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

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1answer
120 views

Ergodicity for the mean of a linear process without finite second moment

Suppose that $\{X_k:k\in\mathbb Z\}$ is a linear process, i.e. a sequence of random variables such that $$ X_k=\sum_{j=0}^\infty\psi_j\varepsilon_{k-j} $$ for each $k\in\mathbb Z$, where ...
0
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0answers
72 views

Uniqueness of a Integro-parabolic differential equation?

Let $r, q,\lambda,\sigma,\kappa,\mu$ are positive real numbers and let $c(t)$ is a differential function of $t$. $\Gamma(\eta)$ is a probability density function. When I consider price of American ...
2
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2answers
182 views

Uniqueness in martingale representation theorem

Dudley's martingale representation theorem states that if $W=\{W_t,\mathcal{F}_t;0\le t<+\infty\}$ is a standard one-dimensional Brownian motion, $0<T<+\infty$ and $\xi$ is ...
1
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0answers
59 views

Stochastic Resonance in Infinite Dimensions

I'll ask this from the point of view of physics more than of theoretical mathematics. I'm searching for a mathematical discussion of stochastic resonance interpreted in a PDE sense. This is a good ...
0
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0answers
24 views

Density of $\int_{B}\frac{|1-|B_{T}|^{2}|}{|y-B_{T}|^{3}}dS(y)$

For $B\subset \partial B(0,1)))$ and random variable $B_{T}\in Int(B(0,1))$ with density $p_{T}$, is there a density for $\int_{B}\frac{|1-|B_{T}|^{2}|}{|y-B_{T}|^{3}}dS(y)$? Context The original ...
3
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1answer
78 views

Could quadratic variation determine distribution?

Let $M=\{M_t,\mathcal{F}_t;0\le t<+\infty\}$, $N=\{N_t,\mathcal{F}_t;0\le t<+\infty\}$ be two continuous local martingales with $M_0=N_0=0\text{ a.s.}$. If $\langle M\rangle=\langle N\rangle$, ...
0
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0answers
42 views

Finding a stochastic differential equation as limit of a discrete stochastic equation

I'm dealing with the following problem: Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation: $Z_{k+1}-Z_k=P_k(1-2Z_k)$ where $P_k=0$ with probability ...
6
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2answers
260 views

A version of Wald identity

Let $W$ be a standard one-dimensional Brownian motion. Let $T$ be a stopping time with $\mathbb{E}\sqrt{T}<+\infty$. Then $$\mathbb{E}W_T=0\quad \mathbb{E}W^2_T=\mathbb{E}T$$ I can prove these ...
2
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0answers
61 views

Existence of 1-1 mapping/homeomorphism

Let $B$ be a standard 2-D Brownian motion, and $\sigma: \Omega\times \mathbb R^{+} \mapsto \mathbb R^{2 \times 2}$ is an $\mathcal F_{t}$ adapted process satisfying, for some constants ...
2
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0answers
48 views

What is the probability of B.M. hitting two disjoint spheres $(d\geq 3)$?

The hitting probability for spheres centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$, where $|x|>r$. 1)So I was wondering how can one compute ...
3
votes
1answer
127 views

Do we need Feller condition if the process jumps?

Consider the SDE: \begin{equation} dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} \end{equation} It describes a process $v_t$ which is a strictly positive if the drift is stronger enough, i.e. ...
1
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0answers
115 views

How to show that two linear combinations of Bernoulli random variables have jointly Gaussian distribution (and more)

Let $X_1,\ldots,X_n$ be independent Bernoulli random variables such that $\mathbb{P}(X_i=\pm 1)=1/2$ and consider two collections of real numbers $a_1,\ldots,a_n, b_1,\ldots, b_n$. For the moment let ...
2
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0answers
99 views

Speed of Approach to Invariant Measure

Let $X_t$ represent a continuous-time Markov process on $\mathbb{R}^d$, say a diffusion with locally Lipschitz coefficients. Suppose that there exists a unique invariant measure $\mu$ on the space, ...
2
votes
1answer
135 views

Onsager-Machlup function and most probable path of a diffusion process

Let $X_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation \begin{equation} dX_{t} = f(X_{t})dt + dW_{t}, \end{equation} where $f \in C_{b}^{2}(R)$ is a ...
1
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1answer
131 views

Does very fast convergence in probability imply almost sur convergence for a continuous stochastic process?

I was wondering if someone knows how to prove the following fact (which might not be a fact ;) ): let X being a stochastic process with almost surely continuous sample path, and such that, there ...
2
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0answers
68 views

Property of relative entropy [closed]

For $X$ a measurable space and $P,Q$ two probability measure on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as $$D(Q\|P)=\int_X ...
-1
votes
1answer
119 views

Property of relative entropy [closed]

For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as $$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$ ...
2
votes
0answers
44 views

Local time for drifted Brownian motion and comparison results for reflected diffusion

Suppose $X(t) = x+ \mu t + \sigma W(t)$ where $x\ge 0$, $\mu, \sigma>0$ are real constants, and $W$ is a standard Brownian motion. The Skorohod decomposition of $X(t)$ can be written as $Z(t) = ...
5
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0answers
187 views

A generalization of Jensen's Inequality

Jensen's inequality is well known as $$E\big[f(X)\big]\le f\big(E[X]\big)$$ where $X$ is a integrable random variable and $f: R\to R$ is a bounded concave function, see also ...
3
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61 views

Lorenz attractor power spectrum

If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...
10
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1answer
273 views

Does Brownian motion immediately visit both sides of a Jordan curve?

Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior). ...
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0answers
30 views

sign and absolute value at fixed time of a diffusion process

I have a diffusion in the plane $(X,Y)$ with Feller semigroup such that each coordinate is a standard Brownian motion, $|X|=|Y|$ and $(X,Y), (Y,X)$ have the same law. I want to prove that for a ...
2
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0answers
76 views

Sum of the entries of the inverse covariance matrix

Let $T \in\left(0,1\right)$, $n\in\mathbb{N}$ and $e_n = [1,\ldots,1]\in\mathbb{R}^n$. Consider the covariance matrix $\mathfrak{A}_n = ...
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0answers
81 views

What is the sigma field of the derivative of a process?

When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in ...
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92 views

Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$. Let $X$ be random function defined ...
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1answer
45 views

Does $L^2$ progressive measurable processes form a Hilbert space?

Let $(\Omega, \mathcal F_1, {\mathbb P}, \mathbb F = \{\mathcal F_t\}_{0\le t \le 1})$ is a filtered probability space. Let $L^2_{\mathbb F}$ be a collection of all $\mathbb F$ progressive measurable ...
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0answers
54 views

When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$ ...
3
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1answer
63 views

Density for Translated Process

Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito ...
3
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136 views

Donsker's Theorem for triangular arrays

I should mention that I already posed this question on Math Stack Exchange, but didn't receive much feedback. Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given ...
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0answers
96 views

First passage time of a pure drift process

I am facing the following unusual problem: $Z_t$ is a pure drift process of the form $$ dZ_t = \kappa(X_t - Z_t) dt $$ where $X_t$ is another bounded process. I am interested in computing / ...
3
votes
0answers
95 views

Ask for reference of a stochastic process

I would like to know whether the following stochastic process is well studied. Let $\{U_k: k \ge 1\}$ be a sequence of i.i.d random variable. $U_1$ is uniformly distributed on the unit interval $[0, ...
5
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0answers
161 views

A note on Doob's theorem

I have faced the following problem, regarding to the Martingale Theory. Because this area far from my area I don't know whether this problem is in literature or this can be simple question for ...
1
vote
1answer
88 views

Perturbation of a Bessel process of dimension 2

Bessel process of dimension 2 is defined to be solution of $$ dX_t=dB_t+\frac{1}{2X_t}dt,\quad X_0=x_0>0 $$ where $B$ is a standard 1-dimensional Brownian motion. $X$ can be viewed as the norm of a ...
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0answers
16 views

Is it posible to differentiate the mean function of Gaussian process regression with respect to its h

The mean function $\hat{\mu}(x_*)$ of Gaussian process regression is given by $k(x_*, X)(k(X, X) + \sigma^2_w I)^{-1}Y$ where $k(\cdot, \cdot)$ is a kernel matrix or vector of appropriate size and is ...
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22 views

Literature on the notion of combining two discrete stationary processes with the latter process slowed down

Is there any literature about the following way of combining two stationary processes? Let $X_1, X_2, \dots$ be a discrete-time stationary process. Let $A$ be a subset in its sample space. Let ...
5
votes
2answers
273 views

Regularity of random Fourier series

The following two statements appear to be true (but do correct me if I am wrong): The coefficients of a $C^k$ function on the torus $T^n$ decay at least as fast as $x^{-k}$ (where $x$ is some norm ...
0
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0answers
37 views

Does a singularly perturbed cadlag process has sample paths in a Polish space?

In the theory of stochastic processes it is often said in the broader literature that Polish state spaces are the only important ones appearing in practice. Are there also examples of stochastic ...
0
votes
1answer
89 views

Functional representation of adapted jointly measurable stochastic processes

It seems like the question stated here in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO. Let $X_t : \Omega \to E, \ t \geq 0$ be ...
3
votes
1answer
115 views

Ising model: probability of a long path of minus under plus boundary conditions

Consider for example the Ising model on a square lattice. Fix zero magnetic field and plus boundary conditions. Low temperature, one minus spin. With a Peierls argument one can prove that, given a ...
2
votes
0answers
111 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
0
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1answer
189 views

Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...
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48 views

Question about Skorokhod embedding problem

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion on some probability space. Now for every centered probability distribution $\mu$ on $R$, i.e. $\int_{R}|x|d\mu(x)<+\infty$ and ...
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0answers
51 views

Steady state of a dynamical equation

Suppose we have the following dynamical equation: $P(k+1)=A\bigg(P(k) - P(k)H^T(k)\big(H(k)P(k)H^T(k)+Z\big)^{-1}H(k)P(k)\bigg)A^T+W$ with $P(0)=0$, where $P$, $A$, $H$, $Z$, $W$ are all $N\times N$ ...
3
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0answers
92 views

Numerical Methods for stochastic PDE, from rough paths to backward equations

this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular, Have the numerical implications, if any, of the results in ...
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57 views

Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit: $$ dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC $$ where $W$ is a standard Brownian motion ...
1
vote
1answer
162 views

Diffusion processes with different diffusion coefficients and absolute continuity

I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level. My initial situation is the following. Consider two stochastic ...
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43 views

Karhunen Loeve expansion of $cos(\theta)$ where $\theta$ is a Gaussian random process or Uniform distribution in $[0,\pi/2]$]

I want to expand the random process $\theta$ using KL expansion for uncertainty quantification using stochastic FEM. But my random variable is function of cosine. i.e. $cos(\theta)$. My pde has ...
0
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1answer
82 views

Could somebody recomends a good book or article about numerical methods for Stochastic Partial Differential Equations

Could somebody recomend a good book or article about numerical methods for Stochastic Partial Differential Equations. I'm looking for a good introductory material thanks.
2
votes
1answer
103 views

Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)

I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research ...
7
votes
3answers
320 views

A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst

I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability. A quick google search gave a lot of references on SLE ...