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.
604 questions
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What do we know about Poisson boundaries of arbitrary Riemannian manifolds?
For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
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Self-adjointness of generator and semigroup of an SDE
$
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$\alpha$ stable processes without jumps
Levy processes with jumps can be formulated following the Levy-kinchkine representation, which provide a decomposition of the characteristic function into three factors corresponding to the diffusion (...
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Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)
Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
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Designing an SDE satisfied by $\frac{B(t)}{1+t}$
Let $B$ be the Brownian motion. I want to find a stochastic differential equation satisfied by the process $$X(t) = \frac{B(t)}{1+t}.$$ I am trying to use Itô's lemma for $f(x,t) = \frac{x}{1+t}$ but ...
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The Ornstein-Uhlenbeck process from modified integrand
Suppose that $\alpha > 0$ and $\sigma \in \mathbb{R}$ are fixed. Define $Y(t), t \geq 0$ to be an adapted modification of the Itô integral
$$
Y(t) = \sigma e^{-\alpha t} \int_0^t e^{\alpha s} dB(s)
...
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Existence and moment estimation for a linear stochastic differential equation (SDE) with random coefficients
Let $W$ be one-demensional Brownian motion, and suppose $X$ satisfies the following SDE
$$
\mathrm{d}X_s=(A_sX_s+B_s)\mathrm{d}s+(C_sX_s+D_s)\mathrm{d}W_s, \quad X_0=x_0\in\mathbb{R}^n,
$$
where $A, C\...
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Can an SDE be made to follow the flow lines of a vector field?
Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE
$$dX_t = V(X_t) \, dW_t,$$
where we identify $V(X_t) \in \mathbb R^n$ with ...
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Bound on the radon-nikodym derivative between two stochastic processes at a time point
I have two stochastic differential equations on $\mathbb{R}^d$ adapted to the same filtration evolving for finite time $t\in [0, T]$ at the same start distribution:
\begin{align*}
dX_t &= b(t, X_t)...
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The unique weak solution to some SDE yields the unique strong solution?
For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, ...
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Approximate the adjoint generator of the discretization of an SDE
Let
$d\in\mathbb N$;
$\sigma\in\mathbb R^{d\times d}$;
$p\in C^1(\mathbb R^d)$ be positive with $$c:=\int p(x)\;{\rm d}x<\infty\tag1$$ and $$b:=\frac12\Sigma\nabla\ln p;$$
$(X_t)_{t\ge0}$ denote ...
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Reference for PDEs from system of SDEs
I'm working with a system of SDEs
\begin{align*}
dX_t &= b(X_t, t) + \sigma dB_t\\
dY_t &= c(X_t, Y_t, t) + \sigma dB_t.
\end{align*}
Here, the Brownian motion is the same.
I know that ...
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What happens to an SDE conditional on the underlying Brownian motion being close to $f \in C[0, T]$?
The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\...
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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 ...
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Can the solution to a controlled SDE with additive noise have non full support?
Let $W$ be a standard $d$-dimensional Brownian motion. Consider the following SDE
$$dX_t = b(X_t, u_t) \, dt + dW_t$$
with initial condition $X_0 = 0$ a.s., $b: \mathbb R^d \times \mathbb R^n \to \...
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Girsanov's theorem for Gaussian measures as the Cameron-martin theorem with a random shift
Let $H \subset E$ be the Cameron-Martin space of a Gaussian measure $\mu$ on a separable Banach space $E$. The Cameron-Martin theorem states that for all $h \in E$ we have $h \in H$ if and only if $\...
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When we should integrate on both side over a SDE?
Maybe I am quite stupid, I am quite confused about, when we should use ito formula to solve SDE and when it is appropriate to integrate directly to get the solution?
Specifically, let us consider the ...
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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 ...
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Is it really interesting to prove well-posedness of unsolved SPDE?
Lots of nonlinear SPDE remained open for decades (especially the non-deterministic ones in higher dimensions because of the regularity of the noise) until Hairer's breakthrough (regularity structures),...
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Existence of SDE solution under integrability of Lipschitz coefficients
I am reading the paper Lan and Wu, Stoch. Process. Appl., 2014, on sufficient conditions weaker than Lipschitzianity for the existence of strong solutions of time-inhomoegneous $d$-dimensional SDEs. ...
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When does an Itô diffusion give a semigroup on $L^2$
I would like a reference for when an Itô diffusion generates a strongly continuous semigroup on $L^2(\mathbb{R}^n)$.
I have a time-homogeneous Itô diffusion of the form
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t$...
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Adding a data-dependent term to the porous medium equation while retaining an explicit solution
I am working with the porous medium equation, which I am treating it as a type of Fokker-Planck equation given by:
$
\frac{\partial u}{\partial t} = \Delta(u^m), \quad m > 1
$
For this equation, ...
- 11
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Diffusions vs elliptic operators with dkp coefficients
I am wondering if there is any literature on the relationship between diffusions and elliptic equations. In particular I am interested in literature concerning operators with Dahlberg–Kenig–Pipher ...
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Malliavin calculus for the regularity of the density of the supremum of a process
I am reading Chapter 2 from Nualart's book 'The Malliavin calculus and related topics'.
Proposition 2.1.10 gives the conditions for the law of the supremum of a process to have a density. Condition (...
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Impulse signal detection
Notation: Here $\mathcal Y_t$ denotes the natural filtration of the process $Y_t$, and $\{\cdot\}$ denotes the fractional part of a real number.
This question concerns detecting the presence (or ...
