# Tagged Questions

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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votes

**1**answer

185 views

### multiplication of two ergodic and stationary processes

If X and Y are stationary and ergodic processes, then, is XY a stationary and ergodic process?
I think the answer is true, but I do not know how to find the mean (we do not know Y and X are ...

**3**

votes

**2**answers

284 views

### Convergence of iterated stochastic matrices

It is well-known that for a stochastic aperiodic matrix $M$,
the sequence $(M^n)_n$ converges.
Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices ...

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votes

**0**answers

330 views

### Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...

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vote

**0**answers

89 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:
...

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votes

**2**answers

120 views

### Smoothness of $g(t,x)=\mathbb{E}[f(X_T)|\mathcal{F}_t]$

Assume a process with Itô dynamics of the generic form
$$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$$
and let $f:\mathbb{R}\to\mathbb{R}$ be borel-measurable. Is the following function smooth ?
...

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vote

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104 views

### Arctic Circle Theorems and the Wave Equation

I've seen the following remark in a number of papers but don't know what to make of it. In this paper by Cohn, Elkies and Propp, it is mentioned that the normalized average Height function ...

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vote

**1**answer

321 views

### Strong solutions on SDE (stochastic differential equations) with discontinuous drift and diffusion coefficients

I want to get some advice from you about the existence (and the uniqueness if possible) of a strong solution on my SDE. In fact, due to the structure of the problem that I consider, both the drift ...

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votes

**0**answers

97 views

### Stationary Distribution for Markov-like system?

Let
\begin{equation}
A=
\begin{pmatrix}
0 & a_{1,2} & a_{1,3} \\
a_{2,1} & 0 & a_{2,3} \\
a_{3,1} & a_{3,2} & 0
\end{pmatrix},
\end{equation}
\begin{equation}
B=
...

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votes

**3**answers

522 views

### Maximum of the expectation of maximum of Gaussian variables

Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of
$$\mathbb{E}\max_{1\le i\le n}|X_i|$$
and
...

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votes

**0**answers

89 views

### Learning resources for Probability Distributions/Models [closed]

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics.
I am already ...

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vote

**0**answers

80 views

### asymptotic estimate of random walk involved hitting time and return time

Consider a reversible random walk on (say) $\mathbb{Z}$, are there any estimate for the following probability $\mathbb{P}(\tau_n=m<\tau_0^+)$ where $\tau_n$ is the first hitting time at site n and ...

**2**

votes

**0**answers

81 views

### Existence of a conditional distribution

Let $X$ and $Y$ be standard Borel spaces and let $J$ be an analytic subset of $X\times \mathcal P(Y)$ where $\mathcal P(\Omega)$ is a set of probability measures on a Borel space $\Omega$ endowed ...

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votes

**0**answers

96 views

### Nonlinear Markov process

Consider the following nonlinear $\mathbb{R}$-valued stochastic recursive sequence:
$ X_{n+1} = F(X_n) + W_{n+1}, \quad (W_n)_{n\ge1} \stackrel{ \scriptsize \mathrm{i.i.d.} }{ \sim } \phi. $
How can ...

**13**

votes

**1**answer

396 views

### Fictitious density of paths of diffusion processes outside the Cameron--Martin space

Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$:
$$dX_t = f(X_t)\,dt + dW_t,$$
where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class ...

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votes

**0**answers

57 views

### Cycle removal process

Consider the following stochastic process for generating a forest: start from a complete graph on $n$ vertices and proceed to repeatedly remove the edges of uniformly chosen cycles. Formally, let ...

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votes

**1**answer

72 views

### Why does differencing create wide-sense stationary time series?

In time series analysis, a common assumption made is that the series is wide-sense stationary, ex. that it has time invariant mean and covariance. However, as this is often not the case in real life, ...

**1**

vote

**1**answer

165 views

### On the solution of a stochastic partial differential equation

Consider a simple SPDE as follows:
$\partial_t u(t,x)=\partial_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$,
$u(t,0)=u(t,1)=0$,
$u(0,x)=v(x)$,
where $V$ is a bounded, smooth ...

**5**

votes

**1**answer

159 views

### Escape Time of Fractional Brownian Motion

Let $B(t)$ be Brownian motion with $B(0)=x>0$ and let $A>x$. It is well known that the expected time for $B(t)$ to escape the interval $[0,A]$ is equal to $x(A-x)$.
Is the expected time known ...

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votes

**2**answers

473 views

### Probability of Brownian motion to have a zero in an interval

I have what should be a very simple questions for Brownian motion experts...
Let $[a,b]$ be a given time interval. Let $f(x)$ be the probability that a linear Brownian motion with initial value $x$ at ...

**3**

votes

**1**answer

174 views

### Characterizations of the GOE/GUE family of distributions

This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...

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vote

**1**answer

73 views

### The probability of Levy process staying at a point

Assume $X_{t}$ is a 1-dimensional Levy process on a probability $(\Omega, \mathcal{F}, P)$. For a fixed point $x$ in the state space and fixed $t\neq 0$, what's the value of $ P(\omega: ...

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votes

**1**answer

127 views

### What are all the stationary and pointwise independent random processes?

In the 60's, I. Gel'fand introduced the concept of generalized stochastic processes (Ch. III, Vol. 4 of his work on Generalized functions). For a generalized stochastic process $\Phi$, he defines the ...

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votes

**2**answers

371 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 ...

