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

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79 views

Does a Gaussian process shrink under a contraction map

Let $T \subset \mathbb R^n$, and assume it's a finite set if that helps. Consider the symmetric Gaussian process $(X_t)_{t\in T}$ defined by $X_t = \langle G, t\rangle$, where $G$ is a standard ...
2
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1answer
91 views

Stochastic differential equation associated with an optimal control problem

We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time ...
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22 views

Question about the characteristics of semimartingales

Let $D=D([0,1,R)$ be the space of cadlag (right-continuous with left limits) functions defined on [0,1] and $X:=(X_t)_{t\in [0,1]}$ be the canonical process on $D$, i.e. $X_t(x)=x(t)$ for all $x\in ...
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0answers
45 views

Bounding correlation between blocks of Gaussian stationary process

Let $X_n$ be a stationary Gaussian process with covariance function $\gamma(n)=\mathrm{Cov}[X(n),X(0)]$. Let $\mathbf{X}_p^q=(X_p,\ldots,X_q)$, $s_n^2=\mathrm{Var}(X_1+\ldots+X_n)$, and ...
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126 views

Quadratic variation and predictable quadratic variation for martingales

Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Fix $N$ and consider now a discrete version of this martingale, i.e., the ...
4
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1answer
119 views

Area enclosed by Brownian motion (without winding number)

The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...
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2answers
50 views

SDEs: Bounding the variance of a solution

I've been thinking about something that would seem intuitive, but I haven't really been able to dig a direct answer to. This is a rough draft of it. Let $$X_t = \mu_{X,t} \mathrm{d}t + \sigma_{X,t} ...
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1answer
81 views

Criterion for weak convergence of probability measures on S' or D'

Let $X_n$ in $S'$ and $\mu_n$, $\mu$ in $M(S')$. $S'$ is the space of tempered distributions. I'm looking for a reference that says if $< f, X_n >$ converges in distribution to $< f,X>$ ...
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5answers
874 views

Deep Learning / Deep neural nets for mathematician

I am interested in finding out the math ideas behind the technologies that are under the umbrella of "Deep Learning" or "Deep neural nets". Most of the papers/books that are often quoted in ...
9
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1answer
213 views

a question on 0-1 valued stochastic process

Here's a question on probability theory from a layman (I'm a game theorist). It is very likely that the question will be a straightforward matter for someone who is a probability theorist. I guess I'm ...
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0answers
20 views

a judicious choice of a parametrix expansion for a Kolmogorov equation?

Consider two SDEs: Let $b, \sigma$ be two smooth lipshitz functions on $\mathbb{R}$ and consider $\left \{ X(s), s\geq t \right \}$ the solution of the following SDE $d X(s) = b(X(s)) d s + ...
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202 views

Squaring random Schwartz distributions

Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance $$ \mathbb{E} [\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)} ...
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0answers
64 views

Does the expected spreading of sample paths imply increase in variance?

Consider a sample-continuous stochastic process $\left\{ X_t \right\}_{t \in T}$ s.t. each $X_t$ is real-valued and $$\int_\Omega | X_t(\omega) | ^p \, \mathrm{d} P(\omega)< \infty$$ for all $1 ...
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0answers
39 views

Invariant Girsanov Theorem on a Riemannian manifold

This is somewhat a follow-up on this post. Let $X_t$ be the stochastic process on a compact Riemannian manifold generated by the (possible time-dependent) second-order elliptic operator $L_t$. Let ...
3
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0answers
56 views

Matroid rank decay

Consider a uniform vector matroid $M(0)=U_{m,n}$ of rank $m$ with $n$ points, $n>m>2$ (you can think of it as a set of $n$ points in general position in vector space $F^m$ for some large field ...
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1answer
43 views

Defining a brownian bridge indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
4
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1answer
181 views

Average probability that a random cosine polynomial with bernoulli coefficients is small

Let $P_{n}(t)=\sum_{k=0}^{n}\varepsilon_{k}\cos(kt)$ where $\varepsilon_{i}$ are independent random variables taking values in $\left\{-1,1\right\}$ with equal probability. Is is true that for any ...
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1answer
45 views

Can real-valued Markov processes with continuous surjective sample paths admit a non-trivial “forward-invariant” set?

I have both a more general question (concerning stopping times), and then a more specific application (as described in the title). Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be ...
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0answers
58 views

Is there a generalization of Polya urns to continuous outcome event?

Take for example the simplest model where there are n blue balls and m white balls in an urn. Then, in a first step realization, a white one has been drawn and then c + 1 of this colour had been put ...
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0answers
52 views

Subclass of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ...
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0answers
67 views

Da Prato's notion of Symmetric Operator

For anyone who's familiar with G. Da Prato's books on infinite dimensional analysis, I was wondering if someone could clarify something. In, for instance, "An Introduction to Infinite Dimensional ...
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2answers
199 views

Existence of strong solution to SDEs with non-Lipschitzian drift

Consider the SDE: $$dX_t=b(X_t)dt+dW_t\quad X_0=x$$ If $b$ is bounded Borel function, using Zvonkin's Transform, one can prove there exists a unique strong solution. I want to know if we assume $b$ ...
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34 views

Bounds on moving average process

Let $X_1,X_2,\dotsc$ be a sequence of i.i.d. random variables and define the average process $\{Y_t\}$ as $$ Y_t = \sum_{i=1}^p a_k X_{t-i} $$ with some constants $a_1,\cdots,a_p \in \mathbb{R}$. This ...
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285 views

Minimal expected absolute value of linear combinations of Gaussian random variables

I am interested in the following question. Consider $n$ independent standard normal random variables $g_i$. Cosider a linear combination $w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for ...
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1answer
120 views

Does $E^{x,t}(f(X_T))$ solve a PDE if $f$ is not continuous?