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Identify an SDE on the sphere from its generator
I have a diffusion on the 2-sphere with expression:
$$
(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)
$$
...
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Simulation of SDEs using Karhunen Loeve expansion
A very common and easy way to simulate the solution of a SDE is to use the Euler-Maruyama method. At each time step the only random part comes from the realization of the increment of the Brownian.
It ...
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Uniqueness of the solution to switching coefficient SDEs
Consider the following SDE driven by real-valued Brownian motion $W=(W_t)_{t\ge 0}$:
$$dX_t = \left(\sigma {\bf 1}_{\{X_t>1\}} + \sigma' {\bf 1}_{\{0<X_t\le 1\}}\right)dW_t,\quad \forall t>0,$...
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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 ...
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Are smooth dynamical systems stabilised by "sufficient noisiness"?
Preliminaries.
(See [1] for further details.)
Let $M$ be a compact connected $C^\infty$ Riemannian manifold.
We say that a list $\sigma_1,\ldots,\sigma_n$ ($n \in \mathbb{N}$) of $C^\infty$ vector ...
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Unique weak solution of an SDE for a general initial distribution
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Equivalence of Wind Forces: Intensity vs. Duration [closed]
The strongest tornado in the world happened recently in Greenfield Iowa with winds over 318 mph: https://www.facebook.com/watch/?v=2176728102678237&vanity=reedtimmer2.0
I am curious, are less ...
user531026
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Pathwise Hölder continuity of Ito diffusions - is this result written anywhere?
Let $X$ be the solution to the multidimensional SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $W$ a Brownian motion, and $\mu, \sigma$ Lipschitz continuous with $\sigma$ nowhere zero. I'm ...
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Combination of the Dirichlet and Cauchy problems, find the PDE by which $\mathbb{E}_x M(X_{\tau_D \wedge t})$ is met
$X_t$ is an Itô diffusion process with continuous version, $\mathbb{L}_X$ is its generator. $D$ is a closed set in $\mathbb{R}$. The stopping time $\tau_D$ is the first entry time of $D$, that is $\...
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Stability of Hölder constants of frozen Itô stochastic integrals
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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+...
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Comparison theorem for SDEs driven by a continuous martingale
Consider the well-known comparison theorem for SDEs, versions of which appear in several textbooks, e.g., Karatzas and Shreve, Proposition 5.2.18, or Revuz and Yor, Theorem IX.3.7.
The result states ...
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Smoothness of resolvent of the infinitesimal generator of an Ito diffusion acting on bounded continuous function
Let $dX_t=\sigma(X_t)\,dW_t+\mu(X_t)\,dt$ be an Ito diffusion with Lipschitz coefficients and $\sigma(x)>0$. Let $f(x)$ be a continuous and bounded and non decreasing function. Can we prove that ...
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Are speed, scale function and killing measures of Itô diffusion absolutely continuous respect to Lebesgue measure and do have smooth derivative?
In Borodin and Salminen's Handbook of Brownian motion (MR1912205, Zbl 1012.60003), pages 16–17, they mention the fact that if the three basic characteristics (speed measure, scale function and killing ...
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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 ...
1
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1
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143
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Ornstein Uhlenbeck process with discontinuous drift
This question is a modified version of this unanswered question asked on MSE, which mainly concerns an Ornstein-Uhlenbeck process with discontinuous drift on $\mathbb R^n$(for simplicity let $n=2$ for ...
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Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?
$
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Conditional expectation w.r.t. filtration of Brownian motion as a continuous map of its paths
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which we define Brownian motion $B$ and let us denote by $\mathcal{F}_t$ its natural filtration. Assume we have Itô process $dX_t = \...
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99
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Rate of convergence of mollified distributions in Besov spaces with negative regularity
Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
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How does the first hitting time depend on the drift of drifted Brownian motion?
Let $W$ be a standard Brownian motion, and $a,b:\mathbb R_+\times \mathbb R\to\mathbb R$ be Lipschitz. Consider the stochastic differential equations:
$$X_t=1+\int_0^ta(s,X_s)ds + W_t,\quad\quad Y_t=1+...
- 1,334
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0
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95
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A stochastic optimal control problem with filtering-like dynamics
I want to extend the following stochastic optimal control problem with randomized feedback control to the continuous time case
\begin{align}
\text{minimize}\quad \mathbb{E}_{\mathbb{H}}&\bigg[\...
- 303
9
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173
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Regularity structures vs Renormalization
What are the substantial differences in the theory of "Regularity Structures" versus perturbative renormalization from Quantum Field Theory?
The idea that to treat divergences inherent to ...
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0
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95
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Brownian bridge as a limit of SDEs
Let $B$ be a Brownian motion and with respect to some probability measure $\mathbf{P}$ and filtration $(\mathcal{F})_{t \geq 0}$ and let $S_\epsilon = \{B_1 \in (-\epsilon,\epsilon)\}$.
For every $t \...
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1
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202
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Perturbation methods for stochastic/partial differential equations
I'm asking for a good reference on perturbation methods for stochastic and/or partial differential equations.
Something like this: Perturbation of a stochastic differential equation
I'm familiar with ...
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4
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0
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159
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Estimating $p$th moment bound of error between small noise SDE and ODE
For a $d$-dimensional standard Brownian motion $W$, and a locally Lipschitz function $b: \mathbb{R}^d \rightarrow \mathbb{R}^d$, consider an SDE:
$$dX_t^\varepsilon = b(X_t) dt + \varepsilon^t dW_t,\...
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