_{0\le t\le 1}$, such that for each $n\ge 2$,
\begin{eqnarray}
(1) && X^n_0=0 \mbox{ and } \...
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UMD constant of finite dimensional spaces
For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...
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$\exists c \in\mathbb{R}_+^*,\forall p,r\in \mathbb{R}_+,E[|X_{p+r}-X_r||\mathcal{F}_r] \leq c$ implies the optional stopping theorem
Consider a integrable submartingale $(X_r)_{r \in \mathbb{R}_+}$ relative to $(\mathcal{F}_{r})_{r \in \mathbb{R}_+}$ and such that $$\exists c \in \mathbb{R}_+^*,\forall k \in \mathbb{N},E[|X_{k+1}-...
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Martingale representation of a stopped Brownian motion
This question follows from the previous post Question on the martingale representation theorem which has not been answered. I consider thus a particular case. Let $(B_t)_{t\ge 0}$ be a standard ...
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Can a continuous mean zero process be turned into a semimartingale via a change of measure?
Let $X_t$ be a continuous process such that $E[X_t] = 0$ for all t. Denote by $\mathcal F_t$ the completion of its natural filtration.
Does there exist some $F_{\infty}$-measurable non negative random ...
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1
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Question on the limit of martingales
I am looking for the condition/criterion that yields the convergence of right-continuous martingales, motivated by the following question.
For $M,N\ge 1$, set $I_M:=\{t_m\equiv m/M: 0\le m\le M\}$ ...
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$L^p$-convergence of submartingale
Let $p\geq1.$ Consider a $\mathcal{F}_k$-submartingale $(X_k)_k$ in $L^p.$ We can prove easily that $(X_k)_k$ converges in $L^p$ if and only if $(|X_k|^p)_k$ is uniformly integrable.
If $(X_k)_k$ was ...
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3
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Does there exist an almost surely differentiable martingale?
Does there exist a continuous time martingale $X_t$ not a.s. constant in $t$ that is almost surely everywhere differentiable?
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A semimartingale interpolation problem
This question is a direct extension of this one.
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t_n\}_{n=1}^{N}$ be a ...
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129
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Does a sequence that verifies the assumptions of a square integrable martingale on some event need to be convergent on this event?
I came across this claim by reading some literature on stochastic approximation.
Let $(\Omega, \mathcal{A}, \mathbb{P}$) be a probability space, $(\mathcal{F}_n)$ a filtration on it. Let $(\epsilon_{n}...
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143
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If a process is periodic on average with mutually incommensurable periods, is the process a martingale?
Motivation:
If a continuous function on the real line is periodic with periods $p_1, p_2 > 0$ such that $\frac{p_1}{p_2}$ is irrational, then the function is constant. Is there a probabilistic ...
4
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1
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588
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If the moving average of a process is a martingale, is the process a martingale?
Problem set up:
Let $\mathcal F_t$ be a filtration satisfying the usual conditions. Let $T > 0$ be a fixed real number, and define the filtration $\mathcal H_t := \mathcal F_{T + t}$.
Suppose a ...
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1
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107
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Weaker than martingale condition
Let $\mathcal{F}_n$ be a filtration and $S_n$ be a sequence such that $\mathbb{E}[S_n-S_{n-1}|\mathcal{F}_{n-2}]=0$ for all $n$. This condition is similar to the martingale condition but the ...
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Decomposition of reversed processes
Consider a reversed filtration $(\mathcal{F}_k)_{k \geq 0} $ $(\mathcal{F}_{k+1} \subset\mathcal{F}_k),$ $(X_k)_{k \geq0}$ is a processes in $L^1,\mathcal{F}_k$-adapted.
Is it possible to decompose $...
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On the property of a nonnegative stochastic process "attracted" near zero
Let $\{X_k\}$ be a nonnegative stochastic process satisfying
$$E\left[ X_{k+1} \mid \mathcal{F}_k \right] \leq \rho X_k + c,$$
where $0 < \rho < 1, c>0$. Intuitively, the process is likely to ...
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Almost supermartingale and a.s convergence
After reading a paper on the convergence of almost supermartingale, the following result appeared:
If $(X_k)_k,(Y_k)_k,(W_k)_k$ are three $(\mathcal{F}_k)$-adapted processes taking values in $\mathbb{...
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Moment generating function of a stopped process from Wald's identity
In an exercise I am asked to prove the following Wald's identities: let $S_n$ be a simple random walk and $T$ a stopping time. Then for all $\lambda \in \mathbb R,$
$$
\mathbb E(e^{\lambda S_1}) = 1 \...
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How can we use Martingales to identify an unknown particle?
Suppose there is a particle in a box. We are interested in identifying what type of particle it is, but are not allowed look inside the box. All we can do is observe the particles that are entering ...
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Martingales associated with heat equation
I am trying to learn the connection between Brownian motion and heat equation (in the spirit of Feynman-Kac, for example, here). I read (Michael E. Taylor's PDE book, Volume II, Chapter 11, ...
