All Questions
405 questions
2
votes
0
answers
137
views
Kernel of the adjoint of the infinitesimal generator of Levy SDE
Consider S.D.Es driven by a combination of Brownian and non-Brownian Levy noise (like say Gamma). Then we know that the flow of the density of the S.D.E variable is given by the adjoint of the ...
- 2,246
0
votes
0
answers
53
views
Are the densities of a continuous stochastic process locally positive in time?
Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ a (non-degenerate) interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\...
- 463
2
votes
0
answers
75
views
Is the $\sqrt{{\rm time}}$ spread of a stochastic process about the global minima the ubiquitous phenomenon?
Given a function $f$ with a global minima at $x^*$, consider a stochastic process given as, $x_{t+1} = x_t - \nabla f(x_t) + \xi$ where $\xi$ is a random variable. Now we want to understand the ...
- 2,246
1
vote
1
answer
99
views
Reference to log-transition-density of a diffusion process
Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ given by
$$\mathrm{d}X_t \ = \ b(X_t)\,\mathrm{d}t \ + \ \sigma(X_t)\,\mathrm{d}B_t, \qquad X_0=\xi,$$
with $b$, $\sigma$ smooth, $\xi$ absolutely ...
- 51
1
vote
2
answers
786
views
When does the predictable $\sigma$-algebra $\mathcal{P}$ coincide with the optional $\sigma$-algebra $\mathcal{O}$?
The setup of my question is the following: Suppose that we have a measurable space $(\Omega,\mathcal{F})$ and a filtration $\mathbf{F} = (\mathcal{F}_t)_{t \geq 0}$ on it. Let $\mathcal{P}(\mathbf{F})$...
- 309
2
votes
0
answers
146
views
Exit time for Brownian motion with stochastic barriers
I am interested in the expected exit time of a one-dimensional Brownian particle from a stochastically evolving interval as follows.
Context:
If $L_t$ and $R_t$ denote the distance to the left and ...
- 91
3
votes
1
answer
107
views
Two approaches two SPDEs not equivalent?
I have arrived at needing SPDEs and encountered a strange thing. In the literature, two approaches are mentioned: One where the equation is thought of as an SDE in an infinite dimensional space; an ...
- 183
1
vote
0
answers
61
views
Convergence of empirical measure to Mc-Kean Vlasov equation for mean-field model with jumps
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
- 31
1
vote
1
answer
103
views
BSDE without volatility
Let $(W_t)_{0\leq t\leq 1}$ be a standard Wiener process on $[0,1]$, and let $\mathcal{F}_t$ be the natural filtration. Consider a BSDE
$$
dX_t=f(t,X_t)dt+\sigma(t,X_t) dW_t
$$
with terminal condition ...
- 229
2
votes
1
answer
159
views
Can we show that this transition semigroup preserves a certain Wasserstein space?
Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(...
- 167
2
votes
0
answers
95
views
Itō formula for the solution of a SPDE in the distributional sense
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be open
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...
- 167
1
vote
0
answers
185
views
Ito's Lemma (CVF) on product of Poisson processes
I have the following stochastic differential equation:
$da(t)=\{r(t)a(t)+w(t)−pc(t)\}dt+βa(t)dq(t)$,
with $q(t)$ a Poisson process with arrival rate $λ$ and its increment $dq(t)$ is denoted by:
$dq(t)...
- 11
2
votes
0
answers
215
views
What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
What is the Onsager-Machlup function for $dX(t)=f(B(t)) dt+dB(t)$?
I know that the Onsager-Machlup function for $dX(t)=f(X(t))dt+dB(t)$ is $$L(x,v)=\frac12\left[v-f(x)\right]^2+\frac12f'(x)$$
But ...
- 93
1
vote
1
answer
209
views
What is the drift for a convex combination of Girsanov measures?
Consider two Girsanov measures $\mu_1$ and $\mu_2$ corresponding to drifts $F_1(t)$ and $F_2(t)$ respectively. By this, I mean that we have that $B(t)\sim F_1(t)+\tilde B(t)$ where $\tilde B(t)$ is a ...
