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

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41 views

About the rate for one approximation to the heat kernel

Let $G(t,x)$ be the heat kernel $$ G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}. $$ Here is one approximation of $G(t,x)$: $$ G_\epsilon(t,x)=e^{-t/\epsilon} \sum_{k=1}^\infty ...
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0answers
36 views

A counterpart of Karhunen theorem

According to the Karhunen theorem, if the correlation function of a process $X(t)$ can be represented as $$ R(t,s)= \int_{\Lambda} f(t, \lambda) \overline{f(s, \lambda)}d\nu(\lambda) $$ then the ...
1
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2answers
81 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
2
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1answer
82 views

M/M/1 Queue with probability of new customer leaving

I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = ...
2
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0answers
108 views

Heat kernel and Wiener measure

A theorem by Barry Simon says that for arbitrary open sets $\Omega\subset \mathbb{R}^n$, we have $$[\exp(t\Delta_{\Omega}^D)](x,y) = \mu_{x,y,t}\lbrace \omega \text{ } \vert \text{ } \omega(s) \in ...
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0answers
73 views

number of times Brownian motion hits boundaries

Any experts here please direct me to some appropriate keywords that I can search for. Consider a Brownian motion constrained to an upper and lower boundaries. Let's say I want to know that how many ...
2
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3answers
62 views

a special filtration satisfying $0$-$1$ law

Let $\xi$ be a uniformly random variable on $[0,1]$ defined on some probability space $(\Omega,\mathcal{F})$. Define the process $\xi_t:=\min(\xi,t)$ for $0\le t\le 1$. And let ...
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0answers
42 views

Quantiles moments and Convergence

QUESTION: Let $F$ be an absolutely continuous distribution function with density $f$, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
2
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1answer
73 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 ...
1
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1answer
55 views

A calculation involving a uniform random variable quantile

THE PROBLEM: Let $U$ be a uniform distribution and $U_{n}$ be its nth empirical distribution. Suppose $t\in (0,1)$ and $n\in \mathbb{N}$ are constants. What's the explicit expression to ...
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1answer
65 views

question about uniform continuity under Skorokhod Metric

Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see: http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g ...
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1answer
230 views

simultaneous jumps of independent Levy processes

Suppose I have two independent Levy processes $X_t$ and $Y_t$, both not continuous. Is anyone familiar and can refer me to a result(or a counterexample) which states that ${\displaystyle \sum_{0\leq ...
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1answer
53 views

Running supremmum of a Levy process

Let X be a cadlag Lévy process with $X_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent. 1): $X$ is $L^p$-integrable. 2): $X^*_t= \mathop{\sup}_{0\leq s\leq t} ...
5
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1answer
919 views

Gluing Markov processes

I am looking for a reference on the gluing together of strong Markov processes to get a new one. Here is an example of what I have in mind. Let $B^1, B^2, \ldots $ be independent one-dimensional ...
5
votes
1answer
296 views

On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations)

Suppose that $(\Omega,\mathscr{F},P)$ is a complete probability space equipped a filtration $\{\mathscr{F}_t\}$ satisfying the usual conditions. $B_t$ is a 1-dimentional Brownian motion with respect ...
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24 views

a question about Dambis, Dubins-Schwarz Theorem

Let $M=(M_t)_{0\le t\le 1}$ be a continous $\mathbb{F}=\{\mathcal{F}_t\}_{0\le t\le 1}$-martingale s.t. $M_0=0$. Now my question is whether there exists a Brownin motion $B$ s.t. ...
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1answer
314 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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0answers
53 views

Fredholm Integral Involving Stochastic Process

I wish to solve an integral equation of the form $$g(X) = c\int_0^1 K(X,t)f(t) \ dt $$ where $f\in L^1([0,1])$ and $g$ is some function on finite sequences of random variables. So, $X$ is a stochastic ...
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0answers
19 views

what is the autocorrelation function for a continuous autoregressive model

Suppose the stochastic process h(t) follows the following dynamics: $\dot{h}(t)=a h(t)+z(t)$ where $a<0$ and $z(t)$ is the standard complex Wiener process. What is the autocorrelation for ...
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0answers
18 views

How to get expectation of function of an optimal stopping time

Let $P_t$ be the posterior probability, $p_0$ be the prior probability. The evolving process of $P_t$ is: $ dP_t=\frac{P_t}{P_t+(1-P_t)(1-\lambda^kdt)}-P_t$. The optimal stopping time problem is ...
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1answer
53 views

explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition: $$V_t=V_0+\int_0^tH_sdX_s-K_t$$ where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...
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28 views

a question about the modification of a supermartingale

Let $\mathbf{D}\subset\mathbf{D}([0,1],\mathbb{R}_+)$ denote the space of positive cadlag functions $\mathbf{x}$ defined on $[0,1]$ with $\mathbf{x}(0)=1$. Define the canonical process ...
6
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3answers
234 views

Markov Processes with Given Marginals

Let $\mu_t, t \geq 0,$ be a family of probability measures on the real line. One can assume whatever one wishes about them, although typically they will be continuous in some topology (usually at ...
2
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1answer
72 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 ...
5
votes
1answer
214 views

Convergence rate for product of stochastic matrices

Hi, I have a system of the form $$x(t+1) = A(t + 1) x(t),$$ for $t \geq 1$, and some fixed initial condition $x(1)$. Here $A (t)$ is a time-varying $m \times m$ matrix that is stochastic at all ...
2
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1answer
85 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?
3
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1answer
141 views

Law of Iterated Logarithm for autoregressive process

Suppose that $\{X_i\}$ is an $\mathrm{AR}(r)$, defined by: $X_{i}= h(i) + \varepsilon_i $, $h(i)=\alpha_1 X_{i-1} + \dots + \alpha_{r} X_{i-r}$ where $\{\varepsilon_i\}$ are i.i.d. ${\cal ...
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0answers
71 views

a question about integration by parts

Let $X$ be a cadlag martingale et $Y$ be a process of bounded variation, do we have the integration by parts formulae? $$\int_0^1Y_tdX_t=X_1Y_1-X_0Y_0-\int_0^1X_tdY_t,~ a.s.$$ Thanks for the reply!
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2answers
88 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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61 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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118 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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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 ...
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0answers
84 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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2answers
177 views

Probability of winding number of 2D Brownian Motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
3
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0answers
65 views

Ito Integral of a SDE [on hold]

I would like to get help in solving the following It\^o stochastic equation: $dY_t=-dW_t \, (Y_t^2+1)$ The process $W_t$ is the standard Brownian motion. If possible I would like to get a path ...
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1answer
94 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 ...
2
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0answers
79 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= ...
7
votes
3answers
233 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 ...
3
votes
1answer
136 views

log-likelihood of ito diffusion

Consider a diffusion process: $ \text{d}X_t = f(X_t)\text{d}t + \text{d}W_t$ I've seen it given that the log-likelihood of the path is proportional to the Onsager-Machlup functional $ \int_0^T ...
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66 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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0answers
74 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 ...
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1answer
74 views

Transition probabilities in coupled CTMCs

I know that for a CTMC, 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 CTMCs evolving simultaneously. Each of the $N$ CTMCs can ...
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0answers
58 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
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0answers
63 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 ...
5
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1answer
104 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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39 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 ...
2
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1answer
50 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, ...
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65 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 ...
2
votes
1answer
123 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 ...
6
votes
2answers
393 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 ...