**9**

votes

**1**answer

959 views

### Karhunen–Loève approximation of Brownian motion and diffusions

The Karhunen–Loève theorem says that Brownian motion on the interval [0,1] can be represented as follows:
$W_t = \sum_{n=1}^\infty Z_n \frac{\sin((n-1/2)\pi t)}{(n-1/2)\pi},$
where $Z_n \sim ...

**4**

votes

**2**answers

977 views

### Weierstrass' function and Brownian motion

Is there a known connection between Weierstrass' function
$W_\alpha (x) = \sum_{n=0}^\infty b^{- n \alpha} \cos(b^n x)$
and Brownian motion? Specifically, when $\alpha = 1/2$, the Weierstrass ...

**1**

vote

**2**answers

272 views

### Martingale part of the discontinuous put payoff

I need the martingale part of the put payoff (not $C^2$..). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$
$d[(S_t -K)^+ ]$ ??
I guess I need to use local times but how?

**1**

vote

**0**answers

264 views

### Inverse Skorokhod Embedding Problem

The Skorokhod Embedding Problem is well known and has many documented solutions in the literature.
Now if we are given a Brownian stochastic basis (satisfying usual hypothesis), a diffusion $X_t$ ...

**1**

vote

**1**answer

425 views

### infimum of a set of stopping times

Let $(Y^a: a\in \Lambda)$ be a set of random processes given by
$$Y^a(s) = \int_0^s \sigma^a(r) dW(r)$$
where $W$ is Brownian motion w.r.t. filtered probability space
$(\Omega, \mathcal{F}, P, ...

**0**

votes

**1**answer

290 views

### Local continuous martingale

Hi,
This is a relatively simple result with a simple proof. However, there are 2 things I don't understand:
Why is M a brownian motion?
How is I calculated ("Thus, we get...")?
Any insight would ...

**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 ...

**11**

votes

**1**answer

2k views

### Martingales in both discrete and continuous setting

I am wondering, polynomials like
$S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...

**0**

votes

**2**answers

385 views

### General question about Stochastic analysis [closed]

Dear all,
I'm wondering which university is a good choice for grad school in the field of stochastic analysis, more specifically, stochastic evolution equations.
Well, I know this is not a strict ...

**5**

votes

**1**answer

517 views

### Exact simulation of SDE

Consider a one dimensional SDE of the form $dX_t = \mu(X_t) dt + \sigma dW_t$, where $\sigma>0$ is a constant. Under mild regularity assumptions on $\mu(\cdot)$, one can exactly simulate ...

**4**

votes

**1**answer

357 views

### Time-integral of a smooth, vector-valued function of a planar Brownian bridge

I'm looking for information on how to compute the distribution of the random vector
$$Z = \int_0^t f(B_s) ds$$
where $t>0$ is fixed, $B_s$ is a 2D Brownian bridge with $B_0 = 0$, $B_t=b \in ...

**0**

votes

**1**answer

467 views

### square root processes with correlated deriving Brownian motion

$$dX = \kappa_x (\theta_x - X)dt + \sigma_x \sqrt{X} \,dW_x$$
$$dY = \kappa_y (\theta_y - Y)dt + \sigma_y \sqrt{Y} \,dW_y$$
$$dW_x dW_y = \rho\, dt$$
we know that $X$ and $Y$ are marginally ...

**4**

votes

**1**answer

338 views

### Homogeneous linear stochastic DE with noncommuting coefficients

The system I am studying can be reduced to a Stratonovich vector
stochastic differential equation
$dX = A X \; dt + \sum B_k X \circ dW_k$
with $W_k$, $k=1..m$ the Brownian motion in $m$ dimensions, ...

**1**

vote

**2**answers

1k views

### Change of measure Markov process

We begin with example. For the Poisson process with an intensity $\lambda_1$ there is an equivalent change of measure which makes it intensity to $\lambda_2$.
I would like to find the conditions ...

**5**

votes

**3**answers

349 views

### comparing diffusions

Consider a probability distribution $\pi$ on the real axis that has a density (w.r.t Lebesgue) proportional to $e^{-V(x)}$, where $V(\cdot)$ is a potential function. For any reasonable volatility ...

**2**

votes

**2**answers

210 views

### Has anyone found a way to determine the invariant measure of a one-dimensional jump-diffusion?

I am working with a jump-diffusion on the unit interval, with absorbing endpoints, and I was hoping someone has found a way to determine the invariant measure, similar to that of an Ito process.

