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We prove central and non-central limit theorems for the
Hermite variations of the anisotropic fractional Brownian sheet
$W^{\alpha, \beta}$
with Hurst parameter $(\alpha, \beta) \in (0,1)2$. When $0<\alpha \leq
1-\frac{1}{2q}$ or $0<\beta \leq 1-\frac{1}{2q}$ a central limit theorem
holds for the renormalized Hermite variations of order $q\geq 2$, while
for $1-\frac{1}{2q}<\alpha, \beta < 1$ we prove that these variations
satisfy a non-central limit theorem. In fact, they converge to a random
variable which is the value of a two-parameter Hermite process at time
$(1,1)$.
The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6
(2010)
Let $B$ be a fractional Brownian motion with Hurst parameter
$H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums
for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability.
We show, however, that they do converge in law in the Skorohod space of
c\`adl\`ag functions. Moreover, we show that the resulting stochastic
integral satisfies a change of variable formula with a correction term
that is an ordinary It\^o integral with respect to a Brownian motion
that is independent of $B$.
In this Note we consider a Lipschitz backward stochastic
differential equation (BSDE) driven by a continuous martingale $M$. We
prove (in Theorem \ref{theorem:main}) that if $M$ is a strong Markov
process and if the BSDE has regular data then the unique solution
$(Y,Z,N)$ of the BSDE is reduced to $(Y,Z)$, \textit{i.e.} the
orthogonal martingale $N$ is equal to zero, showing that in a Markovian
setting the "usual" solution $(Y,Z)$ (of a BSDE with regular data) has
not to be completed by a strongly orthogonal component even if $M$ does
not enjoy the martingale representation property.
We extend some recent works by Delong and Imkeller concerning Backward
stochastic differential equations with time delayed generators (delay
BSDE). We provide sharper a priori estimates and show that the solution
of a delay BSDE is in $L^p$. We introduce decoupled systems of SDE and
delay BSDE (which we term delay FBSDE) and give sufficient conditions
for the variational differentiability of their solutions. We connect
these derivatives to the Malliavin derivatives of such delay FBSDE via
the usual representation formulas which in turn give access to several
path regularity results. In particular we prove an extension of the
$L2$-path regularity result for delay FBSDE.
Model Reduction for a Class of Nonlinear Electrical Circuits by Reduction of Linear Subcircuits
(2010)
We analyze a model reduction approach for a class of electrical circuits containing nonlinear resistance.
In our approach
the linear subcircuits are extracted and replaced with linear passive reduced-order models. The resulting nonlinear
reduced-order model preserves admissibility and passivity. Moreover, we derive a-priori bounds for the error between
the input-output map of the original circuit equations and that of our reduced-order model. These bounds
are valid for all inputs. Since only linear subcircuits are reduced, our approach is effective when the number of nonlinear resistances is relatively small. The performance of our approach is illustrated numerically.
This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type
$$
\om\partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,\ldots,n,
$$
with periodicity conditions in time and reflection boundary conditions in space.
We state sufficient conditions on the data $\om, a_j$ and $b_{jk}$
such that the system has a Fredholm like solvability behavior.
Moreover, we state sufficient conditions on the data such that for any right hand side there exists exactly
one solution, that the solution survives under small perturbations of the data, and that the corresponding
data-to-solution-map
is smooth with respect to appropriate function space norms. In particular, those sufficient conditions
imply that no small denominator effects occur. Moreover, we show that perturbations of the coefficients $\om$ and $a_j$ lead to
essentially different results than perturbations of the coefficients $b_{jk}$, in general.
In this article we compare two different calmness conditions which are
widely used in the literature on bilevel programming and on mathematical
programs with equilibrium constraints. In order to do so, we consider convex
bilevel programming as a kind of intersection between both research areas.
The so-called partial calmness concept is based on the function value
approach for describing the lower level solution set. Alternatively,
calmness in the sense of multifunctions may be considered for perturbations
of the generalized equation representing the same lower level solution set.
Both concepts allow to derive first order necessary optimality conditions
via tools of generalized differentiation introduced by Mordukhovich. They
are very different, however, concerning their range of applicability and the
form of optimality conditions obtained. The results of this paper seem to
suggest that partial calmness is considerably more restrictive than calmness
of the perturbed generalized equation. This fact is also illustrated by
means of a dicretized obstacle control problem.