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

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91 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
78 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
159 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
92 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 ...
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166 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
59 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
80 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 ...
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3answers
356 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
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1answer
301 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
234 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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1answer
176 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 ...
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4answers
349 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 ...
3
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0answers
54 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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0answers
97 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
203 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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0answers
71 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 ...
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2answers
314 views

Ruin time for a two-input “risk only” slot machine

Imagine a "risk only" slot machine that takes 'coins' corresponding to some real number fraction of a dollar $p$, returns the coin with probability $p$, and eats the coin with probability $(1-p)$. ...
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1answer
87 views

construction of a approximate martingale

everyone. Given a probabilistic space $(\Omega, \mathcal{F}_t, \mathbb{P})$ and a martingale $(M_t)_{t\leq 1}$ on it. Suppose $$M_1\stackrel{\mathbb{P}}{\sim}\mu$$ where $\mu$ is a probability ...
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0answers
134 views

The problem of the drunkard in a valley [closed]

We consider a Markov chain on a subset of positive integers S = {0, 1, 2, 3, .......N}, with transition probabilities defined as follows: The chain jumps only one unit to the left or right. p(i, j) ...
4
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0answers
118 views

compactness of a probability set

I have a question about the compactness of a set of martingale measures. Let $\Omega=\mathcal{C}[0,1]$ be the space of continuous functions on $[0,1]$ and $\mathcal{M}_{\Omega}$ be the family of ...
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45 views

Correspondence between viscosity supersolution and supermartingale

Suppose $b : \mathbb{R} \to \mathbb{R}$ and $\sigma: \mathbb{R}\to \mathbb{R}$ are Lipschitz and that $(X_t)_{t\ge0}$ is a diffusion with $X_0 = x_0$ and $dX_t = b(X_t)dt + \sigma(X_t)dW_t$ . ...
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1answer
105 views

Kalman filter with long term bias

I was reading about the Kalman filter and I do not understand how it should be used when our measurements have a long term offset like GPS location updates do. As I understand, the Kalman filter ...
2
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1answer
192 views

Stochastic integral with respect to discontinuous martingale

in my research, I need to deal with a stochastic integral with respect to a compensated poisson process, namely, $ \int_0^t f(X_t) dM_t,$ where $M(t) = N(t) - \int_0^t \lambda(s)ds$. The integrand ...
2
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0answers
97 views

Supremum of scaled Brownian motion shifted by bounded smooth function

I would like to ask the following question: Let $e$ be a nonnegative, bounded, smooth function from $\left[ 0,1\right] $ to $\left[ 0,1\right] $, such that $e\left( 0\right) =e\left( 1\right) ...
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0answers
114 views

Brownian particles in a box: the probability that a sphere (of some radius) centered on a particle only contains one particle for a duration of time

Imagine I have a set of $(s_1,...,s_N) \in S$ Brownian particles in a box of sidelength $L$, each with the same coefficient of diffusion $D$. We fix one particle at the center of the box, and draw a ...
4
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185 views

Generalized Markov Processes on CW complexes of dimension > 1

Markov processes have a large variety of applications to physics and chemistry (as well as many other fields). Such processes are formulated on graphs, i.e., CW complexes of dimension one. It is ...
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1answer
104 views

Billingsley Ch: $4$ Asymptotically Independent random elements

A NEW DOUBT arose in a previous question, so I am posting it again since the previous edits did not draw views. Let $X_n$ be random elements of $D$ (space of cad lag functions on $[0,1]$ as domain). ...
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1answer
165 views

Asymptotically independent increments random elements: Billingsley Ch:$4$

Let $X_n$ be random elements of $D$ (space of cad lag functions on $[0,1]$ as domain). $X_n$ has asymptotically independents if $0\leq s_1 \leq t_1 \leq s_2 \leq \ldots < s_r \leq t_r \leq 1$, then ...
4
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1answer
262 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 ...
4
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1answer
286 views

Donsker Theorem Billingsley

Theorems $16.1$ and $16.3$ in Billingsley Convergence of measures. $16.1$ reads : Random variables $u_1,\ldots$ on $(\Omega,\mathcal{B},\mathbb P)$ and are i.i.d. with $0$ mean and finite variance ...
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2answers
532 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 ...
4
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2answers
208 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 ...
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1answer
208 views

Quadratic variation for discrete Martingale

Is there any analogue of continuous martingale quadratic variation for the discrete case? If so, are there any theorems which characterize simple random walk using quadratic variation - similar to ...
4
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1answer
316 views

