All Questions
Tagged with pr.probability stochastic-differential-equations
237 questions
1
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0
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94
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Generator of a Hilbert space valued Wiener process from the solution of a martingale problem
Let $H$ be a separable $\mathbb R$-Hilbert space, $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$ and $(W_t)_{t\ge0}$ be a $H$-valued Wiener process on a ...
- 167
1
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0
answers
276
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Path dependent Markov property
Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded
\begin{align*}
\Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty)
\end{align*}
Then my question is:...
- 159
1
vote
1
answer
259
views
Show an SDE's solution has positive probability to visit every set in the state space
Let $(\Omega, \mathcal{F},\mathbb{P})$ be a filtered probability space, let $b:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n$ be a continuous function and Lipschitz continuous in the space variable. For ...
- 503
3
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0
answers
235
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Probability of a particle surviving forever
Consider a particle whose position is driven by the following equation:
$$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$
where $y>0$, $0<C<1$...
user128095
3
votes
1
answer
234
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Kac-Rice formula and Borell-TIS inequalities for gradient-flow of centered gaussian random field
Let $x\mapsto g(x)$ be a centered gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow
$$
\dot{x}(t) = -\nabla g(x(t)), \;...
- 6,853
1
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0
answers
62
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Reference request for invariance principles
In various places, an example being
https://projecteuclid.org/download/pdf_1/euclid.aoap/1034625254,
the authors consider a discrete-time process (real-valued, say) $(X_n)_{n \in \mathbb{N}}$, define ...
- 111
1
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2
answers
788
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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
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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
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0
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61
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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
2
votes
1
answer
159
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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
1
answer
495
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Stochastic integral with respect to a random field
I came across a generalized Black-Scholes equation formulation in this paper.
Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...
- 77
2
votes
0
answers
95
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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
2
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0
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215
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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
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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
1
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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
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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:
$$
d X_t = \mu(t,X_t)dt + \Sigma(t,X_t)dW_t, ...
- 5,405
5
votes
2
answers
697
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Intuition behind Gubinelli derivative
I apologise for the confusion of the following sentences. I'm lazy to give more information about Rough path theory as Is a fairly broad subject.
On page 14 of "A Course on Rough Paths
With an ...
- 232
1
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1
answer
259
views
Test for OU-Process
Suppose that I'm given a sample from time-series $(x_n)_{n=1}^N$ and want to decide if it comes from an OU process or not. Is there a (rigorous) test I can use?
So far, everything I've seen is hand-...
- 5,405
1
vote
1
answer
170
views
Diffeomorphism for mapping one SDE into another
Let $Y_t,X_t$ be $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$-adapted Markov diffusion processes with valued in $\mathbb{R}^n$. (When) does there exist a diffeomorphism $\phi:\mathbb{R}^n\to \...
- 5,405
4
votes
1
answer
306
views
How to make sense of recursively defined SPDE solutions, like in Hairer's "Solving the KPZ equation" paper?
In Martin Hairer's 2013 paper "Solving the KPZ equation", the process $X_\epsilon^\bullet$ is defined as the stationary solution to
$$
\partial_t X_\epsilon^{\bullet} = \partial_x^2 X_\epsilon^{\...
- 147
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
2
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1
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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
1
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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
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
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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. ...
- 5,405
0
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1
answer
82
views
In smooth stochastic dynamics, if a Lebesgue-like measure is both forward-time and reverse-time stationary, is the measure necessarily incompressible?
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $X$ be a compact connected $C^\infty$-smooth manifold. Let $F \colon \Omega \times X \to X$ and $\bar{F} \colon \Omega \times X \to ...
- 708
0
votes
1
answer
95
views
If a probability measure is stationary in both forward time and reverse time, does this imply that the measure is incompressible?
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $X$ be a compact metric space. Let $F \colon \Omega \times X \to X$ and $\bar{F} \colon \Omega \times X \to X$ be measurable ...
