**1**

vote

**1**answer

53 views

### Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...

**4**

votes

**1**answer

213 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 ...

**0**

votes

**0**answers

47 views

### Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...

**0**

votes

**0**answers

22 views

### Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...

**0**

votes

**0**answers

59 views

### Brownian motion - probability of hitting an open subset of the sphere

Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let $\mathbb{P}_t(A)$ be the probability of the particle striking $A \subset S^{n - 1}$ within time $t$, where $A = \{ (x_1, ...

**1**

vote

**1**answer

83 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 ...

**0**

votes

**1**answer

105 views

### Transition probabilities for the symmetric random walk on the integers

I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...

**0**

votes

**1**answer

83 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) ...

**11**

votes

**0**answers

89 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, ...

**1**

vote

**0**answers

69 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 ...

**0**

votes

**0**answers

51 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:= ...

**6**

votes

**3**answers

667 views

### What are the difference between modeling with stochastic differential equations (SDE) and ordinary differential equations (ODE) with a random force?

There are lots of differences between SDE and ODE. From the theoretical point of view an also from the numerical algorithms used for simulations. But I am interested in knowing if there is a point ...

**4**

votes

**0**answers

193 views

### stochastic control / geometric mean

Consider the following problem:
Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...

**-2**

votes

**1**answer

52 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 ...

**3**

votes

**1**answer

225 views

### explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition:
$$V_t=V_0+\int_0^tH_sdX_s-K_t$$
where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...

**12**

votes

**1**answer

1k views

### surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...

**1**

vote

**1**answer

45 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 ...

**3**

votes

**3**answers

207 views

### Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit:
$$
dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC
$$
where $W$ is a standard Brownian motion ...

**2**

votes

**0**answers

42 views

### Holomorphic solution to SDE

Consider the SDE $dZ_t = \mu(t,x) d_t + \sigma(t,x) dW_t$.
Are there any known (necessary and) sufficient conditions on $\sigma(t,x)$ and on $\mu(t,x)$ guaranteeing that $f(T):=\mathbb{E}[\int_0^T Z_t ...

**3**

votes

**2**answers

154 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

62 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

**1**answer

105 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 ...

**4**

votes

**1**answer

404 views

### weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...

**1**

vote

**1**answer

65 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 ...

**3**

votes

**1**answer

102 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 ...

**4**

votes

**0**answers

103 views

### Geometric Characterization of Martingales

Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as Geodesics in a very large dimensional manifold.
My question is, is there any research studying this idea?
...

**1**

vote

**0**answers

96 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 ...

**8**

votes

**1**answer

660 views

### Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail.
Here is what I mean exactly. ...

**0**

votes

**1**answer

69 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

**0**answers

82 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*}
...

**0**

votes

**0**answers

71 views

### Expected value of product of Ito integrals

Assume that we have a process $F(t,T)$ that fulfills the following SDE.
$$
dF(t,T) = \sigma(t,T)F(t,T)dW(t)
$$
where $t$ is the running time and $T>t$ is called the delivery-time. $\sigma(t,T)$ is ...

**2**

votes

**1**answer

124 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 ...

**2**

votes

**1**answer

76 views

### Differentiability of value function

Suppose $X$ is a process given by -
$dX_t = db_t$
where $b_t$ is a standard Brownian motion with its filtration $(\mathcal{F}_t)$.
Suppose an agent earns a payoff given by
$V(x) = \mathbb{E} ...

**1**

vote

**1**answer

123 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

318 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\}: ...

**0**

votes

**1**answer

106 views

### Change of variable for integration with respect to Haar measure

I know how to estimate the integral* (see the update)
\begin{gather}
\int f(Ub)d\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [2]
\end{gather}
where ...

**0**

votes

**0**answers

32 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 ...

**2**

votes

**0**answers

77 views

### European call option pricing under mean reverting stock return

Consider the stock price process satisfies the following SDE:
$dS_t=\mu_t S_tdt + \sigma S_t dW_t , S_0=s $
and the mean return $\mu_t$ satisfies the following SDE:
$d\mu_t=(a-\mu_t)dt +dB_t, ...

**0**

votes

**0**answers

45 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 ...

**5**

votes

**2**answers

340 views

### What is the optimal growth of the constant in BDG?

Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} ...

**0**

votes

**0**answers

50 views

### Expectation, exponential of an additive functional of Brownian motion

I have a question about an additive functional of Brownian motion.
Let $d \in \mathbb{N}$. Let $b:\mathbb{R}^{d}\to \mathbb{R}$ be a measurable function and $(X_{t})_{t \in [0,\infty[}$ be a ...

**2**

votes

**2**answers

91 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 ...

**1**

vote

**1**answer

64 views

### Hitting time of a stochastically continuous process [closed]

Suppose $X$ is 1-d stochastically continuous process with $X(0) = 0$, i.e.
$X_s \to X_t$ in probability as $s\to t$ for all $t\ge 0$. Let $\tau = \inf\{t>0: |X_t|>1\}$.
[Q.] Is $\tau>0$ ...

**3**

votes

**1**answer

578 views

### Stochastic integrals as honest martingales — exponential damping

We have a given positive martingale ρt, with the dynamics:
$$\textrm{d}\rho_t = \lambda_t \rho_t \textrm{d}W_t$$
where $W_t$ is a standard Brownian motion. Now we have an "exponentially dampened" ...

**3**

votes

**1**answer

439 views

### Stochastic integrals as honest martingales — comparison criterion

We have a given positive martingale $\rho_t$, with the dynamics:
$$\textrm{d} \rho_t = \lambda_t \rho_t \textrm{d} W_t$$
where $W_t$ is a standard Brownian motion. Now we have a "dumped" process p_t:
...

**5**

votes

**2**answers

196 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

54 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 ...

**0**

votes

**0**answers

48 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

100 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 ...

**1**

vote

**0**answers

75 views

### Maximal principle for stochastic heat equation

Consider $\partial_{t}u=\partial_{xx}u$ with Neumann boundary condition
$u_{x}(0,t)=u_{x}(1,t)=0$ and initial condition $u(x,0)=f(x)\geqslant0$.
Then up to time $T$, the maximal value of $u$ should be ...