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

**13**

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105 views

### Does a theory of stochastic differential algebras exist?

My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...

**0**

votes

**0**answers

56 views

### Circular process ergodic?

Let us define a continuous-time Markov process on a circle consisting of $m-$ equally spaced points, i.e. every point has two neighbours.
Now, we define a space of functions $S:= ...

**0**

votes

**1**answer

85 views

### Weak convergence of process

Background:
I am trying to compute the weak limit of the following model from mathematical biology that is supposed to exist:
Let $$L(f)(\eta)= \sum_{x \in \mathbb{Z}}\frac{1}{2}\left(1_{\eta(x+1) ...

**1**

vote

**0**answers

81 views

### Malliavin differentiability of solutions to SDEs

In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...

**3**

votes

**1**answer

104 views

### Carre du Champ, Subunit Paths and CC-metrics

Let the operator $L$ be given by $Lf(x):=\nabla\cdot (A\nabla f(x))$, where $f:\mathbb{R}^d\rightarrow \mathbb{R}$ belongs to a suitable class of functions $\mathcal{A}$. The carre du champ operator ...

**-2**

votes

**1**answer

62 views

### Definition: Grigelionis Process?ch [closed]

Background
I've been reading this article and it keeps referring to "Grigelionis processes", which apparently generalize Levy processes. However the paper does not define these object clearly and ...

**1**

vote

**1**answer

107 views

### Malliavin derivative under change of measure

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$.
Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...

**2**

votes

**1**answer

60 views

### Quadratic variation and the variance of a semimartingales

I will describe an example that seemingly contradicts the following
Theorem
For a local martingale $M$, let $[M,M]_t$ be its quadratic variation at $t$. For any $t$, if $E[[M,M]_t]<\infty$, then ...

**0**

votes

**0**answers

48 views

### Regularity of the entrance measure of SRW

Let $S(n)$ be the discrete sphere of radius $n$ (i.e., the internal boundary of the Euclidean discrete ball $B(n)$) centered in the origin, and consider a simple random walk starting at some ...

**1**

vote

**0**answers

89 views

### Is a local martingale with constant expectation necessarily a martingale?

Suppose $X\in \mathbb R$ is a weak solution to the SDE $dX_t = \sigma(X_t)dW_t$, in which $W$ is a one-dimensional Brownian motion, and $\sigma$ is Borel measurable so that a weak solution exists and ...

**3**

votes

**2**answers

167 views

### Brownian motion in $\mathbb{R}^n$, probability of hitting a set

Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying ...

**1**

vote

**0**answers

81 views

### Example of progressively measurable process that is not predictable

Is there an example of progressively measurable process that is not predictable?
This question is motivated by Revuz-Yor, Continuous Martingales and Brownian Motion ...

**0**

votes

**0**answers

61 views

### Order statistic of Markov chain sample path and related probabilities

Consider a 1D sample path, denoted as $\{X(1), ..., X(t), ..., X(n)\}$, generated from a discrete time finite state (time homogeneous) Markov chain over states $\{1,...,m\}$, with transition ...

**2**

votes

**2**answers

132 views

### Total absolute variation of brownian motion, with different sampling rates

Let $(B_t)$ be a brownian motion on [0,1]. For the following, let $\omega$ be fixed.
Let's compute the total absolute variation when sampling period = $\delta$ is fixed:
$$V(\delta) = ...

**1**

vote

**0**answers

31 views

### Comparison between the entrance measure and the harmonic measure

Consider the standard two-dimensional Brownian motion, and define $\tau(A)$ to be the hitting time of $A\subset \mathbb{R}^2$. Let $hm_A$ be the harmonic measure (from infinity) on $A$. Let $B(r)$ be ...

**3**

votes

**1**answer

112 views

### Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?

Denote by $P_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by ...

**1**

vote

**1**answer

58 views

### “Convergence speed” results for the Langevin process

The Langevin process is defined by the following stochastic differential equation:
$$ \dot X = - \nabla \phi + \sqrt 2 dW_t $$
Its equilibrium distribution is the following:
$$ p_\infty (x) \propto ...

**1**

vote

**0**answers

97 views

### Full version of Soucaliuc's research announcement “Réflexion entre deux diffusions conjuguées”

Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes:
[1] F. Soucaliuc, Réflexion entre deux diffusions ...

**2**

votes

**0**answers

86 views

### Laplace transform of a integral function of CIR/CEV process

The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...

**1**

vote

**1**answer

101 views

### Feller processes / probability generators

I am looking for a example of a function in $C_0(\mathbb{R})$ such that $f',f'' \,\text{and}\, f''' \in C_0(\mathbb{R})$ with
$$ \inf f < \inf (f-a*f''')$$ for some $a>0$, but I couldn't find ...

**0**

votes

**0**answers

105 views

### Comparison of Parameter estimation using maximum likelihood and Maximum entropy

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...

**0**

votes

**0**answers

28 views

### steady state distribution for a jump Markov chain

Consider a queueing process with the following transition matrix:
$\mathbf{P}=\left( \begin{smallmatrix}
1-\lambda & \lambda & & & & & & &\\
\mu & ...

**4**

votes

**1**answer

165 views

### Random walk with continuously distributed steps on [-1,1]

A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability
$$P(S_n \textrm{ reaches } a \textrm{ before} -b) ...

**0**

votes

**1**answer

86 views

### Finite hitting time implies hits at any finite time?

I was wondering about the following problem:
Assume we have a state space $S:=\mathbb{Z}$ and a Markov chain, such that we can go from any state $x$ to some state $y$ with positive probabilities, ...

