The stochastic-calculus tag has no wiki summary.

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### 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 ...

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**1**answer

214 views

### SDE-removal of the diffusion coefficients

from math.stackexchange
I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_{loc}$. If I have
\begin{align}
dX_t=b(X_t)dt+\sigma dW_t,
\end{align}
...

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72 views

### Potentials of class D

A potential $\pi_t$ is a positive supermartingale with the condition that $\mathbb{E}[\pi_t]\rightarrow 0$ as $t \rightarrow 0$. What are the necessary/sufficient conditions for a potential to be of ...

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176 views

### market completion in stochastic volatility model

Hi all,
Consider a stochastic volatility model. As there are two sources of risk and one asset only, this is an imcomplete market. One can complete the market by considering a derivative V1 used to ...

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100 views

### stochastic volatility valuation equation

I'm trying to derive the valuation equation under a general stochastic volatility model. What one can read in the litterature is the following reasonning:
One consider a replicating self-financing ...

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**1**answer

301 views

### A wrong proof of Squared Bessel process

The squared Bessel process with $\delta$-dimension for $\delta>0$,
denoted by $BESQ^\delta(y)$, is given by
$$d Y_t = \delta t + 2 \sqrt{Y_t} d B_t, \ Y_0 = y\ge 0$$
where $B_t$ is BM under ...

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188 views

### Is this process strictly positive?

Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider 1-D stochastic differential equation
$$ dY_t = dt + \sigma(Y_t) dW_t, \ Y_0 = y\ge 0.$$
We assume $\sigma(0) = 0$, and ...

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95 views

### Does this series stopping times marching forward?

Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider stochastic differential equation
$$ dY_t = dt + Y_t dW_t, \ Y_0 = 0.$$
Note that, the above SDE has a strong non-negative ...

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227 views

### Stochastic processes with random matrices

I am currently working on complex networks. I consider a matrix $\cal N$ with random entries $\delta_{ik}$. These entries are varying randomly in time and so I have a sequence of random matrices that ...

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154 views

### References/Papers on analytic solutions to SDEs

Does anyone know of any good references/research papers on finding analytic solutions to stochastic differential equations and/or finding approximating solutions to such a system?
I am particularly ...

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622 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 ...

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109 views

### change the sign of volatility

Assume the time inhomogeneous SDE
$dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$
has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE
...

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228 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} ...

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166 views

### Expected value of a logarithm of a Levy process

I have a strictly positive Levy process $(L_t)$ with no Brownian part, drift $\gamma$ and jump measure $\nu$. Is it possible to calculate the expected value of the logarithm of this process, so ...

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590 views

### karhunen-Loeve expansion of Poisson process

Let $X_t, t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the karhunen-loeve expansion of $X$ in interval [0, T]. How about the KL expansion of the centered process $X_t-\lambda ...

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160 views

### Time reversibility of Stratonovich Diffusion: Reference Request

Please consider the Stratonovich stochastic differential equation (SDE)
$$
dX = b(X)\circ dB
$$
where $B$ is standard Brownian motion and $X(0)=X_0$. This corresponds to the Ito (SDE)
$$
dX = ...

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### Joint distribution of Ito integral and its quadratic varation

Any idea on solving the joint distribution of
$X_T=\int_0^T \alpha_t dZ_t$ and $Y_T=\int_0^T \alpha_t^2 dt$ ? Here $X_T$ is an Ito integral and $Z_t$ is a standard Brownian process. When $\alpha_t$ ...

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350 views

### Calculate $\mathbb{E}[\int_o^T N_{t-}dS_t]$ - what went wrong?

First note, I had asked a similar question here, but the thread seems to have died, so I'll revive it here with more details. As a simplification of my real problem, I want to compute
...

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147 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 ...

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266 views

### Asking for a Fourier inverse transform, which is related to stable laws

Dear friends,
Denote the function
$$
G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;.
$$
It is well known that if $a\in ]0,2]$, ...

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303 views

### Solving SDE's on subsets of $R^n$.

I posted this on mathstackexchange to no avail.
It is well-known (see for instance Oskendal's text) that if $T>0$ and
$$b(\cdot,\cdot): [0,T] \times \mathbb{R}^n \rightarrow ...

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383 views

### Is the Feynman-Kac formula valid for a time-dependent potential

So I'm looking at a diffusion process with killing with a state- and time-dependent killing rate. This is described in Oksendal's Stochastic differential equations pages 143-145 "The Feynman-Kac ...

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347 views

### Itô-like calculus for $\alpha$-stable processes $\alpha \neq 2$.

After searching the net, I couldn't find a suitable reference to some extension of the Itô calculus that involves wilder sources of randomness than mere brownian motion.
Motivation : Itô calculus is ...

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465 views

### Symmetric Feller processes and Dirichlet Forms

Let $(G, \mathcal D)$ be a densely defined operator on $C_0$ (continuous functions vanishing at infinity on some nice topological space) whose closure $\bar G$ generates a Feller semigroup and let $X$ ...

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269 views

### Stochastic Integration via Skorohod Representation

I am interested to know if Ito integrals against Brownian motion can also be constructed via Skorohod representation. By this I mean the following: let $S_n$ be a simple random walk started at zero; ...

