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Questions tagged [stochastic-calculus]

Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.

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Convergence of space time white noise in Besov-Holder spaces

In the following we are working on the $d$-dimensional torus and we fix $\alpha <1-d/2.$ We consider a dyadic partition of unity $(\rho_j)_{j \geq -1}$ and we re-define the Holder-Besov space: $$\...
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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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Continuity dependence and convergence of the renormalized $\Phi^4_2$ model

This question is continuous for the one asked here: Local solutions of renormalized stochastic PDE but it was better to ask it separetely. Again, we are interested in the local behavior of the $\Phi_2^...
mathex's user avatar
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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 $\...
hua's user avatar
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Simulation of Markov processes with exponential timestepping

Let $(Y_t)_{t\ge0}$ be a time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. Numerical simulation of $(Y_t)_{t\ge0}$ can be done in the following way: Choose an initial ...
0xbadf00d's user avatar
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4 votes
1 answer
146 views

Local solutions of renormalized stochastic PDE

To illustrate the problem consider the mild formulation of the $\Phi^4_2$ model on $[0,T]\times \mathbb{T}^d$: $$\phi=P_r\phi_0+\int_0^rP_{r-q}(-\phi^3(q))dq+Y_r \ \ \ \ \ \ (1)$$ where $(P_r)_{r \...
mathex's user avatar
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2 votes
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70 views

Stability of Hölder constants of frozen Itô stochastic integrals

$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\...
Akira's user avatar
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5 votes
2 answers
309 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+...
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Gaussian Hypercontractivity of Chaos based on Gaussian with value in Hilbert spaces?

The classical Gaussian hypercontractivity is stated as following: Suppose $\xi$ is a Gaussian variable and $H_n(\xi)$ is the space of n-th homogeneous Wiener chaos constructed from $\xi$, then for any ...
Inuyasha's user avatar
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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 ...
Stocavista's user avatar
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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 ...
Stocavista's user avatar
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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 ...
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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 ...
painday's user avatar
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Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?

$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\...
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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?

$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\...
Akira's user avatar
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2 votes
1 answer
207 views

Does $X_t$ with $t>0$ admit a density?

$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\...
Akira's user avatar
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3 votes
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Regularity structures-paracontrolled distributions: do they always work for sub-critical SPDE?

Stochastic PDE could be solved using either regularity structures or paracontrolled distributions, as long it's sub-critical. I was wondering if this was proven, that is every sub-critical SPDE could ...
mathex's user avatar
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1 answer
117 views

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 Ito process $dX_t = \...
Bombadil's user avatar
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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[\...
Francis Fan's user avatar
2 votes
1 answer
113 views

Constructing Wiener process on a given probability space

This is just a short question, and may be to basic, but: is there a way to construct a sequece of independent wiener processes on a given probability spaces?
Perelman's user avatar
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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 \...
Salini Mendisi's user avatar
3 votes
0 answers
56 views

Wasserstein bounds of interpolation measures

Assume we are given two densities, $p_0$ and $p_1$ on $\mathbb{R}^d$, and define (up to the normalization constant) the interpolation $p_t \propto p_0^{1-t} p_1^t$, which interpolates between $p_0$ ...
mathguy23123's user avatar
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31 views

Can the optimal stopping problem be expressed in another form by strong Markov property?

$X_t$ is a strong Markov process in $(\Omega, \mathcal{F},\mathcal{F}_t,\mathbb{P})$. $\tau$ is a stopping time, $T>0, \mathbb{E}_x(\cdot)=\mathbb{E}(\cdot|X_0=x)$. By Markov property, $\mathop{\rm{...
hua's user avatar
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61 views

White noise: a tempered distribution version of the stochastic convolution

Let $\xi$ be a space-time white noise, that is a centered Gaussian process with covariance $E[\xi_{f}\xi_h]=\int_{\mathbb{R}_+ \times \mathbb{R}^d}fh,$ for $f,h\in L^2(\mathbb{R}_+ \times \mathbb{R}^d)...
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The limit ratio of two Markov Chain Probability

Suppose there are two given SDE in $\mathbb{R}^d$: $$ \begin{align} \left\{ \begin{aligned} dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...
Francis Fan's user avatar
2 votes
1 answer
60 views

Can a lift satisfy Chen's relation, geometric condition but not be a rough path?

Let $(X,\mathbb X):[0,1]^2\to \mathbb R^d\oplus\mathbb R^{d\times d}$ satisfy the following four properties: \begin{align} &X_{s,t}=X_{0,t}-X_{0,s}\\ &\sup_{t\neq s}\frac{|X_{s,t}|}{|t-s|^\...
user479223's user avatar
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1 vote
0 answers
38 views

White noise, stochastic convolution: $0$–$1$ law of a stopping time

Let $\mathscr{C}^\alpha:=B_{\infty,\infty}^{\alpha}$ be the Besov space with the usual norm and let $C_T\mathscr{C}^\alpha:=C([0,T],\mathscr{C}^\alpha)$ the space of continuous functions from $[0,T]$ ...
mathex's user avatar
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0 answers
107 views

A question about one Malliavin derivative calculation

Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
tsnao's user avatar
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0 answers
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Expectation of $B_u \operatorname{argmax}_t B_t$

This question is a repost from math.stackexchange. The question turned out to be harder than I initially thought, so I decided to try my luck here. Yesterday I asked a question about the joint law of ...
tsnao's user avatar
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Example of $F\in W_0^{1,2}$ a.s. so that the law of $F+B$ is equivalent to that of $B$ but DD exponential isn't integrable?