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votes

**2**answers

123 views

### Initial paper of Gel'fand on Generalized Random Processes

The theory of generalized stochastic processes was introduced independently in the 50's by Ito* and Gel'fand in a short paper. The latter then developed his theory more extensively in the fourth tome ...

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votes

**0**answers

126 views

### right-continuity of filtration

For a natural filtration of a stochastic process (possibaly multi-dimensional) to be right-continuous, what conditions should the process satisfy? Any references?

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vote

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155 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 ...

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vote

**1**answer

74 views

### The jump and the left martingale of semimartingale

Let $X_{t}$ be a semimartingale. Define
$\Delta X_{t} = X_{t}- X_{t-}$.
For fixed $s> 0$, $\Delta X_{s}$ and $X_{s-}$ are two random variable. Are they independent to each other? I think the ...

**3**

votes

**1**answer

123 views

### The regularity of Levy process

There is a property for continuous Markov process that each point $y$ in its state space is hit with positive probability one starting from any interior point $x$.
This property is called the ...

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votes

**0**answers

79 views

### Null sets visited infinitely often by trajectories of the shift dynamical system

Let $(G,\circ)$ be a Polish group, with identity $e$.
Let $\Omega$ be the set of continuous functions $\omega:\mathbb{R} \to G$ such that $\omega(0)=e$.
For each $t \in \mathbb{R}$, define the ...

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votes

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255 views

### Tight lower bound for expected maximum of K sums of T Rademacher random variables

For each $j \in \{1, \ldots, K\}$, let $(\varepsilon_{j,t})_{t=1}^T$ be an independent sequence of iid Rademacher random variables (i.e. taking values $\pm 1$ with equal probability). What is the best ...

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votes

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93 views

### The distribution of Jump gaps of Levy process

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...

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votes

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268 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. ...

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votes

**1**answer

129 views

### Can this two-dimensional process self intersect?

I would like to know more about the two-dimensional processes derived from Brownian motion by the following stochastic differential equation (in the Ito sense)
$$dX_t = f(X_t) dt + ...

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votes

**0**answers

305 views

### Skorohod theorem (weak convergence) on a discrete setting

I have a question about the application of Skorohod representation theorem. The questions arises in this paper about robust hedging in mathematical finance. It is about the very last equation on page ...

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votes

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212 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

**1**answer

239 views

### Generating independent random variable from two correlated random variables

Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...

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vote

**0**answers

130 views

### Stochastically flipping coins until we see a certain number of heads in two possible durations of time

Imagine that I'm flipping a biased coin (at a rate given by a Poisson process with rate $\lambda$), where the probability the coin lands heads-up is $p$ (tails $q$). I keep flipping the coin until I ...

**0**

votes

**0**answers

67 views

### Ito formula for max(X,0) where X is a semimartingale

Has anyone ever applied the Ito formula on $|X^+|^2$ for $X^+ = \max(X,0)$ with
$X(t) = X(0) + M(t) + V(t)$, where $M(t)$ is a local martingale and $V(t)$ is bounded variation process. I found it in ...

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votes

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136 views

### Different definitions of ergodicity for stationary processes

From page 3 of a note:
A stationary process is ergodic if any two variables positioned far
apart in the sequence are almost independently distributed.
A formal definition is the following: ...

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votes

**0**answers

42 views

### Number of not self-intersecting closed paths spanning $n$ iid uniform points

Let $X_1,X_2,\dots,X_n$ be independent uniform variables in the square. What is the number of piece-wise linear paths which vertices are all the $X_i$ and that do not self-intersect? In other words, ...

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vote

**0**answers

72 views

### CLT for a Markov Renewal Process

Suppose $(X,T)=\{(X_n,T_n)\}_{n\geq0}$ is a Markov renewal process, where $X$ is a finite-state, discrete-time Markov chain with state space $\{1,2,...,R\}$. $T$ is the additive component, more ...

**0**

votes

**1**answer

166 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 ...

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votes

**1**answer

125 views

### Domino Shuffling and Warren's process

In this paper by Nordenstam, it is shown that a certain interlacing particle process that arises from uniformly random Aztec diamond tilings is amazingly similar to Warren's process. One of the ...

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votes

**1**answer

126 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 ...

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votes

**1**answer

223 views

### Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived.
Consider a ...

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votes

**1**answer

203 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 ...

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votes

**2**answers

320 views

### Comparing the stopping times of two stochastic processes

Let $f_0$, $g_1$, $g_0$ be $3$ distinct density functions on the real numbers $\mathbb{R}$ with the corresponding distribution functions $F_0$, $G_1$, and $G_0$, respectively. The following relation ...

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vote

**2**answers

194 views

### On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients

Consider the SPDE $$\frac{\partial}{\partial t}u_t(x) = \frac{\kappa}{2}\frac{\partial^2}{\partial x^2}u_t(x) + u_t(x)(K-u_t(x)) + \sigma u_t(x) \xi(t,x),$$
where $(t,x)\in {\mathbb R}_+\times ...

**2**

votes

**1**answer

169 views

### Transition probabilities in coupled Markov chains

I know that for a continuous-time Markov chain, the probability of transition from time $0$ to $t$ is given by $P(t)=e^{Q(t)t}$. I have a system of $N$ interdependent continuous-time Markov chains ...

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vote

**0**answers

334 views

### What is the characteristic functional for Brownian motion on a sphere?

I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to ...