All Questions
405 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 ...
- 6,195
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
14
votes
1
answer
738
views
For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$
Let stable matrix (i.e., its eigenvalues have negative real parts) $B \in \mathbb R^{n \times n}$ and anti-symmetric matrix $T \in \mathbb R^{n \times n}$ satisfy
$$B^\top - T B^\top = B + B T$$
...
- 371
11
votes
1
answer
498
views
Does Brownian motion immediately visit both sides of a Jordan curve?
Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior).
...
- 113
11
votes
2
answers
4k
views
Parameter estimation for stochastic differential equation from discrete observations
Suppose we have a time-series $x(t_i)$ at discrete times $t_i$ and we want to estimate the parameters of an underlying SDE corresponding to this time-series:
$$dx_t = f(x_t,\theta)dt + \sigma(x_t,\...
- 840
8
votes
2
answers
3k
views
Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?
Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$.
It happens that the ...
- 1,927
8
votes
1
answer
2k
views
total variation distance between two solutions of SDE
Suppose we have two stochastic differential equations with the same initial conditions:
$$d X_t^1= b_1(t,X_t^1)dt + dW_t$$
$$d X_t^2= b_2(t,X_t^2)dt + dW_t,$$
$X_0^1=X_0^2=x_0$; $W_\cdot$ is a ...
- 931
7
votes
1
answer
4k
views
Change of time variable in Wiener process
I'm following a solution of an SDE from here
http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf
Start with the SDE
$$
dX_t = \delta dt + 2\sqrt{X_t} dW_t
$$
consider a deterministic time change
$...
- 173
7
votes
2
answers
613
views
Fractional Brownian motion of Riemann-Liouville type is not a semimartingale
Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...
- 199
7
votes
1
answer
816
views
Solve SDE $dX_t=(c+\sigma_\zeta W'_t)X_tdt + \sigma_\epsilon dW_t$
I am trying to solve the following SDE
$$dX_t=(c+\sigma_\zeta W'_tX_t)dt + \sigma_\epsilon dW_t$$
$c\in \mathbb{R}$ is a constant, $X_t$ is a stochastic process, $\sigma_\zeta,\sigma_\epsilon \in \...
- 211
7
votes
1
answer
249
views
Onsager-Machlup functional when drift is time-dependent
Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by
\begin{align}
\mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i},
\end{align}
where $b_i(x) \in \mathcal{C}_b^2(...
- 203
7
votes
0
answers
304
views
Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories
Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
- 167
6
votes
1
answer
2k
views
Intuition about Skorohod integral
I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on.
In particular ...
- 181
6
votes
2
answers
747
views
Does there exist a stochastic time derivative?
The Setup
Suppose I have a stochastic process $f(Z_t)$ where $Z_t$ solve the $d$-dimensional SDE
$$
dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t
$$
and $f$ is a smooth function.
My Question
Is there a ...
- 5,405
6
votes
1
answer
392
views
Does $E^{x,t}(f(X_T))$ solve a PDE if $f$ is not continuous?
Many books [see below for references] explore the connections between partial differential equations and expectation values.
Assume $X$ is a diffusion with generator $A$, then they conclude, that ...
- 237
6
votes
1
answer
547
views
Diffusion processes with different diffusion coefficients and absolute continuity
I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level.
My initial situation is the following. Consider two stochastic ...
- 61
6
votes
1
answer
684
views
Differentiable dependence on the initial condition of the solution of a SDE
Let
$b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-...
- 167
6
votes
1
answer
387
views
Reference Request: Vector-Valued Ito Formula
I know that there exist Ito formulae to understand
$
f(X),
$
where $f: H\rightarrow \mathbb{R}$ is sufficiently nice, $H$ is a Hilbert space and $X$ is an $H$-valued semi-martingale.
However I'm ...
- 5,405
6
votes
0
answers
88
views
Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
- 5,405
6
votes
0
answers
245
views
Second order calculus and rough paths
In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form
$$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$
where $X$ is a semimartingale on a manifold $M$...
- 9,853
6
votes
0
answers
774
views
Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term
Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...
- 167
5
votes
2
answers
919
views
Analytic Solution to SDEs
Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form:
\begin{equation}
dX_t = f(...
- 379
5
votes
2
answers
311
views
A comparison of diffusions
Consider two diffusions given by
$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$
for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
- 128k
5
votes
1
answer
544
views
Under what condition we get back path from signatures in rough path theory?
A link to wikipedia for rough pat theory is: https://en.wikipedia.org/wiki/Rough_path
It appears path and signatures has one to one mapping in many cases. I understand that the signature is not ...
- 495
5
votes
2
answers
369
views
Markov process on a torus with prescribed invariant distribution
In Euclidean space, $\mathbb R^d$, the Langevin diffusion $${\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
774
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 ...
- 6,195
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 ...
- 6,195
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 ...
- 6,195
5
votes
1
answer
289
views
What is the formal definition of a stochastic PDE and a solution to a stochastic PDE?
While searching through this Wikipedia article, I have stumbled uopn the following 'stochastic' heat equation
$$\partial_tu=\Delta u+\xi,$$
where $\xi$ is the space-time white noise. However, I don't ...
- 153
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
388
views
How can we prove that a stochastic process converges to a deterministic value?
As an illustrative example, consider a modified O-U process $dX_t = -X_tdt + \exp(-t)dW_t$. It is not too hard to understand that after a while the behaviour is dominated by the deterministic ...
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
1
answer
1k
views
Correlated Brownian motions across different times and representation with independent processes
This is a more wide-net question of Two increasingly correlated Brownian motions and Williams decomposition.
In our problem we have two correlated Brownian motions $B^1,B^2$ (starting at time $t=0$ ...
- 5,474
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 ...
- 6,195
4
votes
2
answers
376
views
Gibbs measure as stationary distribution of SDEs
I have been trying to understand how one can mathematically explain some of the results from statistical mechanics, especially regarding certain distributions like the Gibbs distribution. It would be ...
- 807
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 ...
- 835
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 $d$-...
- 835
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
489
views
Regularity for Stochastic heat equation with additive noise in d=2
I would greatly appreciate any references were they study the stochastic equation in higher dimensions: $u_{t}=\Delta u+f$ in great detail, especially in dimension 2.
In Hairer's Spde notes , he ...
- 5,474
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 ...
- 6,195
4
votes
1
answer
302
views
Almost sure stability of a scalar, nonautonomous, nonlinear SDE
I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it.
Maybe there is someone here who can either close the gap in one of the existing answers or has ...
- 1,675
4
votes
1
answer
244
views
Infinite-time, Path-Dependent Expected Value of an Orstein-Uhlenbeck process
I am dealing with an Orstein-Uhlenbeck process $X_t$ with its stochastic differential equation being
$$dX_t=(\mu-X_t)dt+\sigma dW_t.$$
I want to show
$$\mathbb{E}\left[\frac{|X_\infty|}{\int_{0}^{\...
- 389
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 ...
- 6,195