Is there an explicit example of progressively measurable $F=\int_0^\cdot f(s) ds\in W_0^{1,2}(0,1)$ a.s. so that the law of $F+B$ on $(0,1)$ is equivalent to that of a Brownian motion $B$ on $(0,1)$ ...
user479223's user avatar
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7 votes
0 answers
135 views

Stochastic analysis on nuclear Fréchet spaces

This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise. A lot of the time in infinite-...
J_P's user avatar
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0 answers
69 views

Derivative with respect to initial condition for the solution of an SDE

Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution): \begin{align} dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t \end{align} and define its ...
GigaByte123's user avatar
1 vote
0 answers
119 views

Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$

I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...
Yifan's user avatar
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0 answers
88 views

Martingale defined by an integral

Consider a probability space $(\Omega,\mathcal{F},P).$ Let $f \in C^{\infty}_{c}(\mathbb{R}^d,\mathbb{R}),p \geq 2.$ $(X_r^{y})_{(r,y) \in \mathbb{R}_+ \times \mathbb{R}^d}$ is a stochastic process ...
mathex's user avatar
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2 votes
1 answer
201 views

Decay estimate of moment of an SDE

We consider an SDE $$ d X_t = b(t, X_t) \, dt + \sigma(t, X_t) \, d B_t, $$ where $(B_t)$ is a $d$-dimensional Brownian motion on $\mathbb R^d$. We fix $p \in [1, \infty)$. Here $b, \sigma$ are ...
Akira's user avatar
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6 votes
0 answers
78 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 $n$-...
ABIM's user avatar
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3 votes
0 answers
73 views

Norm estimate for parabolic SPDE solution

When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
0xbadf00d's user avatar
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3 votes
0 answers
90 views

Explicit example of drift $F$ so that the law of $F+B$ is not absolutely continuous with respect to $B$

Let $\mu_0$ be the law of Brownian motion on the space of continuous functions. If $\mu\sim\mu_0$ agrees on null sets then there is some progressively measurable $F\in W_0^{1,2}$ a.s. so that $\mu$ is ...
user479223's user avatar
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1 vote
1 answer
94 views

Interchange the deterministic and stochastic integrals

We fix $T >0$ and let $\mathbb T$ be the interval $[0, T]$. Let $(X_t, t \in \mathbb T)$ be a continuous adapted process on some filtered probability space $(\Omega, \mathcal A, (\mathcal F_t)_{t \...
Analyst's user avatar
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2 votes
0 answers
73 views

Assumptions for uniform measure of SDE on manifolds

Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
optimal_transport_fan's user avatar
3 votes
1 answer
190 views

Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process

Consider the modified Ornstein–Uhlenbeck process $$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$ for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...
Jean Daviau's user avatar
0 votes
2 answers
121 views

Find the distribution of maximum of $B_t-t$

Let $B_t$ be a standard Brownian motion. It is easy to show that $\sup B_t-t<\infty$ a.s. . The question is, can we determinate the distribution of $\sup_{t\in [0,\infty)}B_t-t$?
Tiancheng's user avatar
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82 views

Elliptic PDEs in BSDEs and in optimal control

This soft/reference question is related to this MO post of a similar nature. What are some examples of elliptic PDEs appearing in control and BSDEs?
ABIM's user avatar
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2 votes
0 answers
87 views

How to estimate the difference between two Ito diffusions?

Suppose $𝑏:\mathbb R^d \to \mathbb R^d, \sigma:\mathbb R^d \to \mathbb R^{d\times d}$ are measurable functions and satisfy \begin{equation*} 2\langle 𝑥−𝑦,𝑏(𝑥)−𝑏(𝑦)\rangle +\|\sigma(𝑥)−\sigma(�...
epsilon's user avatar
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1 vote
0 answers
67 views

On calculating the second quantization operator $\Gamma(A)$ of the Ornstein-Uhlenbeck operator $A$

Let $A$ be a self-adjoint operator on a Hilbert space , and let $d\Gamma(A)$ be the generator of the second quantization of $A$. Consider the following theorem from Segal's "Non-Linear Quantum ...
matilda's user avatar
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2 votes
0 answers
102 views

Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$

Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...
matilda's user avatar
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2 votes
0 answers
59 views

Autocovariance of harmonic oscillator in fluid (Langevin Equation)

I am looking to work out an analytical solution (if it is known) for the autocovariance $Cov[X_s,X_t]$ of a particle which behaves according to the Langevin equation for a Harmonic Oscillator in a ...
SRB121's user avatar
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3 votes
0 answers
190 views

Flow property for semimartingale driven SDE at a stopping time

Let $S$ be an $n$-dimensional semimartingale such that the SDE $$dX_t = \sigma(X_t, t) \, dS_t$$ with $\sigma$ Lipschitz continuous admits a globally defined unique strong solution on $[0, T]$. For $t ...
Nate River's user avatar
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1 vote
0 answers
116 views

Piecewise Ornstein-Uhlenbeck process time integral

Let $X_t$ be a piecewise Ornstein-Uhlenbeck process with infinitesimal variance $\sigma^2$ and (piecewise) infinitesimal mean $\theta_1$ for $x<c$ where $c$ is a constant and $\theta_2$ for $x\geq ...
17miles's user avatar
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4 votes
1 answer
208 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 $d$-...
Akira's user avatar
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