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Is a stopped Ito-integral integrable if the Ito integrand is only square-integrable on an open interval?
Assume a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\in[0;T)}, \mathbb P)$ with an $\mathbb R^n$-valued Brownian motion $\{W_t\}_{t\in[0;T)}$ and the filtration $\{\mathcal F_t\}_{t\in[0;T)...
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Conditions for existence of a semi-martingale representing a system of probability measures
Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$.
Does there exist a semi-martingale $(X_t)_{t\in[0,1]}$ ...
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0
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586
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Local martingale but not martingale
For a 3-dimensional Brownian motion $B = (B_t, t ≥ 0)$ and $x ∈ \mathbb{R}^3 \backslash \{0\}$ define the process
$Y = (Y_t, t ≥ 0)$ via $Y_t =\frac{1}{|B_t+x|}$ how come this is a continuous local ...
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Understanding the space of parameters in a covariance matrix of conditional expectations
Let $\{(Y_n, Z_n)\}_{n=-\infty}^{n=\infty}$ be a zero-mean jointly stationary Gaussian process where $Z$ takes values in $\mathbb{R}$ and $Y$ takes values in $\mathbb{R}^k$. Here, $n$ runs over the ...
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The min of the mean of iid exponential variables
Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
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Characterization of the generator of a Lévy process using martingale problems
Let $(X_t)_{t\ge0}$ be a real-valued Lévy process. Note that $$\mu_t:=\mathcal L(X_t)\;\;\;\text{for }t\ge0$$ is a continuous convolution semigroup$^1$. Let $$\tau_x:\mathbb R\to\mathbb R\;,\;\;\;y\...
2
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Semimartingale decomposition and filtrations
In short: I am trying to understand how the decomposition of a semimartingale into its local martingale and finite variation components depends on the filtration we are using.
So, taking a toy example,...
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2
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551
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Exponential inequality for the sum of martingale differences $X_1, \dots, X_n$ when $\sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2$
Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that
$$
X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2.
$$
Question 1: Does the following hold?
$$...
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Expectation of Brownian motion increment and exponent of it
While reading a proof of a theorem I stumbled upon the following derivation which I failed to replicate myself. Let $\mu$ be a constant and $B(t)$ be a standard Brownian motion with $t > s$. Show ...
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Martingale derivation by direct calculation
I'm reading the proof of a theorem and stumbled across the following derivation which I cannot replicate myself.
Let $W(t)$ be a $Q$-martingale and be given by $W(t) = B(t) + \mu t$ with $B(t)$ a ...
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1
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685
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Identity involving the probability that a random walk stays below a curve
I'm looking for a direct proof of the following identity:
Let $W_n$ be a simple random walk with $W_0=0$. For all $x>0$ we have
$$
\lim _{N\to \infty} \sqrt{N} \cdot \mathbb P \Big( \forall n \le ...
2
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1
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English translation of "Une inégalité pour martingales à indices multiples et ses applications"
Does anyone know of a English translation of "Une inégalité pour martingales à indices multiples
et ses applications" by Renzo Cairoli. Or could translate the statement of the martingale ...
2
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1
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On the speed of divergence of the converse of the Strong law of large numbers
By the converse of the strong law of large numbers, we know that, given a sequence of i.i.d random variables $X_1,X_2,\dots$ such that $\mathbb{P}(X_1 \ge 0)=1$ and $\mathbb{E}X_1= \infty$,
then I ...
1
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0
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109
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Exponentially suppressed events for bounded difference super-martingales
Let $\{ Z_n \mid n = 0,1,..\}$ be a non-negative super-martingale and assume that it is of bounded difference i.e $\exists ~c_i >0$ s.t $\vert Z_{i+1} - Z_i \vert \leq c_i$. Then we know (Azuma-...
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363
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Probability in Chromatic number upper bound of induced subgraph
Let $G=(V, E)$ be a graph with chromatic number $\chi(G)=1000 .$ Let $U \subset V$ be a random subset of $V$ chosen uniformly from among all $2^{|V|}$ subsets of $V$. Let $H=G[U]$ be the induced ...
2
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0
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165
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Non-integer conditional moment of exponential functional of Brownian motion
Let $B_t$ be a standard Brownian motion.
I want to solve the following:
$$
\mathbb{E}\left[\left(\int_0^1 e^{\sigma B_t}dt \right)^{1/(1-\beta) }\mid e^{\sigma B_1}=z \right],
$$
for some fixed $0<\...
0
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1
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1k
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Martingale convergence theorem in Polya's urn
I want to get checked if my attempt is okay.
First off, let me shortly describe what Polya's urn is:
A certain urn initially contains a red and a blue ball. We now repeatedly do the following : we ...
2
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1
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749
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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 $H_2f(x)=\int e^{-g(x,y)} h(x,dy) (f(y)-f(x))$ (...