**6**

votes

**1**answer

534 views

### Change of space-time in Walsh's stochastic integral

One can read about Walsh's construction of martingale integral in the paper (pp.16-23)
www.math.utah.edu/~davar/ps-pdf-files/SPDEBookDK.pdf
For $U,V\in \mathcal{B}(\mathbb{R}\times \mathbb{R}^+), ...

**0**

votes

**0**answers

203 views

### Stochastic Optimal Control - Maximizing convex terminal costs

The theory of stochastic optimal control deals with the following problem:
Find $\quad\sup\limits_{u} \; \mathrm E[g(X^{(u,x)}_T)]$
where $X^{(u,x)}_t$ solves the following controlled SDE:
...

**3**

votes

**1**answer

490 views

### Results for Hitting Times of (Not Stationary) Ito Processes

Let $W_t$ denote the Wiener process and let
$$
dX_t = a(t, X_t) dt + b(t, X_t) dW_t
$$
be an one dimensional Ito SDE (stochastic differential equation, see Ito stochastic calculus).
A hitting time ...

**6**

votes

**3**answers

597 views

### A simple decomposition for fractional Brownian motion with parameter $H<1/2$

Background
Let $X = \{X(t):t \geq 0\}$ be a (standard, real-valued) fractional Brownian motion (fBm) with parameter $H \in (0,1)$, i.e., a continuous centered Gaussian process with covariance ...

**7**

votes

**1**answer

273 views

### When is it possible to construct a joint law from its two-dimensional marginals?

My question is much more specific than the title:
Given a symmetric
distribution $\Xi$ on $\mathbb R^2$, when is it possible to construct a sequence $\xi_1,\xi_2,\dots$ of random variables such ...

**17**

votes

**3**answers

1k views

### Do convex and decreasing functions preserve the semimartingale property?

Some time ago I spent a lot of effort trying to show that the semimartingale property is preserved by certain functions. Specifically, that a convex function of a semimartingale and decreasing ...

**1**

vote

**3**answers

7k views

### Expectation of time integral of Wiener process

I am trying to calculate $E(\int_0^T {W_s ds})$, where $W_s$ is a standard Brownian motion.
Now two approaches I can think of:
1) Take a partition of $[0,T]$. Calculate $E(\sum {W_{t_i}(t_{i+1} - ...

**6**

votes

**2**answers

943 views

### Brownian Motion Winding Number

Take a simple random walk $\gamma$ in the complex plane conditioned to start at point $a$ and end at point $b$. For this random walk, we can define the winding number $W_\gamma(a,b)$ around $b$ in the ...

**1**

vote

**1**answer

713 views

### Need help understanding Mandelbrot and Van Ness Fractional Brownian Motion

I need help understanding the Mandelbot and Van Ness' definition of Fractional Brownian motion
$
B_H( t , \omega ) - B_H( 0 , \omega ) = \frac{1}{\Gamma(H + \frac{1}{2})} \left\( \int_{-\infty}^0 ...

**5**

votes

**1**answer

330 views

### Stieltjes integrals of predictable processes

I am looking for a direct proof of the fact that, roughly speaking, if $S=S_0+A+M$ is an $L^2$ semimartingale, and $M$ (the martingale part) has the martingale representation property, then for any ...

**13**

votes

**4**answers

1k views

### Wiener process related counterexample

The Wiener process is defined by the three properties:
1. $W(0) = 0$,
2. $W(t)$ is almost surely continuous, and
3. $W(t)$ has independent increments with $W(t) - W(s) \sim N(0, t-s)$ (for $0 ≤ s ...

**3**

votes

**0**answers

145 views

### Characterizing polyhedron from Brownian particle collisions with a boundary

Please imagine that we have an ordinary 2-sphere, of radius $r_{sphere}$, and some three-dimensional polygon that has all of its points fixed at positions strictly internal to the sphere's surface. ...

**4**

votes

**2**answers

249 views

### Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator

Let $\xi_t$ be a zero-mean gaussian process on $[0,1]$ with covariance operator $C$.
I would like to better understand the relation between the covariance operator and the regularity of the ...

**2**

votes

**1**answer

633 views

### Taylor expansion to show that for Stratonovich stochastic calculus the chain rule takes the form of the classical one

As nobody seems to be able to give any kind of answer to that problem over there at math.stackexchange I post this question here:
How can I show with a heuristic argument based on a Taylor expansion ...

**1**

vote

**0**answers

303 views

### Change of Time in Stochastic Integral

Hi everyone,
Let's be given $I(0,t)$ a Stochastic Integral with respect to a local martingale $ M_t$ of the form :
$I(0,t)=\int_0^t h(s_-)dM_s$ with $h\in L(M)$ (for example $h$ is an adapted ...

