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

**-3**

votes

**0**answers

45 views

### Random process & probability problem

A random process r obeys the following distribution:
p(r,ṙ)=$\frac{r}{b_0}\exp{(-\frac{r^2}{b})}\frac{1}{\sqrt{2\pi b_2}} \exp{(-\frac{\dot{r}^2}{2b_2})}$, where $\dot{r}$ is the derivative of r in ...

**1**

vote

**1**answer

96 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

14 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

59 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

73 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

181 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

63 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 ...

**-1**

votes

**0**answers

14 views

### Under what conditions does a continuous-time Markov chain is also a Feller process? [closed]

It is known that not all continuous-time Markov chains are Feller, but are there any discussions on the sufficient conditions? For example, is a continuous-time Markov chain with finite number of ...

**0**

votes

**0**answers

29 views

### How the diffusion in the unit ball induce the boundary process on the boundary directly?

This is Example 1.2.3 from Fukushima, Masatoshi, Oshima, Yoichi and Takeda, Masayoshi's book "Dirichlet Forms and Symmetric Markov Processes".
In this example, we define a Dirichlet form in the unit ...

**0**

votes

**0**answers

43 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 $(\Omega,\mathcal{F})$.
$W_{t}$ ist standard Wiener.
This solution is ...

**0**

votes

**1**answer

95 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 ...

**1**

vote

**1**answer

50 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

23 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 ...

**5**

votes

**3**answers

134 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

83 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

148 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

76 views

### Generalized Ito's lemma

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

**1**

vote

**0**answers

48 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

155 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

35 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 ...

**0**

votes

**0**answers

41 views

### methods to analyze martingale conditioned on return in the future

Consider a martingale $S_t$ on $\mathbb{Z}$ starting from 0. Assume that for any $t$, $Var[s_t\, | \, \mathcal{F}_{t-1}] < V$, where $V$ is some positive constant. Fix an $n$ and for $t \leq n$, ...

**0**

votes

**0**answers

43 views

### skorokhod integral as Weak Integral

Is it possible to express the skorokhod integral on a Banach space $B$ as a special case of the weak (or Pettis) integral over an appropriate Banach space $E$?
For example if $E$ is the space of ...

**3**

votes

**1**answer

177 views

### Prediction with positive weights?

Consider a covariance function (positive definite function) on $\mathbb{Z}$:
$$
\gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0.
$$
It is guaranteed to be positive definite by Polya's criterion ...

**1**

vote

**2**answers

72 views

### Understanding the limits of the Ito Process Defintion

I would like to understand what kind of stochastic process are Ito Processes. According to Kuo[p. 102] an Ito Process is a stochastic process of the form
$$dX_t=g(t)dt+f(t)dW(t),$$
where $W(t)$ is a ...

**5**

votes

**1**answer

214 views

### Does every smooth manifold carry a gaussian random field?

Let $M$ be an arbitrary finite-dimensional smooth manifold. For simplicity, let's assume that $M$ has no boundary. Does there always exist a gaussian random field with constant variance on $M$? If ...

**5**

votes

**2**answers

147 views

### Recurrence of Poisson binomial distributed random walk

Let $X_n$ be the outcome of a Bernoulli trial where the probability of getting 1 is $p_n$ and the probability of getting 0 is $1-p_n$, and let $S_n = \sum_{i=1}^n \left(X_i - \textrm{E} X_i \right)$. ...

**4**

votes

**0**answers

102 views

### regularity of zero point

We consider 1-d process $X$
$$ X(t) = b t + J_{t} + M_{t}$$
where $b$ is constant, $M$ is a continuous martingale process with
$M(0) = 0$, and
$J$ is a symmestric $\alpha$-stable process with its ...

**3**

votes

**1**answer

94 views

### Markov-semigroup sobolev inequality

I have a question about the following definition:
A probability measure $\mu$, such that the Markov semi-group $e^{Lt} \in L(L^2)$ exists and is symmetric, satisfies the Sobolev inequality iff for ...

**4**

votes

**1**answer

90 views

### Exponential of approximate quadratic variation of Brownian motion

Let $X_t$ be a Brownian motion or a Brownian Bridge on a (\edit: compact) Riemannian manifold. Let $T>0$ be given.
The question is: Does there exists a constant $C>0$ such that for all ...

**2**

votes

**0**answers

114 views

### Stopping time sigma-fields

Let $(F_n)$ be a discrete Filtration and $S_n,S$ (not necessarily finite) stopping times with $S_n\uparrow S$ (increasing convergence).
Is it true that the associated sigma-fields satisfy ...

**4**

votes

**1**answer

112 views

### Large deviation for Brownian path on $[0,\infty)$

It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path.
If we equip the space of continuous function starting from $0$, ...

**4**

votes

**1**answer

53 views

### Reference request: Urbanik's work on random integrals and Orlicz spaces

Several important papers on Lévy processes are referring to the following paper:
K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces,
Bulletin de l'Académie Polonaise des Sciences, ...

**5**

votes

**1**answer

175 views

### Nearest neighbor for planar Poisson is normally distributed

This was previously asked on MathSE, but was not answered.
Answering a question, I realized that the nearest point for a planar Poisson point process (with constant intensity $\lambda>0$) is ...

