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

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3
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120 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 ...
3
votes
0answers
59 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 ...
0
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0answers
82 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. ...
3
votes
1answer
104 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 + ...
5
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0answers
241 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 ...
6
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0answers
165 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 $\mathbb{E} \sup_{a \leq t \leq b} X_t(X_T-X_t)$, ...
-1
votes
1answer
183 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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0answers
113 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 ...
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0answers
39 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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0answers
67 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: ...
3
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0answers
35 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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0answers
58 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
1answer
117 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 ...
3
votes
1answer
106 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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0answers
56 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 ...
1
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1answer
38 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 ...
3
votes
1answer
106 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 ...
4
votes
2answers
269 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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2answers
121 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
1answer
116 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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0answers
189 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 ...
2
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2answers
149 views

Any suggestions on a rigorous stochastic differential equations book?

I have been looking through some books and they are not very rigorous. Any suggestions would be great.
3
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0answers
40 views

Relative vulnerabilities in SIS epidemic model

Consider the SIS model of epidemic spreading. There is a finite graph $G(V,E)$, link infection rates $\lambda_{ij}$ and node recovery rates $\mu_i$. There are a few initial nodes which are infected at ...
2
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1answer
133 views

A question about stochastic kernels and invariant measures

Suppose that $E$ is a metric space, let $\mathcal{B}_E$ denote the set of its Borel subsets and suppose that $\mu$ is a probability measure on $(E,\mathcal{B}_E)$. In addition, suppose that $p:E\times ...
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0answers
76 views

An extension of first order stochastical dominance property

A random variable $X$ (distributed as $F$) is said to be stochastically larger than $Y$ (distributed as $G$), ($X>_{FSD}Y$) if their distribution functions satisfy $G(y)>F(y)$ for all $y$. It ...
5
votes
1answer
95 views

Deviation bound for the maximum of the norm of Wiener process

Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof: $$ ...
5
votes
1answer
97 views

A question about extensions of Markov semigroups

I'm cross-posting this question from MSE. It's the first time I do this so I'm unsure of etiquette regarding how to cross-post, if this irritates anyone please vote this down and I'll delete the post. ...
3
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1answer
98 views

Generalization of Lévy's continuity theorem for nuclear spaces

I am interested in a generalization of the following finite-dimensional results in infinite dimensional vector-space with nuclear structure, especially for the cases of the spaces of distributions ...
2
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0answers
80 views

On the infinitesimal generator of a 1-dimensional stochastic heat equation: core and explicit form

Denote $E = C([0, 1])$. I am consider a 1-dimentional stochastic heat equation on $h$: $\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x)$, for all $(t, x) \in (0, ...
4
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1answer
148 views

A queuing process where customers must be detected

Imagine a scenario where customers arrive in some queue according to a Poisson process with rate parameter $\lambda_{arr}$, and where the process of responding to the customers has a kind of ...
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58 views

Discreet customers at more or less discrete counters

A bank has $N$ counters in a row, and customers arrive irregularly at an average of 1 per minute (say, according to a normal distribution with variance $\sigma^2$ – but I don't think that is ...
0
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1answer
105 views

On the expected value of a random integral:

Is it possible to find the expected value of $u(t)$ in terms of the following information: $$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$ where: $X_s$ is a wide sense stationary process with known ...
0
votes
1answer
154 views

Bounds on the eigenvalues of a random binary matrix

Consider $A$, a random binary matrix of zeros and ones in $\mathbb{R}^{{M\times N}}$, and $M>N$. We assume that $P(a_{i,j}=0)=P(a_{i,j}=1)=0.5$ (although I appreciate any advice on the case of ...
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0answers
60 views

Modification of a state in a random walk to be partially absorbing after a walker's position is well-approximated by a stationary distribution

Consider a random walk $(X_0, X_1, X_2, ...)$ on the interval $[0, N]$ starting from some position $k$, where $0$ and $N$ are reflecting barriers. The forward $+1$ transition probability is $p$, the ...
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0answers
62 views

Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...
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2answers
142 views

A Claim on Typical Voronoi Cells

I am trying to prove the following claim (may be it has been proven). Claim: Consider a set of points $\phi=\{x_1,x_2,...,x_i,...\}$ generated by a homogeneous PPP with rate $\lambda$ in the 2-D ...
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1answer
71 views

Multiplicative version of Novikov inequality for Ito integral

It is clear that Ito isometry $E(∫^t_0fdW)^2=E(∫^t_0f^2dt)$ can be written in the multiplicative form as $E(∫^t_0fdW\cdot∫^t_0gdW)=E(∫^t_0f⋅gdt).$ Is it possible to obtain the multiplicative version ...
2
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0answers
82 views

Modification of stochastic processes vs. generalized stochastic processes

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $X = (X_t)_{t \in \mathbb{R}^d}$ a classical stochastic process defined on $\Omega$. One says that a process $Y$ defined on $\Omega$ ...
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0answers
50 views

Stochastic process inference from partial observations

Consider a set $U$. My signal is a piece-wise constant "function" $Sig: t \mapsto s$, i.e. the signal at time $t$ equals to some subset $s \subset U$. One can see $Sig(t)$ as a stochastic process. ...
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0answers
70 views

New conditions to ensure martingality of stochastic exponential?

I am currently doing a project with focus on the Girsanov theorem, and I want to include some of the conditions which would ensure that the stochastic exponential $\mathcal{E}(L)=\exp(L-\langle ...
4
votes
2answers
200 views

When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
2
votes
1answer
292 views

Dynamics of Master Equation

I'm going to do research on dynamics of master equation of $n$ states $$\dot p_i=A_{ij}p_j\qquad i=1\ldots n$$ where $p_i$ is the $i$-th component of probability vector and $A_{ij}$ is transition rate ...
6
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1answer
178 views

Markov processes lacking the Feller property

Let $E$ be a LCH second countable topological space and let $\mathcal{E}$ be its Borel $\sigma$-algebra. Let $(P_t)_{t \geq 0}$ be a conservative transition function on $(E, \mathcal{E})$. This ...
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72 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 ...
3
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4answers
290 views

Does the variance of a continuous time, time homogeneous, Markov process starting from one point necessarily not decrease?

Let $x_t$ be a zero mean, time homogeneous Markovian process (chiefly look at the case where the value is in $1$ dimension) over time $t$ starting from $x_0=0$. Is it necessary that, in continuous ...
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0answers
77 views

Master Equation to Fokker-Planck for a Jump-Diffusion

Does anyone know if there is a derivation of the Master Equation approximation by a Kolmogorov backward equation (Fokker-Planck) to a jump-diffusion with a compensated Poissonian integral? If not, can ...
3
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0answers
42 views

Using Crump-Mode-Jagers processes to get logarithmic bound on a random tree height

I am currently pursuing my PhD degree and in my research I came across a family of random trees. I need to prove a logarithmic asymptotic bound for the heights of such trees as their size grows. I ...
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69 views

On numerical approximation to stationary distribution of diffusion process

Suppose a vector-valued diffusion process X satisfies the stochastic differential equation $$dX_t = b(X_t)dt + \sigma(X_t) dW_t,$$ in which $W$ is a Brownian motion and $b,\sigma$ are such that strong ...
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1answer
153 views

dual space of the subspace of the space of probability measures [closed]

I have a question which maybe so naive but I want to know the result about it. Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Then by some materiau such as ...
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54 views

Time change for non-homogeneous Markov processes

Background: Let $C$ be the space of continuous function on $[0,T]$, $f, \sigma \in C$ bounded with $\sigma^2 \geq \varepsilon > 0$ and let $X=(X_t)_{t\in [0,T]}$ be a diffusion process of ...