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

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Compactness if Emery (semi-martingale) topology

The set $\mathscr{S}$, of semi-martingales is a topological vector space under the Emery topology on the space of semi-martingales. There has been some recent research on closures in this topology (...
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123 views

Random walk on $\mathbb{R}$ with “sticky” origin

Let $P_i$, $N_i$, and $Z_i$, $i\in\mathbb{N}$ be r.v.'s with the $P_i$, $N_i$, and $Z_i$ being identically distributed with known pdf's $f_P$, $f_N$, and $f_Z$, respectively; and with no dependence ...
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46 views

On how to think about a BSDE and its solution

Given a drift coefficent $g$ and a finial value $\xi$ one can consider a BSDE which we denote $e_{\xi}(g)$. A solution for such an equation is a pair adapted pair $(Y,Z)$ such that $Y_{t}=\xi + \int_{...
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24 views

Proof differential product correlated brownian motions

I was wondering how to prove/compute the differential of the product of two Brownian motions. I know how to do it in case they are independent as follows: Suppose $dX_t= \mu_t dt +\sigma_t dW_t$ and ...
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19 views

product of right continuous filtrations is right-continuous?

Let $\mathcal{G} = \sigma \lbrace G_{1},..., G_{n} \rbrace$ where $G_{1},..., G_{n}$ are subsets of $\Omega_{1}$ and $(\mathcal{F}_{t})$ is a right-continuous filtration on $\Omega_{2}$. Is $(\mathcal{...
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1answer
48 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 ...
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92 views

Galton Watson tree with various kinds of offspring

As far as I understood, for the Galton-Watson tree process, the offspring are of one type. I am thinking of the case where we have offspring of different types. I have illustrated this in the example ...
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Characteristic function of hitting time

Suppose that $X_t$ is an affine process, $f$ is a convex function with values in $\mathbb{R}$ such that $X_0=0$, and $M>0$. Then what can be said about the hitting time $$ \tau \triangleq \inf\...
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1answer
86 views

Convergence of a stochastic sequence?

I am reading this paper related to an algorithm for nonsmooth optimization problems. After many simplifications, I was able to formalize the method as follows: let $\Bbb B $ denote the unit ball in $\...
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21 views

Finding a queuing model for waste accumulation

I've been tasked with modeling the accumulation of solid waste in an urban setting. In particular, the objective is to find the steady state distribution describing the amount of waste in a given ...
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57 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 ...
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1answer
52 views

Bivariate Poisson-Binomial distribution

Suppose you have $100$ coins whose probabilities of obtaining the outcome "head" are $p_1,\ldots,\,p_{100}$. These probabilities are not necessarily equal each other. Consider the following random ...
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100 views

Spectral gap for the Brownian motion with drift on a compact manifold

Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
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1answer
82 views

On the Cauchy-Schwarz Inequality for trace function of random matrices

In the deterministic case, for two matrices $A$ and $B$ with appropriate matrices, we know that $$tr((A^{T}B)^{2})\leq tr(A^{T}A)tr(B^{T}B)$$ which is the trace form of Cauchy-Schwarz-Inequality (CSI)....
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42 views

Compare KS test and Wasserstein distance or Earth mover's distance

I have tried the following question in couple of exchange sites but I did not get any views or reply. I am asking here as I am kind of desperate for the answer, please be considerate. Any suggestion ...
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26 views

Irreducibility of Markov processes

I have a question about irreducibility of Markov processes. Let $(E,\mu)$ be a metric measure space. Let $X=(X_t,P_x)$ be a $\mu$-symmetric Markov process on $E$. We denote $\{p_t\}$ by the semigroup ...
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93 views

Is there a distinct Ito-Sasaki version of Riemannian stochastic development?