**1**

vote

**0**answers

236 views

### Density of Dolean exponentials in L2 and Wiener Measure

Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by ...

**4**

votes

**3**answers

408 views

### Continuity in intial state of Brownian Motion

$ B = (B_t, \mathcal{F}_t; t\ge 0 ) $ is a 1-d Brownian family on a
measurable space $(\Omega, \mathcal{F})$ with a family of probability
measures $\{\mathbb{P}^x\}$, i.e. $\mathbb{P}^x(B_0
= x) = 1$, ...

**4**

votes

**1**answer

901 views

### Distribution of running maximum of a local martingale

Let $(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}_t)$ be a given
probability space with usual conditions, on which $W$ is a standard
Brownian motion. For $x \ge 0$, consider
$$X(t) = x + \int_0^t ...

**3**

votes

**1**answer

379 views

### Approximation of the law of a stochastic process

Hello Dear fellows,
I thank you in advance for your help and ideas.
I have just read an article and want you to help me understand the rational behind a part of it.
We have two processes $v_t$ and ...

**26**

votes

**9**answers

5k views

### Intuition and/or visualisation of Ito integral/Ito's lemma

Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve.
See e.g. Wikipedia:Riemann sum
The Ito integral has due to the unbounded total variation ...

**3**

votes

**2**answers

833 views

### Is the truncated Brownian motion of the class DL?

Let $W$ be a standard Brownian motion under given probability space.
For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time
$T^a = \inf(t>0:W(t) = a)$. That is, $W^a(t) = ...

**2**

votes

**0**answers

293 views

### Finding jump probabilities from mean-occupancy values for positions on a one-dimensional random walk

Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as ...

**2**

votes

**2**answers

951 views

### Examples of deterministic processes of quadratic variation which are of unbounded variation

In [Föllmer 81] (English translation to be found here) writes: "The class of processes of quadratic variation is clearly larger than the class of semimartingales: Just consider a deterministic process ...

**1**

vote

**3**answers

941 views

### Föllmer: “Calcul d'Ito sans probabilités” in English or German?

Does anybody know a translation of Föllmer: Calcul d'Ito sans probabilités in English or German?
It seems to be a very interesting text - Abstract: "It is shown that if a deterministic continuous ...

**5**

votes

**2**answers

262 views

### how to sample a conditioned diffusion

there are several reasons why we could be interested in sampling conditioned diffusions:
if we observed a diffusion at discrete time and want to do some kind of inference on the parameters of the ...

**2**

votes

**1**answer

973 views

### Stationary Solutions of stochastic differential equations

When does the stationary density of an homogeneous Markov process exist?

**3**

votes

**1**answer

319 views

### initial condition of a diffusion approximation

I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...

**6**

votes

**1**answer

2k views

### Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory

I am confused and don't get the big picture concerning the connection between
Ito integral
Stratonovich integral
Standard results in probability theory concerning skewed distributions.
Example: ...

**4**

votes

**0**answers

492 views

### Dynamic programming principle (DPP)

In stochastic control problem, one shall use the measurable selection theorem to prove DPP. It was discussed in discrete time case in [Bertsekas and Shreve 1978]. Is there unified framework in ...

**0**

votes

**1**answer

435 views

### Looking for a version of Itô's Lemma

Hi Everyone
I have some difficulties deriving the Stochastic Differential Equation for the following problem, any help or reference would be appreciated.
We are given a Brownian Motion $B_t$ and ...

**1**

vote

**2**answers

257 views

### Problem with a Long Range Correlated Time Series

Consider a stochastic process $X_t$ , $t \in 1,2,3,..,N $.
$X_t$ is a Bernoulli variable and $\Pr(X_t=1) = p$ for all $t$.
The Autocovariance function $\gamma(|s-t|)= E[(X_t - p)(X_s -p)]$ is given
...

**3**

votes

**2**answers

374 views

### Systematization of deterministic and stochastic integrals

With this question I try to build up a systematization of different kinds of integrals. The following table differentiates between deterministic and stochastic integrals, the summation processes ...

**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 $\frac{1}{N}$:
$$ ...

**9**

votes

**2**answers

2k views

### Convergence and non-convergence of left-point and mid-point Riemann sums

In standard calculus it is a well known fact that left-point and mid-point Riemann sums do become equal in the limit. When it comes to stochastic integration this is no longer the case. Taking the ...