5,416 research outputs found
Another approach to some rough and stochastic partial differential equations
In this note we introduce a new approach to rough and stochastic partial
differential equations (RPDEs and SPDEs): we consider general Banach spaces as
state spaces and -- for the sake of simiplicity -- finite dimensional sources
of noise, either rough or stochastic. By means of a time-dependent
transformation of state space and rough path theory we are able to construct
unique solutions of the respective R- and SPDEs. As a consequence of our
construction we can apply the pool of results of rough path theory, in
particular we obtain strong and weak numerical schemes of high order converging
to the solution process
A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations
We construct normed spaces of real-valued functions with controlled growth on
possibly infinite-dimensional state spaces such that semigroups of positive,
bounded operators thereon with
are in fact strongly continuous. This result applies to prove optimal rates of
convergence of splitting schemes for stochastic (partial) differential
equations with linearly growing characteristics and for sets of functions with
controlled growth. Applications are general Da Prato-Zabczyk type equations and
the HJM equations from interest rate theory
Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case
We consider stochastic (partial) differential equations appearing as
Markovian lifts of affine Volterra processes with jumps from the point of view
of the generalized Feller property which was introduced in
e.g.~\cite{doetei:10}. In particular we provide new existence, uniqueness and
approximation results for Markovian lifts of affine rough volatility models of
general jump diffusion type. We demonstrate that in this Markovian light the
theory of stochastic Volterra processes becomes almost classical.Comment: Revised version with several improvements and corrections. We are
grateful to Sergio Pulido and an anonymous referee for pointing out
inaccuracies. In particular the structure of path properties for generalized
Feller processes is clear no
Smooth perfectness through decomposition of diffeomorphisms into fiber preserving ones
We show that on a closed smooth manifold equipped with fiber bundle
structures whose vertical distributions span the tangent bundle, every smooth
diffeomorphism of sufficiently close to the identity can be written as
a product , where preserves the -fiber. The
factors can be chosen smoothly in . We apply this result to show that
on a certain class of closed smooth manifolds every diffeomorphism sufficiently
close to the identity can be written as product of commutators and the factors
can be chosen smoothly. Furthermore we get concrete estimates on how many
commutators are necessary
Discrete Time Term Structure Theory and Consistent Recalibration Models
We develop theory and applications of forward characteristic processes in
discrete time following a seminal paper of Jan Kallsen and Paul Kr\"uhner.
Particular emphasis is placed on the dynamics of volatility surfaces which can
be easily formulated and implemented from the chosen discrete point of view. In
mathematical terms we provide an algorithmic answer to the following question:
describe a rich, still tractable class of discrete time stochastic processes,
whose marginal distributions are given at initial time and which are free of
arbitrage. In terms of mathematical finance we can construct models with
pre-described (implied) volatility surface and quite general volatility surface
dynamics. In terms of the works of Rene Carmona and Sergey Nadtochiy, we
analyze the dynamics of tangent affine models. We believe that the discrete
approach due to its technical simplicity will be important in term structure
modelling
Fourier transform methods for pathwise covariance estimation in the presence of jumps
We provide a new non-parametric Fourier procedure to estimate the trajectory
of the instantaneous covariance process (from discrete observations of a
multidimensional price process) in the presence of jumps extending the seminal
work Malliavin and Mancino~\cite{MM:02, MM:09}. Our approach relies on a
modification of (classical) jump-robust estimators of integrated realized
covariance to estimate the Fourier coefficients of the covariance trajectory.
Using Fourier-F\'ejer inversion we reconstruct the path of the instantaneous
covariance. We prove consistency and central limit theorem (CLT) and in
particular that the asymptotic estimator variance is smaller by a factor
in comparison to classical local estimators.
The procedure is robust enough to allow for an iteration and we can show
theoretically and empirically how to estimate the integrated realized
covariance of the instantaneous stochastic covariance process. We apply these
techniques to robust calibration problems for multivariate modeling in finance,
i.e., the selection of a pricing measure by using time series and derivatives'
price information simultaneously.Comment: revised and slightly shortened final versio
The proof of Tchakaloff's Theorem
We provide a simple proof of Tchakaloff's Theorem on the existence of
cubature formulas of degree for Borel measures with moments up to order
. The result improves known results for non-compact supports, since we do
not need conditions on st moments
The G\"{a}rtner-Ellis theorem, homogenization, and affine processes
We obtain a first order extension of the large deviation estimates in the
G\"{a}rtner-Ellis theorem. In addition, for a given family of measures, we find
a special family of functions having a similar Laplace principle expansion up
to order one to that of the original family of measures. The construction of
the special family of functions mentioned above is based on heat kernel
expansions. Some of the ideas employed in the paper come from the theory of
affine stochastic processes. For instance, we provide an explicit expansion
with respect to the homogenization parameter of the rescaled cumulant
generating function in the case of a generic continuous affine process. We also
compute the coefficients in the homogenization expansion for the Heston model
that is one of the most popular stock price models with stochastic volatility
An elementary proof of the reconstruction theorem
The reconstruction theorem, a cornerstone of Martin Hairer's theory of
regularity structures, appears in this article as the unique extension of the
explicitly given reconstruction operator on the set of smooth models due its
inherent Lipschitz properties. This new proof is a direct consequence of
constructions of mollification procedures on spaces of models and modelled
distributions: more precisely, for an abstract model of a given regularity
structure, a mollified model is constructed, and additionally, any modelled
distribution can be approximated by elements of a universal subspace of
modelled distribution spaces. These considerations yield in particular a
non-standard approximation results for rough path theory. All results are
formulated in a generic Besov setting
Finite dimensional Realizations of Stochastic Equations
This paper discusses finite-dimensional (Markovian) realizations (FDRs) for
Heath-Jarrow-Morton interest rate models. We consider a d-dimensional driving
Brownian motion and stochastic volatility structures that are non-degenerate
smooth functionals of the current forward rate. In a recent paper, Bj\"ork and
Svensson give sufficient and necessary conditions for the existence of FDRs
within a particular Hilbert space setup. We extend their framework, provide new
results on the geometry of the implied FDRs and classify all of them. In
particular, we prove their conjecture that every short rate realization is
2-dimensional. More generally, we show that all generic FDRs are at least
(d+1)-dimensional and that all generic FDRs are affine. As an illustration we
sketch an interest rate model, which goes well with the Svensson curve-fitting
method. These results cannot be obtained in the Bj\"ork-Svensson setting.
A substantial part of this paper is devoted to analysis on Fr\'echet spaces,
where we derive a Frobenius theorem. Though we only consider stochastic
equations in the HJM-framework, many of the results carry over to a more
general setup
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