All Questions
250 questions
2
votes
0
answers
95
views
Itō formula for the solution of a SPDE in the distributional sense
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be open
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...
- 167
1
vote
0
answers
185
views
Ito's Lemma (CVF) on product of Poisson processes
I have the following stochastic differential equation:
$da(t)=\{r(t)a(t)+w(t)−pc(t)\}dt+βa(t)dq(t)$,
with $q(t)$ a Poisson process with arrival rate $λ$ and its increment $dq(t)$ is denoted by:
$dq(t)...
- 11
2
votes
0
answers
215
views
What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
I know that the Onsager-Machlup function for $dX(t)=f(X(t))dt+dB(t)$ is $$L(x,v)=\frac12\left[v-f(x)\right]^2+\frac12f'(x)$$
But ...
- 93
1
vote
1
answer
209
views
What is the drift for a convex combination of Girsanov measures?
Consider two Girsanov measures $\mu_1$ and $\mu_2$ corresponding to drifts $F_1(t)$ and $F_2(t)$ respectively. By this, I mean that we have that $B(t)\sim F_1(t)+\tilde B(t)$ where $\tilde B(t)$ is a ...
- 7,514
2
votes
0
answers
41
views
If a stochastic flow is Fréchet differentiable in the spatial parameter, does the induced transition semigroup preserve differentiability?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $X:\Omega\times[0,\infty)\times E\to E$ be $(\mathcal A\otimes\mathcal B([0,\infty))\otimes\...
- 167
1
vote
0
answers
80
views
Large deviations estimate for arbitrary continuous function
Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of ...
- 5,380
1
vote
1
answer
90
views
Probability that a stochastic flow is near $0$
Fix $\epsilon>0$ and let $(\Omega,F,F_t\mathbb{P})$ be a stochastic base. Is there a (Markov) diffusion process $X_t$ satisfying an SDE of the form:
$$
d X_t = \mu(t,X_t)dt + \Sigma(t,X_t)dW_t, ...
- 5,405
0
votes
3
answers
639
views
Non-smooth Ito lemma for semi-martingales
Is there an extension of Ito's Lemma where $X_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth?
I've been looking but have not found much, any ...
- 63.9k
2
votes
1
answer
596
views
Question about the exit time of a time-homogeneous Itô diffusion
Consider a one-dimensional Itô diffusion:
$$\mathrm{d} X_{t}=b\left(X_{t}\right) \mathrm{d} t+\sigma\left(X_{t}\right) \mathrm{d} B_{t}$$
where $X_0 = 0$ and $B_t$ is the standard Brownian Motion. ...
CommunityBot
- 1
1
vote
0
answers
237
views
On the level of measure theory, what does it mean for a drift to be deterministic?
Given a drift $F\in W^{1,2}([0,T])$ adapted to the filtration of a Brownian motion $B(t)$ on Wiener space $(C[0,T],\mathcal B(\|\cdot \|_\infty)$ with Wiener measure $\mu_0$, there is another measure $...
- 53
2
votes
0
answers
250
views
SDE conditional expectation
Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
- 159
1
vote
0
answers
73
views
conditional expected value and in Stochastic differential equations
Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
- 159
-2
votes
1
answer
138
views
Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?
I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find ...
- 29.8k
2
votes
0
answers
120
views
Taking limits in stochastic partial differential initial value problems
Background: A (stochastic) Cauchy problem I am interested in looks like this:
$$
(1) \hspace{0.5cm} \frac{\partial u}{\partial t}+A(u) \cdot \frac{\partial u}{\partial x} =\nu \cdot \frac{\partial^2 ...
- 657
1
vote
0
answers
106
views
Domain of a reflected stochastic differential equation
I am currently investigating the domain of the infinitesimal generator of a reflected stochastic differential equation (for a smooth and bounded domain) with Lipschitz coefficients. Namely SDEs of the ...
- 63.9k
2
votes
1
answer
182
views
What's the role of commutation relations in stochastic mechanics?
In a stochastic context, we can understand a term like
$$ \int_0^T \frac{d q(t)}{dt} dq $$
either as the (Ito) limit
$$ \lim_{N\to\infty} \sum_{i}^N dq(t_i) \frac{d q(t_i)}{dt} $$
or the (Anti-...