- 93
6
votes
1
answer
684
views
Differentiable dependence on the initial condition of the solution of a SDE
Let
$b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-...
- 167
2
votes
0
answers
41
views
If a stochastic flow is Fréchet differentiable in the spatial parameter, does the induced transition semigroup preserve differentiability?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $X:\Omega\times[0,\infty)\times E\to E$ be $(\mathcal A\otimes\mathcal B([0,\infty))\otimes\...
- 167
1
vote
0
answers
80
views
Large deviations estimate for arbitrary continuous function
Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of ...
- 5,405
1
vote
1
answer
90
views
Probability that a stochastic flow is near $0$
Fix $\epsilon>0$ and let $(\Omega,F,F_t\mathbb{P})$ be a stochastic base. Is there a (Markov) diffusion process $X_t$ satisfying an SDE of the form:
$$
d X_t = \mu(t,X_t)dt + \Sigma(t,X_t)dW_t, ...
- 5,405
1
vote
1
answer
259
views
Test for OU-Process
Suppose that I'm given a sample from time-series $(x_n)_{n=1}^N$ and want to decide if it comes from an OU process or not. Is there a (rigorous) test I can use?
So far, everything I've seen is hand-...
- 5,405
1
vote
1
answer
170
views
Diffeomorphism for mapping one SDE into another
Let $Y_t,X_t$ be $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$-adapted Markov diffusion processes with valued in $\mathbb{R}^n$. (When) does there exist a diffeomorphism $\phi:\mathbb{R}^n\to \...
- 5,405
2
votes
2
answers
255
views
Process with covariance $E[Y_{t}Y_{s}]=a_{1}-a_{2}|t-s|$
We have a centered Gaussian process $X_{t}$ where we have exact equality $$E[X_{t}X_{s}]=a_{1}-a_{2}|t-s|$$ for $|t-s|<\epsilon_{0}\ll \frac{a_{1}}{a_{2}}$ and $a_{i}>0$.
Q: I am curious if ...
- 5,474
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 $...
- 11
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
2
votes
1
answer
490
views
Absolute value of a diffusion
Suppose $B_t$ is a standard Brownian motion on a filtered probability space $\langle \Omega, \mathcal F, \{\mathcal F_t\}_t, \mathbb P\rangle$. Consider two SDEs below.
Suppose, $X_0 = Y_0 = 0$
\...
- 553
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. ...
- 331
1
vote
0
answers
15
views
About the role of total variation measure on boundary reflected stochastic processes
I am reading this paper about stochastic differential equations with reflecting boundary conditions. In page 165, an example equation with an explicit solution is presented. However, I can't see that ...
- 11
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
0
votes
1
answer
206
views
Stochastic invariant subset
Let us consider a stochastic differential equation (SDE),
$$
dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}%
$$
and a compact set $C\subset\mathbb{R}^{n}$.
Given a stochastic ...
- 61
0
votes
0
answers
59
views
How to find the PDE for the following transition density
Suppose I have the following two stochastic differential equations ($t\geq 0$)
$$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \ \ \text{ and } \ \ dZ_t =dt,$$
where $X = (X_t)$, $Z = (Z_t).$
Note that
$W=(...
- 345
-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 ...
1
vote
0
answers
659
views
"Expected Value" of a solution to a differential equation
I'm going to write this question in a very informal way as I'm looking for guidance, rather than a specific answer to a specific problem. So I took a course on stochastic processes and Martingales ...
- 597
1
vote
1
answer
206
views
Rough paths theory for Non-Markovian processes
I would like to know whether there is a suitable extension of the theory of rough paths that could be useful to solve Non-Markovian problems.
I would appreciate any example or also any other theory (...
- 13
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-...
