All Questions
250 questions
23
votes
5
answers
3k
views
What phenomena are better modelled by SDE instead of ODE?
Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE ...
- 6,215
23
votes
1
answer
1k
views
Does a theory of stochastic differential algebras exist?
My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...
- 231
8
votes
2
answers
3k
views
Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?
Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$.
It happens that the ...
- 1,927
7
votes
1
answer
4k
views
Change of time variable in Wiener process
I'm following a solution of an SDE from here
http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf
Start with the SDE
$$
dX_t = \delta dt + 2\sqrt{X_t} dW_t
$$
consider a deterministic time change
$...
- 173
7
votes
1
answer
249
views
Onsager-Machlup functional when drift is time-dependent
Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by
\begin{align}
\mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i},
\end{align}
where $b_i(x) \in \mathcal{C}_b^2(...
- 203
6
votes
1
answer
2k
views
Intuition about Skorohod integral
I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on.
In particular ...
- 181
6
votes
2
answers
748
views
Does there exist a stochastic time derivative?
The Setup
Suppose I have a stochastic process $f(Z_t)$ where $Z_t$ solve the $d$-dimensional SDE
$$
dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t
$$
and $f$ is a smooth function.
My Question
Is there a ...
- 5,405
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
6
votes
1
answer
387
views
Reference Request: Vector-Valued Ito Formula
I know that there exist Ito formulae to understand
$
f(X),
$
where $f: H\rightarrow \mathbb{R}$ is sufficiently nice, $H$ is a Hilbert space and $X$ is an $H$-valued semi-martingale.
However I'm ...
- 5,405
6
votes
0
answers
88
views
Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
- 5,405
6
votes
0
answers
245
views
Second order calculus and rough paths
In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form
$$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$
where $X$ is a semimartingale on a manifold $M$...
- 9,853
6
votes
0
answers
774
views
Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term
Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...
- 167
5
votes
2
answers
919
views
Analytic Solution to SDEs
Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form:
\begin{equation}
dX_t = f(...
- 379
5
votes
2
answers
369
views
Markov process on a torus with prescribed invariant distribution
In Euclidean space, $\mathbb R^d$, the Langevin diffusion $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1,$$ where $\sigma:\mathbb R^d\to\mathbb R^{d\times k}$, $$b:=\frac{\Sigma+U}2\nabla\ln p+...
- 167
5
votes
1
answer
775
views
Best textbooks/resources for "advanced" probability theory?
When I say "Advanced Probability", I mean for a person acquainted with the measure-theoretic foundations of probability theory, that wants to learn about Stochastic Processes from there, in ...
5
votes
3
answers
878
views
Perturbation of a stochastic differential equation
Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively
\begin{align}
dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\
dx &= -(k_0(t)+\epsilon ...
- 2,239
5
votes
2
answers
437
views
A Stochastic Taylor Expansion/Asymptotics
Question:
Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and
$$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$
$(\mu,\sigma)$ obeys the linear growth ...
- 2,239
5
votes
1
answer
392
views
Uniqueness of the solution to some SDE
Consider the stochastic differential equation as follows:
$$X_t=X_0+t+\int_0^t\frac{dW_s}{1+m(s)},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable and $m(t)=\mathbb P[...
- 1,334
5
votes
2
answers
556
views
Conditioning an SDE on the event that the driving noise is small
Let $X$ be the solution to the one dimensional SDE
$dX_t = \mu(t, X_t)dt + \sigma(t, X_t) dW_t$, for $t \in [0, T]$.
with $X_0= x_0$ a.s. for some $x_0 \in \mathbb R$.
Here $W_t$ denotes a standard ...
- 6,215
5
votes
1
answer
828
views
Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$
I am considering the transition semigroup $P_t$ associated with the Ito diffusion process
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$
where the coefficients are assumed to be Lipschitz continuous.
I hope to ...
- 503
5
votes
1
answer
334
views
Does the entropy of a SDE with nondegenerate noise always increase?
Let $W$ be a standard Brownian motion, and let $X$ be the solution to the one dimensional SDE
$$dX_t = \sigma(t, X_t) \, dW_t$$
with initial condition $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. We ...
- 6,215
5
votes
1
answer
336
views
Joint distribution of drawdown time and value of geometric Brownian motion
Let $X$ be a geometric Brownian motion, satisfying the SDE
$$dX_t = \sigma X_t \, dW_t, X_0 = 1.$$
for $W$ a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Define the ...
- 6,215
5
votes
1
answer
372
views
Reference: Stochastic Analysis on Hilbert Manifolds
I'm looking for a reference to a book which develops an It\^{o} lemma for semi-martingales with values in infinite dimensional Hilbert-Manifolds. I expect the techniques to be the same but still I ...
- 5,405
5
votes
1
answer
820
views
Onsager-Machlup function and most probable path of a diffusion process
Let $X_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation
\begin{equation}
dX_{t} = f(X_{t})dt + dW_{t},
\end{equation}
where $f \in C_{b}^{2}(R)$ is a ...
- 83
5
votes
1
answer
531
views
Riemannian metric induced by a stochastic differential equation
Following this paper, a diffusion process in $\mathcal{R}^d$
$$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$
with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be ...
5
votes
0
answers
400
views
Uniform bound for the occupation time of a diffusion
Note: We denote by $\mathcal L(U)$ the Lebesgue measure of a set $U$.
Let $\mu: \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R^{d} \to \mathbb R^{d \times d}$ be Borel functions.
Suppose the ...
- 6,215
4
votes
1
answer
403
views
When are the transition densities of an SDE symmetric?
We fix $T>0$. Let $b:[0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $\sigma:[0, T] \times \mathbb{R}^d \rightarrow \mathcal{M}^\text{sym}_{d \times d}(\mathbb{R})$ be measurable and ...