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296 views

### Tanaka stochastic differential equation and Kolmogorov equation

Given Tanaka sde
$$dX_t=[a{\rm sign}(X_t)+b]dW_t$$
is there associated a diffusion process and so a Kolmogorov (Fokker-Planck) equation? What is this equation?
References answering the question are ...

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501 views

### A class of Ito integrals

I am currently working on stochastic processes and I have met a stumbling block in the Ito integral
$$\int_{t_0}^tdt'G(t')[dW(t')]^\alpha$$
with $\alpha\in\mathbb{R}$ and $\alpha>0$. Textbooks ...

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380 views

### Finding a stochastic differential equation as limit of a discrete stochastic process

I stumbled upon a problem that seems simple but I cannot tackle it. Let $X_n$ be a discrete process defined by the following algorithm.
Choose $X_0\in[0,1]$, set $\kappa>0$ small enough and
...

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### 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
...

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264 views

### White noise in Lie group

The matter is in the title.
Is there a means to define the white noise process in Lie group. A basic definition
link text
Question:
Can we replace $\mathbb{R}^n$ by a Lie group?
In fact, I would ...

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### How to deal with the vector norm item as a denominator in this expectation?

Hello, everyone.
I want to calculate the expectation shown in the following formula, where $X$ follows a standard $d$-dimensional multi-variable normal distribution as ...

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350 views

### X-harmonic and mean value property

We know in elliptic equation theory(or related area) that harmonic function has mean value property. Roughly speaking, harmonic function function at point x is equal to its average on the spherical ...

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219 views

### Observing drift of a Levy process

It is a well known fact, that it is very difficult to estimate the drift of a Brownian motion with drift from looking at a single path over a finite interval $[0, T]$. Is it the case with Levy ...

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### How to calculate this expectation with logarithm?

If $\mathbf{x}\sim\mathbf{N}(\mathbf{0},\mathbf{I})$, and assume that $A$ is a symmetric positive definite matrix, how can I calculate the following two expectations where there is a logarithm in it?
...

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### How to calculate this expectation where the random variable is restricted on a sphere? [closed]

Hello! I have a question about how to calculate the expectation of a quadratic form as follows, where $X$ is a random variable that uniformly distributed on the unit sphere:
$$
E_X[(\mathbf{x}^\top ...

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226 views

### Getting $B_t$ from its local times $L^x_t$

Hi
Given a Brownian Motion $B_t$ is it possible to reconstruct it from the knowledge of the local times $L^x_t$ ?
Using occupation time formula this would mean solving for some $f$ the following ...

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343 views

### Joint law of the time integral of Brownian motion and its maximum

Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively:
$$M_t=\max_{0\leq s\leq t}\,W_s$$
...

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461 views

### Measure changes for gamma process

GENERAL THEORY
In his book Ken-Iti Sato ("Lévy Processes and Infinitely Divisible Distributions") provides the theory for measure change for Lévy processes in Theorems 33.1 and 33.2.
It can be ...

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967 views

### Generalized Ito's formula

Consider classical statement of Ito's formula: Let $X$ be a continuous
semimartingale and $F \in C^2(\mathbb{R}^d, \mathbb{R})$; then $F(X)$
is a continuous semimartingale and
$$F(X_t) = F(X_0) + ...

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448 views

### Stochastic Green-Gauss Theorem

Is there a stochastic analog for the Green-Gauss theorem? I'm looking for an expression that relates the flux (or statistical moments of the flux) through a random surface to the divergence of the ...

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484 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" ...

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### 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:
...

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353 views

### Positive martingale representation with jumps

I am looking for a martingale representation theorem for positive semimartingales. Using the answer to this question:
Martingale representation theorem for Levy processes
My best guess is (subject to ...

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### Levy jump measure vs. Levy measure vs. sum of jumps

This question might be a bit basic, but I am struggling to understand the connection between various versions of the Ito's lemma for Levy processes (and semimartingales in general). Could someone ...

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613 views

### Reachability for Markov process, continuous time

Let $X$ be a strong Markov process in the continuous time with a state space $\mathbb R^n$. Consider a reachability problem for this process, i.e.
$$
v(x):=\mathsf P_x(X_t\in A\text{ for some }0\leq ...

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### Martingale representation theorem for Levy processes

Is there an equivalent of martingale representation theorem for Levy processes in some form? I believe there is no such theorem in generality, but maybe there are some specific cases?

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### Fractional Brownian motion and Laplacian

Having read this link on math stackexchange, I would like to submit to your wisdom the following questions.
Is it possible, mutatis mutandis, to repeat the same reasoning for a fractional Brownian ...

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1k views

### Time-dependent Markov process: infinitesimal generator

If one talks about homogeneous Markov diffusion
$$
\mathrm dX_t = \mu(X_t)\mathrm dt+\sigma(X_t)\mathrm dw_t
$$
with $\mu,\sigma$ sufficiently differentiable and of appropriate dimensions, there is ...

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455 views

### Stochastic Integrals and Cauchy Variables

I hope there is a straighforward literature-pointer here.
If I were interested in $\sum_{t=1}^{n} f(t) X_{t}$, where $X_{t}$ consists of independent normal random variables, I could approximate the ...

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### law of iterated logrithm

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. By law of iterated logarithm, one has
...