- 281
0
votes
1
answer
119
views
Convergence rate estimates of Monte-Carlo first-passage time estimates
Setup
Let $X_t$ be a $d$-dimensional diffusion process solving the Ito-stochastic differential equation
$$
X_t = x+ \int_0^t f(X_t,u_t)dt + \int_0^t \sigma dW_t,
$$
where $x \in \mathbb{R}^d$, $u_t$ ...
- 5,405
1
vote
0
answers
99
views
Large Deviations Principle for First Exit time of a Diffusion Process
Let $b:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be a smooth Lipschitz function, $x \in \mathbb{R}^d$, $\sigma >0$, and consider the solution to the SDE $X_t^x$ defined by
$$
dX_t^x = b(X_t^x)dt + \...
- 5,405
3
votes
1
answer
121
views
Hölder continuity for discrete time process
Let $(X_n)_{n\in\mathbb N}$ be a discrete time stochastic process taking values in a Banach space $E.$ Suppose there exist constants $C,\alpha,\beta>0$ such that $\mathbb E\|X_n-X_m\|^\alpha\leq C|...
- 1,879
0
votes
0
answers
293
views
Malliavin derivative of Ito process
Let $X_t= X_0 + \int_0^t \mu(s,X_s)ds + \int_0^t \sigma(s,X_s)dW_s$ where $\mu$ and $\sigma$ are $C^1$ functions satisfying the usual growth restriction and $W_t$ is a $d$-dimensional Brownian motion. ...
- 5,405
2
votes
1
answer
336
views
Is this a "contradiction" on stochastic Burgers' equation? How to understand it?
For the stochastic Burgers' equation with linear noise, I can deduce two results. Both of them can be applied to same initial data, but the first result means the global existence with high ...
-1
votes
1
answer
2k
views
The probability distribution of "derivative" of a random variable
Disclaimer: Cross-posted in math.SE.
Let me set the stage;
Consider a stochastic PDE, which has to following form
$$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$
where $H$ is a deterministic function, ...
- 133
1
vote
2
answers
413
views
Backward stochastic differential equation
Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and
$$
dX_t=f_tdt+B_tdW_t
$$
where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated ...
- 229
2
votes
0
answers
77
views
Extension of probability space problem: Hilbert space valued process V.S. random field
Maybe the question should be "Understanding the measurability: Hilbert space valued process V.S. random field"
Consider the SPDE $${\rm d}u+\cdots{\rm d}t=\sigma(t,u){\rm d}W.$$
Consider the ...
4
votes
1
answer
417
views
An application of Itô's formula to an SDE on a Lie group
I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t)...
- 41
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:
$$
(1) \hspace{0.5cm} \frac{\partial u}{\partial t}+A(u) \cdot \frac{\partial u}{\partial x} =\nu \cdot \frac{\partial^2 ...
- 657
2
votes
0
answers
139
views
Lebesgue Integral in SDE
In the context of proving existence of solutions of S(P)DEs, I've found that few (if any) texts offer significant mention to the deterministic drift term of the form
$$
\int_0^t f(s,X(s))ds.
$$
If we ...
- 379
2
votes
0
answers
94
views
Defining weak solutions to infinitely many SDEs on the same probability space
Suppose I have an SDE of the form
$$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$
which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ...
- 121
2
votes
0
answers
220
views
How to judge the solution process of an SDE to lie on the sphere?
Consider the following SDE on $\mathbf R^d$:
\begin{equation}\tag{*}
dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d,
\end{equation}
where $W = (W^1,W^2,...
- 261
3
votes
1
answer
289
views
Smoothness of expectation
Suppose that $X_t$ is a strong solution to the SDE,
$$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by ...
- 553
0
votes
0
answers
48
views
Characterization of Time-homogeneous flows for conditional expectation
Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
- 5,405
3
votes
1
answer
345
views
Why control a continuous approximation of stochastic gradient descent instead of just the SGD?
In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in
$$x_{k+1} = x_k - \eta u_k \nabla f_{\...
- 677
3
votes
0
answers
569
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 ...
