**1**

vote

**0**answers

83 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 http://www.springer.com/gb/book/...

**3**

votes

**1**answer

123 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

112 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

99 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 ...

**0**

votes

**1**answer

85 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 $0$....

**0**

votes

**0**answers

110 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

159 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

82 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} [\...

**0**

votes

**0**answers

116 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

196 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\}: ...

**0**

votes

**1**answer

140 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 $f:S^{n-1}(\...

**3**

votes

**0**answers

94 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, \...

**2**

votes

**0**answers

72 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 ...

**2**

votes

**1**answer

82 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 ...

**0**

votes

**0**answers

85 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 $d$-...

**1**

vote

**1**answer

77 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 $\tau>0$ ...

**2**

votes

**2**answers

105 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 ...

**0**

votes

**1**answer

252 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

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 \...

**0**

votes

**0**answers

55 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

106 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 ...

**0**

votes

**0**answers

63 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
$$dX_t=\mu(t,X_t)...

**1**

vote

**1**answer

84 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

**1**answer

436 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 ...

**0**

votes

**0**answers

88 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

**0**

votes

**0**answers

58 views

### Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...

**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

**0**answers

70 views

### integrability of Brownian motion stopped at some stopping time

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion starting at zero and denote by $S=(S_t)_{t\ge 0}$ its running maximum, i.e. $S_t=\sup_{0\le s\le t}B_s$. Given a fixed number $p>1$, define the ...

**-2**

votes

**1**answer

83 views

### About the boundary conditions of the Black-Scholes-Merton PDE [closed]

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve.
Let $c(t,x)$ be the value of the ...

**0**

votes

**0**answers

41 views

### Systems of stochastic differential equations with non-Lipschitz coefficients

I am looking for references to any literature which might consider the existence / behavior / regularity of solutions to systems of stochastic differential equations with non-Lipschitz coefficients.
...

**4**

votes

**0**answers

50 views

### Existence of martingales given some constraint on laws

Let $X=(X)_{0\le t\le 1}$ be a continuous martingale starting at $0$, then denote by $\mu$ and $\nu$ the probability laws of $\int_0^1X_t \mathrm{d}t$ and $X_1$. Then it is easy to see that the couple ...

**0**

votes

**0**answers

40 views

### Proof of Linear Stochastic Sate-Space Model is Gaussian Process

I would like to prove that the vector-linear stochastic state space model
$$
\dot{x}(t)=A(t)x(t)+B(t)u(t)+G(t)q(t) \\ y(t)=C(t)x(t)+D(t)u(t)+F(t)r(t)
$$
corresponds to a particular multi-output ...

**2**

votes

**0**answers

130 views

### The existence of stationary measures for certain Markov process

My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time ...

**1**

vote

**0**answers

76 views

### Definition of Ito Integral

In Karatzas and Shreve, the integral for Bounded Progressively measurable processes is defined first. Then, for Bounded measurable and adapted processes ($f(t,\omega)$), the authors say that there ...

**1**

vote

**1**answer

157 views

### Proof of no bound for stochastic integral

I have Ito integral $X=\int_0^T f(t) dW(t)$ and I would like to proof that $P(X>K)>0$ for all $K$ provided $f(t) > \epsilon > 0$.
My idea was $\int_0^T f(t) dW(t) \sim \int_0^T \epsilon dW(...

**1**

vote

**0**answers

102 views

### Asymptotics of Variable Drift Ornstein–Uhlenbeck Process

The Ornstein–Uhlenbeck process is defined as the stochastic process that solves the following SDE:
$dx_t = \theta (\mu-x_t)\,dt + \sigma\, dW_t$
where $\theta>0$, $\mu$ and $\sigma>0$ are ...

**2**

votes

**0**answers

58 views

### Deriving HJB equation (why $\frac{dZ_t}{dt}=0$?)

I am trying to derive the HJB equation in a stochastic setting. Let
me exemplify my problem with the simplest case where there is no control,
just one state variable. Assume the payoff is given by
$$
...

**0**

votes

**0**answers

54 views

### Strong Markov Property of the joint process $(B_t,L_t)_{t\ge 0}$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion and $L=(L_t)_{t\ge 0}$ be its local time in zero. Given two strictly increasing functions $\phi_1$, $\phi_2: \mathbb R_+\to\mathbb R$ such that $\phi_1(0)=\...

**1**

vote

**0**answers

175 views

### Joint law of a standard Brownian motion and its local time at a nonzero level

Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is
$$
P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\...

**0**

votes

**0**answers

53 views

### Recursive parameter estimation for partially observed Ito SDEs

I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a ...

**2**

votes

**1**answer

188 views

### Stochastic differential equation associated with an optimal control problem

We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time $\...

**2**

votes

**0**answers

62 views

### Question about the characteristics of semimartingales

Let $D=D([0,1,R)$ be the space of cadlag (right-continuous with left limits) functions defined on [0,1] and $X:=(X_t)_{t\in [0,1]}$ be the canonical process on $D$, i.e. $X_t(x)=x(t)$ for all $x\in D$....

**5**

votes

**0**answers

418 views

### Quadratic variation and predictable quadratic variation for martingales

Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$.
Fix $N$ and consider now a discrete version ...

**1**

vote

**2**answers

78 views

### SDEs: Bounding the variance of a solution

I've been thinking about something that would seem intuitive, but I haven't really been able to dig a direct answer to. This is a rough draft of it.
Let
$$X_t = \mu_{X,t} \mathrm{d}t + \sigma_{X,t} \...

**4**

votes

**2**answers

449 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?

**5**

votes

**0**answers

267 views

### Squaring random Schwartz distributions

Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance
$$
\mathbb{E}
[\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)}
\widehat{g}(\xi)}{|\xi|^{d-2[\...

**2**

votes

**2**answers

310 views

### Existence of strong solution to SDEs with non-Lipschitzian drift

Consider the SDE:
$$dX_t=b(X_t)dt+dW_t\quad X_0=x$$
If $b$ is bounded Borel function, using Zvonkin's Transform, one can prove there exists a unique strong solution.
I want to know if we assume $b$ ...

**1**

vote

**2**answers

255 views

### $\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?

I am trying to prove the following Lemma, which seems intuitive, but I still have doubts:
Lemma
Given a Brownian motion $\{W_t,\mathcal F_t:0\le t \le1\}$, two bounded processes, $\mu$ and $\sigma$, ...