Many books [see below for references] explore the connections between partial differential equations and expectation values. Assume $X$ is a diffusion with generator $A$, then they conclude, that ...
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2answers
85 views

Existence of an invariant measure on an infinite dimensional space via Lyapunov functional

Set-up. Assume that we have a complete separable metric space $\mathcal{X}$ that is not locally compact. Let $V: \mathcal{x} \to [0; +\infty]$ be a functional such that $K_r :=\{x \in \mathcal {X} : V ...
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2answers
78 views

Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)

Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have $$X_n = A_n \cdots ...
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0answers
90 views

How to decide a value of learning rate for Stochastic Gradient Descent?

I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation, $w_{i+1} = w_i + -\eta \nabla ...
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0answers
58 views

Do the Birkhoff averages of a measurable stationary homogeneous Markov process in continuous time “converge to the right limit”?

Let $\,(P_x^t)_{x \in \mathbb{R} , t \geq 0}\,$ be a measurable Markovian family of transition probabilities - that is, a family of Borel probability measures $P_x^t$ on $\mathbb{R}$ such that for ...
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2answers
197 views

$\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?

I am trying to prove the following Lemma, which seems intuitive, but I still have doubts: Lemma Given a Brownian motion $\{W_t,\mathcal F_t:0\le t \le1\}$, two bounded processes, $\mu$ and $\sigma$, ...
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63 views

How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
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82 views

Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mu$ be a probability measure ...
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72 views

Rolling map as a diffeomorphism?

Let $M$ be a (compact) Riemannian manifold and $x \in M$. For a piecewise smooth path $\gamma: [0, T] \longrightarrow M$, we can define Cartan's development map (or rolling map) $$(\Phi\gamma)(t) = ...
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92 views

When an integral with respect to a Poisson point process is finite?

Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ ...
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0answers
56 views

probabilistic interpretation of elliptic equation with mixed boundary condition

I would like to understand the probabilistic interpretation of the following elliptic problem with mixed Dirichlet-Neumann boundary conditions: Let $B := \{ x \in \mathbb{R}^n, \quad \| x \|_2 \leq 1 ...
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64 views

Consistency Conditions of the Kolmogorov Extension Theorem

Kolmogorov's extension theorem allows for the construction of a variety of measures on infinite-dimensional spaces, and its conditions are supposedly "trivially satisfied by any stochastic process". ...
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1answer
152 views

Change of time variable in Wiener process

I'm following a solution of an SDE from here http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf Start with the SDE $$ dX_t = \delta dt + 2\sqrt{X_t} dW_t $$ consider a deterministic time change ...
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30 views

References for symmetric α-stable process (SSP) for $a>2$

Many properties of Brownian motion have been extended to SSP's for $0\leq \alpha\leq 2$ and so it is quite easy to find literature on them. However, I am currently studying the SSP for $\alpha>2$ ...
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0answers
92 views

Equivalence of two non-degenerate Gaussian measures on Banach space

The motivation of this question is to show that two probabilities on $C_{0}^{n}(0,1)$ (the space of continuous $\mathbb R^{n}$ valued process on $[0,1]$ starting from zero) induced by two ...
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41 views

Killing a Feller Process

Given a canonical Feller process $(X_t,P_x)$ with Feller semigroup $P$. Let $T$ a (good) stopping time, for example $T=\inf\{u\ge 0 : X_u=0\}$. I'm looking for a proof of the following claim ...
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160 views

Version of Ito's lemma applied to a stochastic function

The Ito's formula stated in most books in stochastic calculus is in the form $F(t,X_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and ...
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2answers
119 views

Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds

I am studying BM on Riemannian manifolds and I am curious how this theory started. In the references below (esp. in Hsu's exposition), you will find many applications of that theory such as a ...
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1answer
70 views

Correlation between two continuous-time stochastic processes

Consider two continous-time stochastic processes $\{A(t)\}_{t \ge 0}$ and $\{B(t)\}_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For ...
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87 views

Convergence in distribution of random measures

Let $M$ denote the space of real Radon measures on $\mathbb{R}$ as the topological dual of $C_c(\mathbb{R})$ equipped with the inductive limit topology (for possibly unbounded Radon measures) or ...
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110 views

On the decay of correlations of an ergodic sequence over the set $X_{0}=0$

The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...
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3answers
338 views

Reference request: a guide through quantum probability

Could you point out a comprehensive reference book (or more than one, if it is the case) on Quantum Probability that introduces the subject and then gradually builds up to the edges of contemporary ...
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31 views

Reference request for specific POMDP examples

Following is strictly for discrete-time discrete-space Markov chain. Consider a partially observed Markov decision process (POMDP) $P = \{X,O,A,P,B_a\}$. Here $X = \{x_1, \cdots, x_n\}$ refers to ...
3
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1answer
243 views

Convergence of random variables with hypergeometric distribution

This is a very interesting conjecture of large scale property of hypergeometric distribution. Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...
0
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1answer
225 views

Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$ where ...
0
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1answer
108 views

Question about the stochastic integral of martingales

Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $R$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e. ...