**3**

votes

**0**answers

45 views

### $X_t = B_t^q$, $X_t = (\sin B_t)^q$, $X_t = B_t^q (\sin B_t)^r$, $dM_t = R_t\,M_t\,dB_t$ [closed]

What are the SDE's satisfied by the following processes?
$X_t = B_t^q$
$X_t = (\sin B_t)^q$
$X_t = B_t^q (\sin B_t)^r$
Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the ...

**5**

votes

**0**answers

44 views

### Full distribution of FPTs in random walks on graphs

There is a lot of published research on the mean passage passage time (FPT) for random walks on various types of graphs. How about the variance of the FPT and higher momenta? In fact, I would be ...

**4**

votes

**2**answers

54 views

### Distribution of the RKHS norm of the posterior of a Gaussian process

In a classical noisy regression setting, let $\big(f(x)\big)_{x\in\cal X}$ be a centered Gaussian process of covariance $k$ on a compact $\cal X$, and $\mathcal{F}_n$ be the filtration generated by ...

**0**

votes

**0**answers

72 views

### Example of an adapted measurable process which is not Progressively Measurable

In this question
Progressively measurable vs adapted, one finds a discussion on the subject of adapted processes versus progressively measurable processes.
Counter-examples can be readily given. We ...

**6**

votes

**1**answer

181 views

### Bound on expectation, not a really simple process, circumvent using Itō's lemma?

Assume that $H_t$ is a progressively measurable process such that with probability one $|H_t| \le k$ for all $t$. Let$$Z_t = \int_0^t H_s\,dB_s.$$How do I see that for all $s < t$, $\lambda \in ...

**0**

votes

**0**answers

63 views

### What is the success probability of this stochastic process?

Suppose you have $k$ black balls and $X\cdot k$ white balls.
The procedure start with you having a bag containing $y\le k$ white balls (e.g. $k+1,\ldots k+y$).
In every iteration:
A single white ...

**0**

votes

**0**answers

53 views

### Law of motion when initial condition is perturbed

We know how to find the law of motion (Ito process) of the value function:
$$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$
such that
...

**2**

votes

**1**answer

73 views

### Solve SDE $dX_t=(c+\sigma_\zeta W'_tX_t)dt + \sigma_\epsilon dW_t$

I am trying to solve the following SDE
$$dX_t=(c+\sigma_\zeta W'_tX_t)dt + \sigma_\epsilon dW_t$$
$c\in \mathbb{R}$ is a constant, $X_t$ is a stochastic process, $\sigma_\zeta,\sigma_\epsilon \in ...

**1**

vote

**1**answer

79 views

### Continuity of expected payoff from a diffusion

Fix a discount rate $r>0$, and let $m,v,f:\mathbb{R} \rightarrow \mathbb{R}$ be bounded measurable functions of locally bounded variation, with $v$ globally bounded below by some strictly positive ...

**0**

votes

**0**answers

32 views

### How to implement conjugate gradient method to minimize this nonlinear action?

Given a 2D stochastic differential equation:
\begin{align}
\dot{x}_{i}=f_{i}(\textbf{x})+g_{ij}\xi_{j}(t),
\end{align}
where $i=2$, $g_{ij}g_{jk}=2\epsilon\delta_{ik}$ and ...

**0**

votes

**1**answer

155 views

### Time Change of a Brownian motion

We know that for if $X$ is a stochastic integral of the form below -
$X_t = \int_0^t v(s,\omega) db(s,\omega)$.
then we can use time change formula to claim that
$X_t = W_{\alpha(t)}$ where $W$ is ...

**1**

vote

**2**answers

117 views

### The Levy measure of the compound Poisson distribution

The compound Poisson distribution is defined as(see Levy processes and infinitely divisible distributions page: 18):
Let $c>0$ and $\sigma$ be a measure on $\mathbb{R}$ with $\sigma(\{0\})=0$, a ...

**0**

votes

**0**answers

72 views

### strong law of large number for semimartingale

I just want to know if for semimartingale X we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{<X>_{t}}=0$ or when it is possible. i know it is true for brownian motion.
Thanks

**3**

votes

**1**answer

116 views

### Brownian bridge on a Lie group as a stochastic differential equation

Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation
$$dg_t = dB_t \circ g_t$$
where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes ...

**2**

votes

**0**answers

50 views

### Using compactness method to prove the existence of a pathwise solution to an SPDE

For given initial data $u_0\in H^k$ for some $k$, I want to prove the existence of solution to some PDE with multiplicative white noise. I modify the SPDE by regularizing it and then use the ...

**2**

votes

**0**answers

55 views

### Strong solution to an SDE with a discontinuous diffusion term

I am having an SDE for which I would be in trouble if there were no strong solution.
The SDE is -
$ dX = \mu(x) dt + \sigma_1 (x) db_{1t} + \sigma_2(x) db_{2t}$
where $b_1$ and $b_2$ are two ...

**5**

votes

**1**answer

174 views

### Optimisation of betting strategy

Consider integers $r \geq 1$ and $k \geq 1$ and consider the following game:
We start with $r$ tokens and at each round we choose $i \in \{1,...,r\}$ tokens to bet (if we have $N<r$ tokens we ...