Questions tagged [stochastic-differential-equations]
Stochastic ordinary and partial differential equations generalize the concepts of ordinary and partial differential equations to the setting where the unknown is a stochastic process.
558
questions
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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*}
\...
3
votes
0
answers
315
views
An SDE version of a Fokker Planck Equation
Assume $\rho$ is probability density defined on $\mathbb{R}^d\times\mathbb{R}^d$. I am interested in the Wasserstein gradient flow of a functional:
\begin{equation*}
\mathcal{E}(\rho)=\iint_{\mathbb{...
5
votes
2
answers
401
views
Is it possible to compare Rough path theory and White noise Theory?
Please accept apology if this question is vague. (Would you please comment rather then downvote, I may be stopped to ask more questions. I will delete my question if required.)
It is related to the ...
0
votes
0
answers
75
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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 $...
5
votes
3
answers
758
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
votes
1
answer
355
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,\...
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 ...
1
vote
0
answers
63
views
Martingale covariation operator in infinite-dimensions
Let
$(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ be a filtered probability space
$U,H$ be separable $\mathbb R$-Hilbert spaces
$(e_n)_{n\in\mathbb N}$ and $(f_n)_{n\in\mathbb N}$...
6
votes
2
answers
2k
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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 $$d|X_t| = sgn(X_t)dX_t+...
4
votes
0
answers
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Optimal control of SDEs
I've set up a system of stochastic differential equations that I'd like to control. I'm new to optimal control theory and SDEs (and, admittedly, weak on PDEs), so I'm not certain if I've set this up ...
3
votes
1
answer
355
views
Estimate for the composition of two Hilbert-Schmidt operators
Let
$U$, $H$, $\tilde H$ be infinite-dimensional separable $\mathbb R$-Hilbert spaces
$Q$ be a self-adjoint and nonnegative nuclear linear operator on $U$
$\Psi$ be a Hilbert-Schmidt operator from$^1$...
9
votes
1
answer
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Is there any reason to use paracontrolled calculus over regularity structures?
Paracontrolled calculus was developed by Gubinelli, Imkeller and Perkowski as a way of treating singular stochastic PDEs such as KPZ, $\Phi_3^4$ or PAM, around the same time regularity structures were ...
3
votes
1
answer
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Stopping time property
Hi, I am reading a textbook about SDE, and am very confused about the transition
$$X_T 1_{T\lt t} + E\{X_T 1_{T\geq t} | F_{t\wedge T}\}$$
$$= X_T 1_{T\lt t} + E\{X_T | F_t\} 1_{T\geq t}$$
I ...
4
votes
1
answer
295
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Almost sure stability of a scalar, nonautonomous, nonlinear SDE
I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it.
Maybe there is someone here who can either close the gap in one of the existing answers or has ...
2
votes
1
answer
275
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Walker whose Velocity is a Brownian Bridge
Consider a continuous random walk $x (t) $, in which the velocity $v (t) = \mathrm dx/\mathrm dt $ rather than the position is described by Brownian motion, so that $v (t) = B_t $ where $B_{t+\epsilon}...
3
votes
1
answer
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Walk with randomised boosts
The classical random walk can be described as the evolution of the position $X_t$ of a walker for integers $t \geqslant 0$, where $X_0 = 0$ and $X_t = X_{t-1} + V_t$ for $t \geqslant 1$, where the "...
5
votes
3
answers
987
views
How to define (and solve) the diffusion equation with a sticky boundary at the origin?
For the diffusion equation $\frac{\partial} {\partial t} P_t(x)=D \frac{\partial^2} {\partial x^2} P_t(x)$, a reflecting boundary at the origin for example, means: $\frac{\partial} {\partial x} P_t(...
4
votes
1
answer
241
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Infinite-time, Path-Dependent Expected Value of an Orstein-Uhlenbeck process
I am dealing with an Orstein-Uhlenbeck process $X_t$ with its stochastic differential equation being
$$dX_t=(\mu-X_t)dt+\sigma dW_t.$$
I want to show
$$\mathbb{E}\left[\frac{|X_\infty|}{\int_{0}^{\...
3
votes
0
answers
79
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Generalisation of Lyapunov time to stochastic dynamical systems
Might there be useful generalisations of the Lyapunov time to stochastic dynamical systems? In particular, I'm interested in methods for calculating confidence intervals around stochastic analogues of ...
3
votes
0
answers
197
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Reference request on connection between PDE problems
I am trying to find references in the literature that connect solutions of any two of the problems given bellow. I study deterministic and stochastic conservation laws. Problems that I am interested ...
2
votes
0
answers
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What type of boundary (if any) problem for this family of elliptic PDEs? "half boundary"?
Classic literature for a general elliptic PDE with Dirichlet boundary condition is typically studied with the following set up: Let $\Omega \subset R^n$ be some open bounded domain and $\partial \...
4
votes
1
answer
316
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Asymptotic form of pdf of Escape Time of arithmetic fBm
I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems ...
2
votes
1
answer
481
views
Time interval of existence of an SDE solution with locally Lipschitz drift
Consider the stochastic ODE $$
dX = F(X) \, dt + dB
$$
where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "...
2
votes
0
answers
70
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 ...
7
votes
2
answers
694
views
Probabilistic interpretation for Fokker-Planck equation
It is well known that if $X_t$ is a stochastic process that solves the SDE
$$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$
with $W_t$ a Wiener process, then the associated ...