- 16.5k
2
votes
2
answers
557
views
Is the stochastic integral invariant under equivalent change of probability?
Let $(\Omega,\mathcal F, \mathbb F,\mathbb P)$ be a filtered probability space under the usual conditions and suppose $\mathbb Q\sim\mathbb P$ is an equivalent probability measure. Let $X$ be a $\...
CommunityBot
- 1
0
votes
0
answers
294
views
Malliavin derivative of Ito process
Let $X_t= X_0 + \int_0^t \mu(s,X_s)ds + \int_0^t \sigma(s,X_s)dW_s$ where $\mu$ and $\sigma$ are $C^1$ functions satisfying the usual growth restriction and $W_t$ is a $d$-dimensional Brownian motion. ...
- 63.9k
3
votes
2
answers
380
views
Large deviation bound for O-U process
Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X_t = -\alpha X_t dt + \sigma dB_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max_{t\le T} |X_t| \ge z
\...
- 325
-1
votes
1
answer
2k
views
The probability distribution of "derivative" of a random variable
Disclaimer: Cross-posted in math.SE.
Let me set the stage;
Consider a stochastic PDE, which has to following form
$$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$
where $H$ is a deterministic function, ...
- 11
4
votes
1
answer
418
views
An application of Itô's formula to an SDE on a Lie group
I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t)...
- 6,045
2
votes
0
answers
220
views
How to judge the solution process of an SDE to lie on the sphere?
Consider the following SDE on $\mathbf R^d$:
\begin{equation}\tag{*}
dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d,
\end{equation}
where $W = (W^1,W^2,...
- 261
3
votes
1
answer
345
views
Why control a continuous approximation of stochastic gradient descent instead of just the SGD?
In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in
$$x_{k+1} = x_k - \eta u_k \nabla f_{\...
- 4,361
3
votes
0
answers
570
views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
1
vote
0
answers
59
views
Existence and uniqueness of the asymptotic distribution of $x(k+1) = Ax(k) + v(k)$
Consider the linear discrete-time stochastic systems:
\begin{equation}
x_{k+1} = Ax_k + v_k,
\end{equation}
with time-instants $k \in \mathbb{N}$, state $x_k \in \mathbb{R}^n$, stochastic process $v_k ...
- 53
3
votes
0
answers
90
views
Mutual dependencies of BSDE solutions with markovian drivers with different starting points
Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$.
Let ...
- 335
2
votes
0
answers
74
views
Floquet stochastic process
Let $X_t$ be defined by the SDE
$$
dX_t = A(t, X_t)dt + dW_t
$$
where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...
- 205
2
votes
0
answers
140
views
Is there a distinct Ito-Sasaki version of Riemannian stochastic development?
Given a smooth manifold $M$ with a linear torsion-free connection on its tangent bundle, the Eells-Elworthy-Malliavin stochastic development provides a way of transforming a semimartingale $X$ defined ...
- 319
4
votes
0
answers
276
views
Exit time of a stochastic process defined by a SDE
Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation
\begin{align*}
\...
- 205
5
votes
2
answers
437
views
A Stochastic Taylor Expansion/Asymptotics
Question:
Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and
$$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$
$(\mu,\sigma)$ obeys the linear growth ...
- 2,239
3
votes
0
answers
89
views
Why is the Jain Monrad condition the right condition on general Gaussian processes?
Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process.
Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...
CommunityBot
- 1
2
votes
1
answer
391
views
Is there an Itō formula for random functions in infinite-dimensions?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$T>0$
$I:=(0,T]$
$(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\...
CommunityBot
- 1
1
vote
0
answers
235
views
Associative law of the stochastic integral in Hilbert spaces
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$T>0$
$I:=(0,T]$
$(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$
...
- 133
4
votes
0
answers
145
views
Regularity of martingales with respect to spatial parameters
In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...
- 167
5
votes
3
answers
878
views
Perturbation of a stochastic differential equation
Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively
\begin{align}
dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\
dx &= -(k_0(t)+\epsilon ...