- 708
3
votes
1
answer
302
views
Core of the generator of squared bessel process in $L^2(\mathbb{R}_+)$
Consider the squared bessel process with generator $$Gf(x)=xf''(x)+f'(x), \ \ x\in\mathbb{R}_+.$$ It is known that the Lebesgue measure is an invariant measure for this process and thus, can be ...
- 407
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
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
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1
answer
111
views
Conditioning on an irrelevant variable in a martingale control problem
Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued $\...
- 553
0
votes
0
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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
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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
$$x_{k+1} = x_k - \eta u_k \nabla f_{\...
- 677
3
votes
0
answers
569
views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
1
vote
1
answer
508
views
Divergence form degenerate pde and Feynman Kac
Consider
$$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$
and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...
- 5,474
1
vote
0
answers
127
views
Gradient bound for the Markov semigroup generated by the solution to an Langevin SDE
Let
$h\in C^2(\mathbb R)$ with $$h''\ge\rho\tag1$$ for some $\rho>0$ and $$\int\underbrace{e^{-h}}_{=:\:\varrho}\:{\rm d}\lambda=1$$
$\mu$ be the measure with density $\varrho$ with respect to the ...
- 167
0
votes
1
answer
204
views
How is the Cauchy-Schwarz inequality used in the proof of Lyapunov's criterion in the book "Analysis and Geometry of Markov Diffusion Operators"
Let $(E,\mu,\Gamma)$ be a full Markov triple (see definition below), $J\in\mathcal A$ with $J\ge1$ and $g\in\mathcal A_0$. In the proof of Theorem 4.6.2 of the book "Analysis and Geometry of Markov ...
- 167
1
vote
1
answer
305
views
Existence of a Lyapunov function for a log-concave measure
Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1$$ and $\mu$ denote the measure with density $e^{-f}$ with respect to the Lebesgue measure on $\...
- 167
1
vote
1
answer
435
views
How to calculate the probability of 2 events happening in time series under only cdf information?
In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
- 29
1
vote
1
answer
154
views
Is there solution to a backward stochastic differential equation with $yz$ in the generator?
Please consider the following backward stochastic differential equation:
$$ Y(s)=\xi+\int_{s}^{T}a(u)Y(u)+b(u)Y(u)Z(u)du-\int_{s}^{T}Z(u)dW(u)$$
Here $a(s)$, $b(s)$ are square-integrable stochastic ...
- 11
3
votes
1
answer
177
views
Convergence of SDEs
Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...
- 5,405
3
votes
0
answers
95
views
"Expanding" around the invariant measure
In the spde literature we have results of the form
$$|P_{t}F(x)-\mu(F)|\leq O(g(t)),\text{for all } x\in H, F\in S$$
where $P_t$ is a semigroup, $H$ some Hilbert space, $F\in S$ some function space, $...
- 5,474
4
votes
1
answer
351
views
Gaussian free field limiting distribution of additive Stochastic heat eqn bounded domain
Hairer in his spdes notes on pg.6, says that GFF is the stationary solution of $u_{t}(z)=\Delta u(z)+\xi(z,t)$, where
$\xi$ is the space-time white noise
$$\xi(x,t)=\sum \sqrt{\lambda_{k}} B_{k}(t)e_{...
- 5,474
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
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,\...
- 167
1
vote
0
answers
134
views
Moment Estimate
Let $\epsilon > 0$ be a small parameter and consider the following lemma.
Lemma. Let $B(t)$ be a bounded, continuous, $R^{n \times n}$-valued function defined on a time interval $[0,T]$ such that ...
- 31
6
votes
2
answers
2k
views
Tanaka-Meyer formula
I have a simple question about Tanaka-Meyer formula, I am having difficulty applying it. Let $X$ be a continous martingale vanishing at zero. From Tanaka-Meyer formula it holds $$d|X_t| = sgn(X_t)dX_t+...
- 69