The Fourier cosine expansion (COS) method is used for pricing European options numerically very fast. To apply the COS method, a truncation interval for the density of the log-returns need to be provided. Using Markovs inequality, we derive a new for
mula to obtain the truncation interval and prove that the interval is large enough to ensure convergence of the COS method within a predefined error tolerance. We also show by several examples that the classical approach to determine the truncation interval by cumulants may lead to serious mispricing. Usually, the computational time of the COS method is of similar magnitude in both cases.
We develop a Hilbert-space approach to the diffusion process of the Brownian motion in a bounded domain with random jumps from the boundary introduced by Ben-Ari and Pinsky in 2007. The generator of the process is introduced by a diffusion elliptic d
ifferential operator in the space of square-integrable functions, subject to non-self-adjoint and non-local boundary conditions expressed through a probability measure on the domain. We obtain an expression for the difference between the resolvent of the operator and that of its Dirichlet realization. We prove that the numerical range is the whole complex plane, despite the fact that the spectrum is purely discrete and is contained in a half-plane. Furthermore, for the class of absolutely continuous probability measures with square-integrable densities we characterise the adjoint operator and prove that the system of root vectors is complete. Finally, under certain assumptions on the densities, we obtain enclosures for the non-real spectrum and find a sufficient condition for the non-zero eigenvalue with the smallest real part to be real. The latter supports the conjecture of Ben-Ari and Pinsky that this eigenvalue is always real.
Let $Omegasubset mathbb{R}^n$ be a bounded $C^1$ domain and $p>1$. For $alpha>0$, define the quantity [ Lambda(alpha)=inf_{uin W^{1,p}(Omega),, u otequiv 0} Big(int_Omega | abla u|^p,mathrm{d}x - alpha int_{partialOmega} |u|^p ,mathrm{d} sBig)Big/ in
t_Omega |u|^p,mathrm{d} x ] with $mathrm{d} s$ being the hypersurface measure, which is the lowest eigenvalue of the $p$-laplacian in $Omega$ with a non-linear $alpha$-dependent Robin boundary condition. We show the asymptotics $Lambda(alpha) =(1-p)alpha^{p/(p-1)}+o(alpha^{p/(p-1)})$ as $alpha$ tends to $+infty$. The result was only known for the linear case $p=2$ or under stronger smoothness assumptions. Our proof is much shorter and is based on completely different and elementary arguments, and it allows for an improved remainder estimate for $C^{1,lambda}$ domains.
We show that the eigenvalues of the intrinsic Dirac operator on the boundary of a Euclidean domain can be obtained as the limits of eigenvalues of Euclidean Dirac operators, either in the domain with a MIT-bag type boundary condition or in the whole
space, with a suitably chosen zero order mass term.
We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner openings,
while only rough estimates were available for the next eigenvalues. Under some geometric assumptions, we go beyond the critical eigenvalue number and give a precise asymptotics of any individual eigenvalue by establishing a link with an effective Schrodinger-type operator on the boundary of the domain with boundary conditions at the corners.
Let $Omegasubsetmathbb{R}^N$, $Nge 2,$ be a bounded domain with an outward power-like peak which is assumed not too sharp in a suitable sense. We consider the Laplacian $umapsto -Delta u$ in $Omega$ with the Robin boundary condition $partial_n u=alph
a u$ on $partialOmega$ with $partial_n$ being the outward normal derivative and $alpha>0$ being a parameter. We show that for large $alpha$ the associated eigenvalues $E_j(alpha)$ behave as $E_j(alpha)sim -epsilon_j alpha^ u$, where $ u>2$ and $epsilon_j>0$ depend on the dimension and the peak geometry. This is in contrast with the well-known estimate $E_j(alpha)=O(alpha^2)$ for the Lipschitz domains.
We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schrodinger operators with attractive $delta$-interactions supported by infinite cones. Under the assumption that the cones ha
ve smooth cross-sections, we prove that such operators have infinitely many eigenvalues accumulating below the threshold of the essential spectrum and we express the accumulation rate in terms of the eigenvalues of an auxiliary one-dimensional operator with a curvature-induced potential.
Let $Omega_-$ and $Omega_+$ be two bounded smooth domains in $mathbb{R}^n$, $nge 2$, separated by a hypersurface $Sigma$. For $mu>0$, consider the function $h_mu=1_{Omega_-}-mu 1_{Omega_+}$. We discuss self-adjoint realizations of the operator $L_{mu
}=- ablacdot h_mu abla$ in $L^2(Omega_-cupOmega_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_mu$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Sigma$) and study some properties of its unique self-adjoint extension $mathcal{L}_mu:=overline{L_mu}$. If $mu e 1$, then $mathcal{L}_mu$ simply coincides with $L_mu$ and has compact resolvent. If $n=2$, then $mathcal{L}_1$ has a non-empty essential spectrum, $sigma_mathrm{ess}(mathcal{L}_{1})={0}$. If $nge 3$, the spectral properties of $mathcal{L}_1$ depend on the geometry of $Sigma$. In particular, it has compact resolvent if $Sigma$ is the union of disjoint strictly convex hypersurfaces, but can have a non-empty essential spectrum if a part of $Sigma$ is flat. Our construction features the method of boundary triplets, and the problem is reduced to finding the self-adjoint extensions of a pseudodifferential operator on $Sigma$. We discuss some links between the resulting self-adjoint operator $mathcal{L}_mu$ and some effects observed in negative-index materials.
For $alphain(0,pi)$, let $U_alpha$ denote the infinite planar sector of opening $2alpha$, [ U_alpha=big{ (x_1,x_2)inmathbb R^2: big|arg(x_1+ix_2) big|<alpha big}, ] and $T^gamma_alpha$ be the Laplacian in $L^2(U_alpha)$, $T^gamma_alpha u= -Delta u$,
with the Robin boundary condition $partial_ u u=gamma u$, where $partial_ u$ stands for the outer normal derivative and $gamma>0$. The essential spectrum of $T^gamma_alpha$ does not depend on the angle $alpha$ and equals $[-gamma^2,+infty)$, and the discrete spectrum is non-empty iff $alpha<fracpi 2$. In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle $alpha$. In particular, there is just one discrete eigenvalue for $alpha ge frac{pi}{6}$. As $alpha$ approaches $0$, the number of discrete eigenvalues becomes arbitrary large and is minorated by $kappa/alpha$ with a suitable $kappa>0$, and the $n$th eigenvalue $E_n(T^gamma_alpha)$ of $T^gamma_alpha$ behaves as [ E_n(T^gamma_alpha)=-dfrac{gamma^2}{(2n-1)^2 alpha^2}+O(1) ] and admits a full asymptotic expansion in powers of $alpha^2$. The eigenfunctions are exponentially localized near the origin. The results are also applied to $delta$-interactions on star graphs.
Let $Lambdasubset mathbb{R}^d$ be a domain consisting of several cylinders attached to a bounded center. One says that $Lambda$ admits a threshold resonance if there exists a non-trivial bounded function $u$ solving $-Delta u= u u$ in $Lambda$ and va
nishing at the boundary, where $ u$ is the bottom of the essential spectrum of the Dirichlet Laplacian in $Lambda$. We derive a sufficient condition for the absence of threshold resonances in terms of the Laplacian eigenvalues on the center. The proof is elementary and is based on the min-max principle. Some two- and three-dimensional examples and applications to the study of Laplacians on thin networks are discussed.
