Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.

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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 $a\in ]0,2]$, ...
6
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3answers
310 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 ...
1
vote
1answer
449 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 ...
5
votes
2answers
396 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 ...
5
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2answers
470 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 $X$ ...
6
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0answers
283 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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0answers
312 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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1answer
511 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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1answer
427 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 ...
5
votes
3answers
525 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 ...
3
votes
1answer
273 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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0answers
273 views

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 ...
1
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1answer
355 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 ...
3
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0answers
223 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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2answers
1k views

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? ...
2
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2answers
323 views

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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2answers
232 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 ...
8
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1answer
439 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: $$M_t=\max_{0\leq s\leq t}\,W_s$$ ...
0
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1answer
520 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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2answers
1k 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) + ...
5
votes
2answers
454 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 ...
2
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1answer
505 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
1answer
382 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: ...
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1answer
363 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 ...
5
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0answers
1k views

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 ...
3
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1answer
617 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 ...
5
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1answer
1k views

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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1answer
542 views

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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1answer
2k 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 ...
2
votes
3answers
466 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 ...
4
votes
2answers
483 views

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 ...
4
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1answer
1k views

What does progressively measurable actually entail?

There is a definition that has always left nagging questions in my mind. To set it up, let $(\Omega, \cal{A}, (\cal{F}_t)_{t\geq 0}, P)$ be a filtered probability space. From Comets & Meyre's ...
9
votes
1answer
860 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
2answers
932 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 ...
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2answers
270 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?
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0answers
263 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
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1answer
391 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
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1answer
289 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 ...
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2answers
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 ...
10
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1answer
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). ...
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2answers
377 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
1answer
495 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
1answer
354 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
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1answer
462 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
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1answer
337 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, ...
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2answers
977 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
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3answers
346 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 ...
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2answers
206 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
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1answer
530 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}^+), ...
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0answers
199 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: ...