**0**

votes

**1**answer

82 views

### Weak existence for modified Tanaka SDE

Tanaka's theorem (wikipedia) implies that $X_t = |B_t|$ is a weak solution to the SDE
$dX_t = dW_t + dL_t^0(X_t)$,
where $W_t$ is a Brownian motion and $L_t^0(X_t)$ is the local time of $X_t$ at ...

**0**

votes

**1**answer

110 views

### Markov chain with Feller property

Does anybody know whether there is an analysis of when the monotone decreasing chain has the Feller-property?
The monotone decreasing is defined as a chain on $\mathbb{N}$ and the rate of going down ...

**0**

votes

**0**answers

52 views

### hitting time of an Ornstein-Ulhenbeck

If we consider a nice Ornstein Uhlenbeck process
$d x (t) = - \gamma x(t) dt + \sigma d w (t)$
with $x(0) = x_0 \in (-L,L)$.
Here $\gamma, \sigma$ are positive constant and $w(t)$ is a Wiener ...

**7**

votes

**1**answer

152 views

### approximate stationary distributions of a doubly stochastic matrix and its supports

Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...

**4**

votes

**0**answers

168 views

### Existence or construction of a sequence of orthogonal matrices with three properties

This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help ....
Any pointers or suggestions are appreicated!
...

**2**

votes

**0**answers

55 views

### Reference for branching processes

A popular model of a continuous time branching process was introduced around 1970, which is now called the Crump-Mode-Jagers branching process, was introduced here:
A General Age-Dependent Branching ...

**2**

votes

**1**answer

150 views

### using Feynman-Kac formula

I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ...

**3**

votes

**0**answers

39 views

### Memorylessness of residence times for a Markov process

I'm stuck on the trivial problem of showing memorylessness of holding (residence) times for a continuous time homogeneous Markov chain on finite state space.
I have a homogeneous Markov process ...

**0**

votes

**0**answers

67 views

### Random process & probability problem met in wireless communication

A random process r obeys the following distribution:
$p(r,ṙ)=rb_0\exp(−\frac{r^2}{2b_0})\sqrt{2πb_2}exp(−\frac{\dot{r}^2}{2b_2})$, where $\dot{r}$ is the derivative of r in the time domain.
You can ...

**2**

votes

**1**answer

178 views

### General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot ...

**0**

votes

**0**answers

17 views

### Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup.
Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...

**0**

votes

**0**answers

94 views

### Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info
For standard vector-valued diffusion processes the following result is well-known:
Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by
\begin{align*}
...

**1**

vote

**1**answer

169 views

### Intuition about Skorohod integral

I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on.
In particular ...

**0**

votes

**1**answer

331 views

### What is the derivative of this integral?

I have asked this question here
http://math.stackexchange.com/questions/1536018/how-to-find-derivative-of-this-intergral
but still has no response.
Might I ask it here ?
Let $\alpha(t)\in\{0,1\}: ...

**1**

vote

**1**answer

108 views

### Differentiability of stochastic process

Is it possible to construct a stochastic process $X_t$ where the limit
$\lim_{\Delta \rightarrow 0} \rm{Var}\left(\frac{X_{t_0+\Delta}-X_{t_0}}{\Delta}\right)$
does not exist but the sample paths ...

**2**

votes

**0**answers

71 views

### Non-existence for a sort of probability measures

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$.
$W_{t}$ is standard Wiener.
This solution is ...

**0**

votes

**1**answer

131 views

### Is it safe to work on a Cadlag modification of a Feller process?

Let $f$ be a continuous bounded function.
$X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write
$$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...

**2**

votes

**1**answer

79 views

### Compactness of cadlag martingales w.r.t. to the point-wise topology

Given a sequence of cadlag (right-continuous with left limits) martingales $X^n=(X^n_t)_{0\le t\le 1}$, we may use the well known criteria to determine whether it is weakly convergent, i.e. subtract a ...

**1**

vote

**1**answer

68 views

### Problem of random scheduling of queues of tasks

Consider $L$ queues in a discrete time system. At each time $n=0,1,2,\ldots$, one task would arrive at one of the queues with equal probability $\frac{1}{L}$. Immediately after that, a task scheduler ...

**6**

votes

**3**answers

157 views

### Product and convex combination of two stochastic matrices

Let $K_1$ and $K_2$ be two $N \times N$ stochastic matrices (hence non-negative and rows adding up to one) with zeros on the diagonal. If $\alpha \in (0,1)$, is it possible to have
$$K_1 K_2 = ...

**2**

votes

**2**answers

104 views

### A question about Skorokhod embedding problem

The Skorokhod Embedding Problem is well known and has many solutions. Now let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be an embedding to the centered distribution $\mu$, i.e. the ...

**4**

votes

**1**answer

312 views

### Blumenthal and Kolmogorov 0-1 law

Blumenthal's 0-1 law see theorem 5.8/5.9 tells us that an event in the germ $\sigma-$ algebra has either probability zero or one with respect to a measure induced by a Brownian motion starting in some ...

**0**

votes

**1**answer

207 views

### Generalized Ito's lemma

I have the following quantity:
$$
g(t)=(f(t))^{M_{t}},
$$
where $M_{t}$ is a jump process which is neither Markovian nor Levy, and $f(t)$ is a positive, increasing but limited, right-continuous ...

**1**

vote

**0**answers

58 views

### Modify exponential family representation to a semimartingale

Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a class of probability measures $P=\{P_{\theta}:\theta \in \Theta\}$ definied on the filtered space.
We ...

**5**

votes

**2**answers

227 views

### Brownian motion in $n$ dimensions

Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in ...

**1**

vote

**0**answers

47 views

### Factorization of the Fokker-Planck semigroup

I have posted this question on stackexchange over four month ago, but didn't get an answer: "In the classical theory of Markov processes, the Fokker-Planck semigroup $\{T_t:t\ge 0\}$ can be factored ...