1
vote
0
answers
229
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 $(\Omega,\mathcal A)$
...
2
votes
0
answers
166
views
Singularity of the solution of a PDE whose coefficients have zeros
The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post):
$$\mathcal{A}p=0, \quad p\in C^2(\...
14
votes
1
answer
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For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$
Let stable matrix (i.e., its eigenvalues have negative real parts) $B \in \mathbb R^{n \times n}$ and anti-symmetric matrix $T \in \mathbb R^{n \times n}$ satisfy
$$B^\top - T B^\top = B + B T$$
...
2
votes
0
answers
101
views
Stochastic stability of "open" continuous-time stochastic systems: reference request
I'm looking for results on the stability of stochastic systems, e.g. SDEs, whose coefficients depend on a different process that is not necessarily stable. I'm calling those systems "open" here, but ...
4
votes
1
answer
504
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Conditional stochastic integration
Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g.
$$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$
What is the ...
1
vote
1
answer
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Generator of spacetime Markov Process is Parabolic?
Suppose one considers some non-autonomous SDE thereby the Markov transition function is not homogeneous. In order to "recover" some homogeneity, one can consider the "spacetime" or "lifted" ...
2
votes
2
answers
496
views
Is the stochastic integral invariant under equivalent change of probability?
Let $(\Omega,\mathcal F, \mathbb F,\mathbb P)$ be a filtered probability space under the usual conditions and suppose $\mathbb Q\sim\mathbb P$ is an equivalent probability measure. Let $X$ be a $\...
3
votes
1
answer
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Solution of multivariate Geometric Brownian Motion?
It is known how to solve the SDE $dX=X\,dW$ to get a closed form expression of $X(t)=\exp(W_t-\frac{t}{2})$. The question is, is there also a way to solve
\begin{equation} \begin{cases}
dX=X \, dW_1+...
3
votes
1
answer
446
views
Regularity for Stochastic heat equation with additive noise in d=2
I would greatly appreciate any references were they study the stochastic equation in higher dimensions: $u_{t}=\Delta u+f$ in great detail, especially in dimension 2.
In Hairer's Spde notes , he ...
2
votes
0
answers
60
views
Assertion of Local Martingale
I am currently reading a proof of the Feynman-Kac representation theorem. The main step in the proof is to consider an "interpolation martingale" which has the form $$M_s := \varphi(t-s, x+B_s)\exp \...
3
votes
0
answers
72
views
Uniqueness of a stationary measure for a simple ODE
How to prove uniqueness of the stationary measure using coupling approach to the following process: $u_{n+1}=S(1)u_n+\eta_{n+1}(\omega)$, where
a) $S(1)y_0:=y(1)$, $y(t)$ solves the following Cauchy ...
1
vote
0
answers
104
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 ...
0
votes
0
answers
349
views
stochastic recurrence relation "convergence"
Consider a recurrence relation $x_n = f(x_{n-1})$. Suppose $f(x_n)<0$ for some "large enough" $n$. Now consider a "stochastic variant" $x_n = f(x_{n-1})+y_n$ where $y_n$ is a sequence of random ...
5
votes
0
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views
Uniqueness of a SDE with non-negativity constraint
I am working on the following SDE (but we will dealing only with deterministic object: $\omega\in\Omega$ is fixed):
\begin{equation}\label{sde}%sde
x_t=\underbrace{\xi_0+\int_0^tb(s,x_s)\,ds+\int_0^t\...
3
votes
1
answer
498
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 ...
4
votes
1
answer
381
views
Existence of normal number except random numbers
For normality, see https://en.wikipedia.org/wiki/Normal_number. For random number/sequence, see https://en.wikipedia.org/wiki/Algorithmically_random_sequence.
Now, is there any number that is normal ...
1
vote
0
answers
83
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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: $$ \mathcal{L}(x,v) =...
3
votes
2
answers
2k
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Kolmogorov continuity theorem and Holder norm
The Kolmogorov Continuity theorem (see for example the Wikipedia page) lets us prove that a stochastic process $X_t$ (on some complete metric space $(S,d)$) is Holder continuous almost surely provided ...
1
vote
0
answers
334
views
Construction of the quadratic variation for Hilbert space valued local martingales
Let
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a ...
4
votes
1
answer
312
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Uniqueness of a SDE with positivity constraint
We start by fixing some notation.
If $x\in\Bbb R^N$, we denote the usual euclidean norm in $\Bbb R^N$ with $\|x\|$: we omit the reference to the space $\Bbb R^N$ or to the dimension $N$ since it ...
3
votes
1
answer
651
views
Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined
Remark: I've asked this question on MSE as well.
Let
$T>0$
$I:=[0,T]$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in I}$ be a complete and right-continuous ...
3
votes
2
answers
344
views
Large deviation bound for O-U process
Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
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
\...
3
votes
1
answer
78
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}(...
2
votes
1
answer
456
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.
$$\int_{\mathbb R^d}|x|~\mu(dx),\quad \int_{\mathbb R^d}|x|~\nu(dx) \quad <\quad +\infty.$$
$\mu$...
1
vote
0
answers
226
views
Ito's formula for jump diffusions
Suppose I have $dP_t^i = (r^i + h_i^{\mathbb{P}})P_t^i dt - P_{t-}^i dH_t^i$ where $H_i(t) = \mathbb{1}_{\tau_t \leq t}$ denotes a default indicator process of i. $\tau_i$ is the default time and $h_i$...