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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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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**2**answers

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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**1**answer

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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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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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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**3**answers

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?

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**1**answer

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 ...

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**1**answer

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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**1**answer

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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**4**answers

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 ...

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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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**0**answers

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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**1**answer

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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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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**2**answers

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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**1**answer

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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**0**answers

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) ...

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**0**answers

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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**1**answer

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 ...

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**1**answer

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 ...

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**0**answers

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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**0**answers

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 ...

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**0**answers

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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**1**answer

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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**1**answer

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 ...

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**1**answer

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 ...

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**1**answer

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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**2**answers

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 ...

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**2**answers

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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**1**answer

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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**1**answer

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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**1**answer

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 ...

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**1**answer

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 ...

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votes

**1**answer

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 ...

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**1**answer

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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**1**answer

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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**1**answer

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 ...

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**0**answers

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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**2**answers

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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**0**answers

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 ...

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**1**answer

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 ...

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**1**answer

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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**2**answers

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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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 ...

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votes

**1**answer

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 ...

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**2**answers

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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**1**answer

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 ...