- 825
4
votes
1
answer
350
views
Reference request: showing that solution of an Ito SDE stays bounded with positive probability
Assume that we have a (well-posed) Ito SDE of the form $$\mathrm{d} X_t = b(X_t)\,\mathrm{d} t + \sigma(X_t)\,\mathrm{d}W_t \label{1}\tag{1},$$ where $b \colon \mathbb{R}^d \to \mathbb{R}^d$, $\sigma \...
- 730
4
votes
1
answer
249
views
Weak uniqueness of an SDE with locally Lipschitz drift and additive noise?
Consider the $d$-dimensional SDE, $d > 1$,
$$dX_t = b(X_t) \, dt + \sqrt 2 \, dW_t$$
where
$b$ is locally Lipschitz such that $|b(x)| \le C |x|^2$ for $x \in \mathbb R^d$.
$W$ is a standard $d$-...
- 825
4
votes
1
answer
509
views
Conditional stochastic integration
Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g.
$$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$
What is the ...
4
votes
1
answer
509
views
What work has been done on SDE with diffusion coefficients of bounded variation in $\mathbb R^d$?
Consider the $d$-dimensional SDE, $d > 1$,
$$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t$$
where $W$ is a standard $d$-dimensional Brownian motion.
I am interested in the case where $\sigma: \mathbb ...
- 6,215
4
votes
1
answer
315
views
Impulse signal detection
Notation: Here $\mathcal Y_t$ denotes the natural filtration of the process $Y_t$, and $\{\cdot\}$ denotes the fractional part of a real number.
This question concerns detecting the presence (or ...
- 6,215
4
votes
1
answer
417
views
An application of Itô's formula to an SDE on a Lie group
I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t)...
- 41
4
votes
1
answer
521
views
Stochastic process with discontinuous drift
While studying a portfolio optimization problem, I came across the process
$$dX(t) = X(t)\,\Big(\,\big(\mu - \alpha\,1_{\{X(t)\,\geq\,C\}}\big)\,dt + \sigma\,dW(t) \Big)$$
which has a discountinuous ...
- 169
4
votes
1
answer
610
views
Malliavin derivative under change of measure
Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$.
Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...
- 141
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 $\...
- 355
4
votes
1
answer
107
views
Identify an SDE on the sphere from its generator
I have a diffusion on the 2-sphere with expression:
$$
(L\phi)(u):=\frac{1}{2{N(u)}}\Big(f(u)\Delta_{\mathbb S^2}\phi+
2g\left( \nabla_{\mathbb S^2}\phi, \nabla_{\mathbb S^2}f\right)\Big)
$$
...
- 73
4
votes
1
answer
181
views
Small noise limits with irregular drift
Let $W$ be a standard $d$-dimensional Brownian motion.
Suppose $b: \mathbb R^d \to \mathbb R^d$ is measurable and bounded. Consider, for every $\varepsilon > 0$, the solution $X^\varepsilon$ on $[0,...
- 6,215
4
votes
1
answer
343
views
Convergence of a continuous time stochastic gradient descent algorithm
Let $f: \mathbb R \to \mathbb R$ be a $C^1$ convex function, satisfying the growth conditions
$$\lim_{x \to -\infty} \nabla f(x) = -\infty, \lim_{x \to \infty} \nabla f(x) = \infty.$$
and let $\...
- 6,215
4
votes
1
answer
146
views
Time Integral over the (positive) Innovations of a Stochastic Process
Consider the Itô-Process
$$X(t) = X_{0} + \int_{0}^{t}\mu(s,X(s))\,ds + \int_{0}^{t}\sigma(s,X(s))\,dW(s)$$
where you can safely assume that the drift $\mu$ and the volatility $\sigma$ satisfy the ...
- 169
4
votes
0
answers
122
views
Finiteness of the moments of the Malliavin derivative of the stochastic heat equation
I am studying section 2.4.2 from Nualart's book "The Malliavin calculus and related topics" on the stochastic heat equation. I have some questions on the validity of some estimates for the ...
- 61
4
votes
0
answers
328
views
Convergence to unique stationary distribution for SDEs and Markov processes
I am interested in understanding the behavior of solutions to stochastic differential equations (SDEs) and continuous-time Markov processes with constant coefficients. Specifically, I would like to ...
- 807
4
votes
0
answers
306
views
A notion of SDE via the martingale representation theorem
$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
- 11.8k
4
votes
0
answers
167
views
Occupation time of SDE
Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
- 93
4
votes
0
answers
145
views
Regularity of martingales with respect to spatial parameters
In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...
- 167
4
votes
0
answers
276
views
Exit time of a stochastic process defined by a SDE
Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation
\begin{align*}
\...
- 205
4
votes
0
answers
414
views
Definition of the Stratonovich integral in Hilbert spaces
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
$B$ be a (standard, real-...
- 167
3
votes
2
answers
2k
views
Kolmogorov continuity theorem and Holder norm
The Kolmogorov Continuity theorem (see for example the Wikipedia page) lets us prove that a stochastic process $X_t$ (on some complete metric space $(S,d)$) is Holder continuous almost surely provided ...
- 175
3
votes
2
answers
380
views
Large deviation bound for O-U process
Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
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
\...
- 719
3
votes
1
answer
628
views
Asymptotic behavior of an integral of OU process
Let $X=(X_t)_{t\ge 0}$ be a stochastic process (Ornstein-Uhlenbeck process) determined by
$$dX_t=-aX_tdt+\sigma dW_t,$$
where $X_0=0$, $a>0$ and $\sigma>0$ are constants, and $W=(W_t)_{t\ge 0}$...
- 1,835