# Tagged Questions

**16**

votes

**5**answers

2k views

### Brownian motion and spheres

Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$:
$$ W\left(\frac{k}...

**5**

votes

**3**answers

579 views

### A non-degenerate martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which
$\mathcal{F}_t$ is filtration satisfying general conditions.
$W_t$ is
a standard Brownian motion.
Let $Y_t$ be a martingale given by
$$...

**4**

votes

**2**answers

453 views

### Average Value of Area Closed by Brownian Motion

Two dimensional brownian motion will intersect its own path infinitly many times. What is the average value of area, closed by curve during an intersection in brownian motion?

**3**

votes

**2**answers

404 views

### How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration?
$$
\int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$
This post is related to my previous question here , ...

**5**

votes

**1**answer

350 views

### 'Nonclassical' abstract Wiener space

Is it possible to construct an abstract Wiener space $(W,H,\mu)$ such that $C^{0,\frac{1}{2}}(\Omega)\subset H$ and $W$ is a normed function space such that the convergence in norm implies convergence ...

**2**

votes

**1**answer

148 views

### Numerical computation of Skorokhod integral

How can I numerically compute the Skorokhod integral of a non-adapted process? If it is adapted, that is easy since the integral is just an Ito integral.
I have found that computing the Malliavin ...

**5**

votes

**2**answers

243 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

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 ...

**1**

vote

**2**answers

1k views

### The only continuous martingales with stationary increments are Brownian motions

Hi,
I know that the above statement is true, but I can't demonstrate it.
It's a pretty powerful theorem, here is its mathematical formulation:
Theorem: The only continuous martingales with ...

**0**

votes

**2**answers

476 views

### Representation theorem for continuous uniformly integrable martingales

For some time $u$ and positive continuous process $a_t$ adapted to $\mathcal{F}_t$ I have a (continuous-time) martingale defined as:
$$M_t(u) = \mathbb{E}[a_u | \mathcal{F}_t]$$
for $t\leq u$. I ...