- 2,239
2
votes
1
answer
503
views
Generalisation of Strassen's (Kellerer's) Theorem
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^d$ with finite first movements, i.e.
$$\int_{\mathbb R^d}|x|~\mu(dx),\quad \int_{\mathbb R^d}|x|~\nu(dx) \quad <\quad +\infty.$$
$\mu$...
- 345
0
votes
0
answers
76
views
Ornstein-Uhlenbeck type process with thresholding
(Edited) I met a univariate Ornstein-Uhlenbeck type process but with self soft-thresholding:
$$
dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\mu}\big]_+ dt + \sigma dB(t), \quad X(0)=0,
$$
where $...
- 31
2
votes
1
answer
280
views
Walker whose Velocity is a Brownian Bridge
Consider a continuous random walk $x (t) $, in which the velocity $v (t) = \mathrm dx/\mathrm dt $ rather than the position is described by Brownian motion, so that $v (t) = B_t $ where $B_{t+\epsilon}...
- 6,045
4
votes
1
answer
610
views
Stochastic differential equation associated with an optimal control problem
We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time $\...
- 1,675
2
votes
0
answers
74
views
Convergence of empirical measure in case of proliferation
I am currently working on the theory of mean field limits of interacting particles. Here are two slides of a talk from an Italian researcher:
I don't understand why he calls $u(t,x)$ a time dependent ...
- 1,228
3
votes
1
answer
83
views
Filtering Mixed Discrete and Continous
Suppose I have signal process $\lambda_t$ following the dynamics
\begin{equation}
\begin{aligned}
\zeta_t&=\mu^{\zeta}(t,{\zeta}_t)dt+\sigma^{\zeta}(t,{\zeta}_t)dW^{\zeta}_t\\
\xi_t&=\mu^{\xi}(...
- 1,675
2
votes
0
answers
61
views
Assertion of Local Martingale
I am currently reading a proof of the Feynman-Kac representation theorem. The main step in the proof is to consider an "interpolation martingale" which has the form $$M_s := \varphi(t-s, x+B_s)\exp \...
- 21
3
votes
1
answer
528
views
Why would one work with Kushner-FKK equation over Zakai equation?
In stochastic filtering you are interested in a process called the optimal filter $\pi_t$ which is a probability measure(d stochastic process). You can consider the unnormalized version $V_t$.
The ...
- 188k
1
vote
1
answer
242
views
Non-commutative Ito Formula
Does there exist a formula of the Ito lemma for matrix valued processes under matrix multiplication?
That is where
$$
\Delta X_{t+\Delta t} \neq X_{t+\Delta t} - X_t
$$
but instead
$$
\Delta X_t = ...
CommunityBot
- 1
3
votes
2
answers
2k
views
Kolmogorov continuity theorem and Holder norm
The Kolmogorov Continuity theorem (see for example the Wikipedia page) lets us prove that a stochastic process $X_t$ (on some complete metric space $(S,d)$) is Holder continuous almost surely provided ...
- 8,992
1
vote
0
answers
90
views
Onsager-Machlup Function of a Killed Diffusion Process
Given a diffusion process $ X_t $ on a Riemannian manifold $(M,g)$, with an infinitesimal generator $\mathcal{G}=\Delta_g/2 + b$, the Onsager-Machlup function is well-known to be: $$ \mathcal{L}(x,v) =...
- 213
1
vote
0
answers
340
views
Construction of the quadratic variation for Hilbert space valued local martingales
Let
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a ...
- 167
1
vote
0
answers
79
views
Stochastic Control with Stochastic Cost-functional
Is there any literature dealing with a stochastic control problem whose cost-functional $J_t$ is stochastic also?
That is, let $X_t^u$ is the solution to a controlled SDE
$$
dX_t = \mu(t,u_t,X_t^u)dt ...
- 5,405
3
votes
1
answer
110
views
Sequence of diffusions
Can every càdlàg semi-martingale be written as a sequence of diffusions? That is, is the set of continuous semi-martingales dense in some Skorohod space?
- 724
2
votes
0
answers
260
views
Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?
If this is too basic for MathOverflow... say the word and I shall move it to Math.SE
First consider this system of ODEs. Say I have two variables $u$ and $a$, following
$$
\dot u = -u + f(a)
$$
$$
\...
- 19.6k