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11
votes
1answer
744 views

Bochner integral of stochastic process = path by path Lebesgue integral?

After some helpful comments, I realized that I had to repost this question in a more systematic way. On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
2
votes
1answer
147 views

contraction property for conditioned SDEs

Consider a strictly convex potential $U: \mathbb{R}^d \to \mathbb{R}$ and the Langevin diffusion $$dX = -\nabla U(X) dt + dW \qquad (*)$$ where $W$ is a standard Brownian motion. If $(X_t)_{t \geq 0}$ ...
1
vote
1answer
312 views

Solving an Ornstein-Uhlenbeck-like SDE $y(t,T)=H_t + \mathbb{E}[\int_t^T y(s-,T)dX_s|\mathcal{F}_t]$

I have asked a similar question involving some finance background some time ago here math.stackexchange, however no really good answer came up. I was able to find a solution at least for a special ...
0
votes
2answers
316 views

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 ...
1
vote
1answer
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} ...
1
vote
0answers
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 ...
1
vote
2answers
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 ...
1
vote
0answers
101 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 ...
2
votes
1answer
305 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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 ...
0
votes
1answer
228 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 ...
1
vote
1answer
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 ...
11
votes
0answers
631 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 ...
2
votes
1answer
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 ...
4
votes
1answer
229 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
0answers
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 ...
3
votes
1answer
592 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 ...
3
votes
0answers
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 = ...
3
votes
0answers
171 views

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 $\alpha_t$ ...
0
votes
1answer
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 ...
3
votes
0answers
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 ...
3
votes
3answers
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 $a\in ]0,2]$, ...
6
votes
3answers
304 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
384 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
352 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
votes
2answers
466 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
votes
0answers
271 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; ...
1
vote
0answers
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 ...
1
vote
1answer
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 ...
1
vote
1answer
382 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
483 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
265 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 ...
2
votes
0answers
242 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
vote
1answer
351 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
votes
0answers
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 ...
1
vote
2answers
964 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
votes
2answers
312 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 ...
1
vote
2answers
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 ...
4
votes
0answers
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: $$M_t=\max_{0\leq s\leq t}\,W_s$$ ...
0
votes
1answer
463 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 ...
4
votes
2answers
978 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
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 ...
2
votes
1answer
485 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
373 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: ...
1
vote
1answer
355 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
votes
0answers
952 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
votes
1answer
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 ...
5
votes
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?
4
votes
1answer
497 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 ...