Expectation of the time t standard brownian motion stopped at itself's square

I have a one dimensional standard brownian motion $W$ defined under a stochastic basis with probability $\mathbf{Q}$ and filtration $\left(\mathscr{F}\right)_{t\in{\mathbf{R}}_{+}}$, and I want to ...
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1answer
129 views

Continuity of caglad process

Consider a non increasing, caglad process $(X_t)_{t\geq0}$ such that, for each $t$, the distribution function $F_t(x):=P(X_t\leq x)$ is a continuous function of (real) $x$. Are there any sufficient ...
3
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1answer
131 views

Example of Girsanov change of density with finite relative entropy, but with infinite integral over squared changed drift

Let $(\Omega, (\mathcal F_t), \mathbb P)$ denote the usual Wiener space where $\Omega = C[0,\infty)$, etc., and where $(W_t)_{t \geq 0}$ denotes the Wiener process. Let $Z \in L^1(\mathbb P)$ with $Z ...
3
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1answer
269 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 ...
2
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1answer
190 views

Finding the Levy triplet of a Levy process

I know the levy triplet of a Poisson process $N_t$- $(0,0,\lambda\delta_{1}(y))$ and its characteristic function is ...
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1answer
106 views

Is the linear span of the Neumann eigenfunctions dense in $C(\overline{D})$

Let $D\subset R^d$ be a bounded Lipschitz domain. We know that the Neumann eigenfunction lies in $C(\overline{D})$ (i.e. continuous up to the boundary). This can be seen from the fact that ...
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1answer
195 views

$L^2$ convergence of a tight sequence [closed]

Let $(X_n,n\geqslant 1)$ be a tight sequence of stochastic processes defined on the same probability. Suppose $\lVert X_n\rVert_{L^2}$ converges to $\lVert X\rVert_{L^2}$. Under what conditions do we ...
2
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0answers
237 views

distribution of integral of exponential of wiener process

I am absolute newbie to stochastic calculus and have to solve a weighted hazard rates integral, where the hazard rates are stochastic, their logarithm governed by arithmetic Ornstein-Uhlenbeck (OU) ...
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2answers
412 views

Does the strong law of Large Number hold for an infinite dimensional Brownian motion?

For finite-dimensional Brownian motion $W_t$, it is well known that \begin{equation} \lim_{t\to \infty}\frac{W_t}{t}=0,\text{ a.s. }\ \ \ \ \hspace{1cm} \langle 1\rangle \end{equation} Now suppose we ...
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0answers
150 views

Upper bound on expectations of the sum of product of a martingale difference sequence with a predictable sequence, weighted by certain random weights

Let $(\mathcal{F}_i)_{i\geq 1}$ be a filtration. Let $0\leq p_i\leq 1$, be a random variable measurable w.r.t. $\mathcal{F}_i$. Consider two sequences of random vectors ...
3
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1answer
330 views

Path integrals for stochastic equations

Does there exist a rigorous mathematical proof for path integral representations given in the physics literature? See for example http://arxiv.org/abs/hep-ph/9912209v1 For imaginary time rigorous ...
2
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1answer
172 views

Total variation distance between diffusion processes with different volatility coefficient

Preamble: This question is similar to the one in total variation distance between two solutions of SDE . The difference is that in my case the drift is the same but there are different diffusion ...
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2answers
113 views

Distribution similar to PPP

According to the definition of Poisson Point Process, I can't define a certain number of nodes which are distributed in an area as PPP. Is there a distribution (a certain number of nodes distributed ...
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0answers
110 views

Is there a theory of SDEs whose coefficients are themselves adapted processes (i.e. “may depend on the past”)?

Is there an existence and uniqueness theorem for SDEs of the following type: $dW_{t}=d\tilde{W}_{t}+\mu\left(\left(W_{s}\right)_{0\le s\le t},t\right)dt$, where $\tilde{W}_{t}$ is say ...
2
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1answer
180 views

Uniform bound on the rate of convergence of the renewal measure

Consider a renewal process whose holding times are given by a continuous random variable $X$ supported on $[0,1]$. It is known (e.g. Stone '65) that the renewal function $m(t)$ converges to ...
11
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2answers
391 views

Error term for renewal function

Consider a sequence of independent uniform $[0,1]$ random variables, and for nonnegative real $t$, let $m(t)$ be the expected number of terms in the first partial sum that exceeds $t$. For instance ...
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1answer
266 views

Markov Chain: state reduction

Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following: Firstly we have a Markov chain $\{Y_k\}$ with finite ...