Given a smooth manifold $M$ with a linear torsion-free connection on its tangent bundle, the Eells-Elworthy-Malliavin stochastic development provides a way of transforming a semimartingale $X$ defined ...
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27 views

Reference request on theory about Stochastic Riemann problem

I am trying to find references in the literature that deal with the Stochastic Riemann problem. Let me explain it a bit. On one hand, in the literature it is not hard to find books or papers that deal ...
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1answer
74 views

Properties of Cameron Martin Space

In the case that I'm working with a separable Hilbert space, $H$, on which I have a trace class operator, $K$, that's coming from a Gaussian (i.e., $K$ is self-adjoint, and for simplicity, has trivial ...
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1answer
86 views

Multiple Wiener-Ito integral distribution

Distribution of standard Ito integral is well known: $$I_1(f) = \int_0^T f(t)dB(t) \sim \mathcal{N}\bigg(0, \int_0^T f^2(t)dt\bigg).$$ Is it possible to find the distribution of multiple Wiener-Ito ...
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108 views

Predictability of countably valued accessible stopping times on complete and cadlag filtrations

The following question is motivated by this part of the proof of Lemma 2 on page 107 of the book Stochastic integration and differential equations of Philip Protter. Lemma 2. Let $T$ be a totally ...
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24 views

Mean and correlation of product of two random processes

I have two random process: $$A(at)$$ $$\cos(2\pi f_0t+\Phi)$$ with these hypothesis: $a$ and $f_0$ are constant $\Phi$ is uniformly distributed in $[0,\pi)$ $A(at)$ is WSS I must calculate the ...
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1answer
76 views

how to derive stationary distribution of maximal entropy random walk

I was reading the paper 0810.4113v2, burda, which analyzed the stationary distribution maximal entropy random walk on the irregular lattice. I am confused on some of the steps. Description: The ...
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27 views

Probability of exiting on the boundary for a monotone Lévy-type process

Let the continuous function $\ell:\mathbb R \times(0,\infty)\to[0,\infty)$ be a Lévy-type kernel, such that $$ \sup_{x}\int_0^\infty \min\{1,y\}\ell( x, y)\,dy<\infty, $$ and suppose that $\...
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1answer
93 views

Can we transform $\int_\rho^1 (W_t - W_{t-\rho}) dW_t$ to make its law $\rho$-invariant?

I just bumped into the stochastic integral $$ \int_\rho^1 (W_t - W_{t-\rho}) dW_t $$ where $0 < \rho < 1$ is a constant and $W$ is a standard Wiener process. It would be nice if we have a closed-...
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Continuous Local Martingales under time change under what conditions are they still local martingales?

This question is motivated by reading a section in Continuous Martingales and Brownian Motion by Daniel Revuz, Marc Yor. In Chapter V there is a section on time-change: Definition: A time change $C$...
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1answer
56 views

Is the law of $\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}}$ atomless?

Let $B_t$ denote a standard Brownian motion, and $0 < l < u$. I am wondering if the law of $\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}}$ is atomless, that is, $\mathbb{P}\left(\sup_{l \leq t \...
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72 views

Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail. ...
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32 views

Random solute transport equation

After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...
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Generators and Covariance Operators of Diffusions

For a constant coefficient Ornstein-Uhlenbeck process, how should I think about the relationship between the infinitesimal generator of the process and the covariance operator of the process (or, ...
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31 views

Differentiability of a stochastic process depending on a spatial parameter

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $T>0$ $I:=(0,T]$ $d\in\mathbb N$ $M:\Omega\times\overline I\times\mathbb R^d\to\mathbb R$ such that $M(\;\cdot\;,\;\cdot\;,x)$ is $\...
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Question about Protter's proof of the Ito's formula

The following is a question about a notation that Protter uses in the proof of the Ito's formula for cadlag processes of finite variation (FV) that appears on Stochastic Integration and Differential ...
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238 views

Counting Hamiltonian cycles in $n \times n$ square grid

I wonder if anyone has counted these curves, either exactly or asymptotically? Let $S_n$ be an $n \times n$ subset of $\mathbb{Z}^2$ consisting of $n^2$ lattice points: a lattice square. Define a ...
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Does completion of natural filtration preserve the Markov property?

