The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory

@article{Friz2016TheJC,
  title={The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian H{\"o}rmander theory},
  author={Peter K. Friz and Benjamin Gess and Archil Gulisashvili and Sebastian Riedel},
  journal={Annals of Probability},
  year={2016},
  volume={44},
  pages={684-738}
}
We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46–57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions… 
View PDF on arXiv
40 Citations
Highly Influential Citations
1
Background Citations
10

Figures from this paper

40 Citations

Citation Type

Citation Type

Ito formulae for the stochastic heat equation via the theories of rough paths and regularity structures
In this thesis, we develop a general theory to prove the existence of several Ito formulae on the one-dimensional stochastic heat equation driven by additive space-time white noise. That is denoting…
Topics in Stochastic Analysis and Riemannian Foliations
This dissertation contains three research directions. In the first direction, we use rough paths theory to study stochastic differential equations and stochastic partial differential equations. We…
Discrete rough paths and limit theorems
In this article, we consider limit theorems for some weighted type random sums (or discrete rough integrals). We introduce a general transfer principle from limit theorems for unweighted sums to…
Itô’s formula for Gaussian processes with stochastic discontinuities
We introduce a Skorokhod type integral and prove an Ito formula for a wide class of Gaussian processes which may exhibit stochastic discontinuities. Our Ito formula unifies and extends the classical…
Convergence of trapezoid rule to rough integrals
Rough paths techniques give the ability to define solutions of stochastic differential equations driven by signals $X$ which are not semimartingales and whose $p$-variation is finite only for large…
From Rough Path Estimates to Multilevel Monte Carlo
TLDR
This paper investigates, from a numerical analysis point of view, stochastic differential equations driven by Gaussian noise in the aforementioned sense, and focuses on numerical implementations, and more specifically on the savings possible via multilevel methods.
A support and density theorem for Markovian rough paths
We establish two results concerning a class of geometric rough paths $\mathbf{X}$ which arise as Markov processes associated to uniformly subelliptic Dirichlet forms. The first is a support theorem…
Discretisations of rough stochastic partial differential equations
This thesis consists of two parts, in both of which we consider approximations of rough stochastic PDEs and investigate convergence properties of the approximate solutions. In the first part we use…
Taylor Expansions and Castell Estimates for Solutions of Stochastic Differential Equations Driven by Rough Paths
We study the Taylor expansion for the solutions of differential equations driven by $p$-rough paths with $p>2$. We prove a general theorem concerning the convergence of the Taylor expansion on a…
Density Bounds for Solutions to Differential Equations Driven by Gaussian Rough Paths
We consider finite-dimensional rough differential equations driven by centered Gaussian processes. Combining Malliavin calculus, rough paths techniques and interpolation inequalities, we establish…
...
1
2
3
4
...

References

SHOWING 1-10 OF 61 REFERENCES
Malliavin-Skorohod calculus and Paley-Wiener integral for covariance singular processes
We develop a stochastic analysis for a Gaussian process $X$ with singular covariance by an intrinsic procedure focusing on several examples such as covariance measure structure processes,…
A simple proof of distance bounds for Gaussian rough paths
We derive explicit distance bounds for Stratonovich iterated integrals along two Gaussian processes (also known as signatures of Gaussian rough paths) based on the regularity assumption of their…
A generalized Fernique theorem and applications
We prove a generalisation of Fernique's theorem which applies to a class of (measurable) functionals on abstract Wiener spaces by using the isoperimetric inequality. Our motivation comes from rough…
Di erential equations driven by rough signals
This paper aims to provide a systematic approach to the treatment of differential equations of the type dyt = Si fi(yt) dxti where the driving signal xt is a rough path. Such equations are very…
Convergence rates for the full Gaussian rough paths
Under the key assumption of finite {\rho}-variation, {\rho}\in[1,2), of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are…
From Rough Path Estimates to Multilevel Monte Carlo
TLDR
This paper investigates, from a numerical analysis point of view, stochastic differential equations driven by Gaussian noise in the aforementioned sense, and focuses on numerical implementations, and more specifically on the savings possible via multilevel methods.
Smoothness of the density for solutions to Gaussian rough differential equations
We consider stochastic differential equations driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields satisfy Hormander's bracket condition, we demonstrate that…
Rough Stochastic PDEs
In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too high spatial roughness for…
Densities for rough differential equations under Hormander's condition
We consider stochastic differential equations dY = V (Y) dX driven by a multidimensional Gaussian process X in the rough path sense [T. Lyons, Rev. Mat. Iberoamericana 14, (1998), 215-310]. Using…
Stochastic partial differential equations: a rough path view
We discuss regular and weak solutions to rough partial differential equations (RPDEs), thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many previous works on…
...
1
2
3
4
5
...