Let $(\Omega, \mathcal{A}, P)$ be complete probability space and $(X_t)_{t \geq 0}$ a Markov process on a general measurable space $(E, \mathcal{E})$ meaning that the Markov property $P(X_{t+h} \in B ...
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38 views

Stationary recursive sequence and nonzero probabilities

A while ago I posted the following problem: Suppose I have a two sided stationary sequence of random variables $\ldots,X_{-1},X_0,X_1,\ldots$ such that all finite dimensional joint densities $f(x_1,\...
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38 views

Reflected SDE with vanishing diffusion term at the boundary

I have a reflected SDE of the form $dX_{t} = a(X_{t})dt + b(X_{t})dW_{t}+dL_{t}, \quad X_{0}=x_{0}\in (0,\infty) $ where $L$ keeps the process from going below zero. The coefficients $a$ and $b$ ...
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Why is the Jain Monrad condition the right condition on general Gaussian processes?

Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process. Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...
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1answer
72 views

Expected time of distinguishability of a series of Poisson processes bounded by each other

Consider a system of $n$ "bounded" Poisson processes over the integers, $X_1, \ldots X_n$, all incrementing at rate $\lambda$. Initially all the processes begin at $0$. The process $X_i$ is inactive ...
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1answer
138 views

Are Holder Condition and signal to noise ratio (SNR) related?

This question was posted in https://math.stackexchange.com but I got hardly any view. If posting here is an objection please let me know I would delete it immediately. This question has evolved from ...
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Reference request: semimarkov processes

What are some good modern introductions to the theory of semimarkov processes? To be clear, by a semimarkov processes, I mean a Markov chain, together with "waiting times" between transitions, the ...
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67 views

Hitting times for stochastic convolution integral

Let $N$ be a Poisson measure of intensity 1 on $\mathbb{R}$. To remind you, this is a counting measure on $\mathbb{R}$ such that $$ N(s,t]\sim \mathrm{Pois}(t-s)\quad \forall t>s\in \mathbb{R}$$ ...
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339 views

Is the ito integral $\int_0^t \operatorname{sign}(W_s)\mathrm{d}W_s$ a Brownian motion?

Consider the ito integral of the sign of the Brownian motion $W_s$ from $0$ to $t$: $$\int_0^t \operatorname{sign}(W_s)\,dW_s$$ This appears for instance in the Tanaka formula. I think this is a ...
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Semimartingale property up to a specified time

I have a question about semi-martingale property of stochastic processes. Let us consider the following cusp \begin{equation*} C=\{(x,y)\in \mathbb{R}^2 \mid |y| \le x^{\gamma}\},\quad 0<\gamma&...
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1answer
49 views

order of the singularity of a Green's function to the fractional Laplacian

I was looking at a problem which involves the Green's function of a fractional Poisson equation. To fix notation, let $D\subset \mathbb{R}^n$ very nice, i.e. a hypercube, and \begin{equation} \begin{...
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1answer
117 views

fractional Brownian Motion driven stochastic integrals

We consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$ With a fractional Brownian motion $B^H_{t}$. If $H\neq\...
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14 views

variance of a recursive distribution

I have the following recursive equation, $\tilde{S}(k+1) = \beta \bar{r}(k) + (1-\beta)\tilde{S}(k)$ where $\bar{r}(k) \sim \mathcal{N}(0, I)$. How can I calculate the variance of $\tilde{S}(k)$? (...
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35 views

Ergodicity of differentiated processes

Let $S$ be a vector space, and $X$ a jointly-measurable random process/field with two parameters: $$ X: [0,\infty)\times\mathbb{R}\times\Omega\to S,$$ i.e. $X_{t,\theta}:\Omega\to S$ are random ...
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89 views

Regularity of martingales with respect to spatial parameters

In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...
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26 views

Are Matérn class kernels universal kernels or not?

This is a question that I can't find the solution. I don't know it is a open question or it is a well-known result that can be attained from several lemmas. Here are the definition of Matérn class ...
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1answer
80 views

Under what condition we get back path from signatures in rough path theory?

A link to wikipedia for rough pat theory is: https://en.wikipedia.org/wiki/Rough_path It appears path and signatures has one to one mapping in many cases. I understand that the signature is not ...