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We examine the empirical performance of some parametric and nonparametric estimators of prices of options with a fixed time to maturity, focusing on variance-gamma and Heston models on one side, and on expansions in Hermite functions on the other side. The latter class of estimators can be seen as perturbations of the classical Black-Scholes model. The comparison between parametric and Hermite-based models having the same "degrees of freedom" is emphasized. The main criterion is the out-of-sample relative pricing error on a dataset of historical option prices on the S&P500 index. Prior to the main empirical study, the approximation of variance-gamma and Heston densities by series of Hermite functions is studied, providing explicit expressions for the coefficients of the expansion in the former case, and integral expressions involving the explicit characteristic function in the latter case. Moreover, these approximations are investigated numerically on a few test cases, indicating that expansions in Hermite functions with few terms achieve competitive accuracy in the estimation of Heston densities and the pricing of (European) options, but they perform less effectively with variance-gamma densities. On the other hand, the main large-scale empirical study show that parsimonious Hermite estimators can even outperform the Heston model in terms of pricing errors. These results underscore the trade-offs inherent in model selection and calibration, and their empirical fit in practical applications., Comment: 35 pages, 2 figures

Let $U,H$ be two separable Hilbert spaces and $T>0$. We consider an SDE which evolves in the Hilbert space $H$ of the form \begin{align} dX(t)=AX(t)dt+\widetilde{\mathscr L}B(X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align} where $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup $(e^{tA})_{t\geq0}$, $W=(W(t))_{t\geq0}$ is a $U$-cylindrical Wiener process defined on a normal filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in [0,T]},\mathbb{P})$, $B:H\to H$ is a bounded and $\theta$-H\"older continuous function, for some suitable $\theta\in(0,1)$, and $\widetilde{\mathscr L}:H\to H$ and $G:U\to H$ are linear bounded operators. We prove that, under suitable assumptions on the coefficients, the weak mild solution to the equation depends on the initial datum in a Lipschitz way. This implies that pathwise uniqueness holds true. Here, the presence of the operator $\Lambda$ plays a crucial role. In particular the conditions assumed on the coefficients cover the stochastic damped wave equation in dimension $1$ and the stochastic damped Euler--Bernoulli Beam equation upto dimension $3$ even in the hyperbolic case., Comment: arXiv admin note: text overlap with arXiv:2308.05415

The current standard model of cosmology successfully describes a variety of measurements, but the nature of its main ingredients, dark matter and dark energy, remains unknown. Euclid is a medium-class mission in the Cosmic Vision 2015-2025 programme of the European Space Agency (ESA) that will provide high-resolution optical imaging, as well as near-infrared imaging and spectroscopy, over about 14,000 deg^2 of extragalactic sky. In addition to accurate weak lensing and clustering measurements that probe structure formation over half of the age of the Universe, its primary probes for cosmology, these exquisite data will enable a wide range of science. This paper provides a high-level overview of the mission, summarising the survey characteristics, the various data-processing steps, and data products. We also highlight the main science objectives and expected performance., Comment: Accepted for publication in the A&A special issue`Euclid on Sky'

In this paper, we consider the linear evolution equation $dy(t)=Ay(t)dt+Gy(t)dx(t)$, where $A$ is a closed operator, associated to a semigroup, with good smoothing effects in a Banach space $E$, $x$ is a nonsmooth path, which is $\eta$-H\"older continuous for some $\eta\in (1/3,1/2]$, and $G$ is a non-smoothing linear operator on $E$. We prove that the Cauchy problem associated with the previous equation admits a unique mild solution and we also show that the solution increases the regularity of the initial datum as soon as time evolves. Then, we show that the mild solution is also an integral solution and this allows us to prove a It\^o formula.

Given an abstract Wiener space $(X,\gamma,H)$, we consider an open set $O\subseteq X$ which satisfies certain smoothness and mean-curvature conditions. We prove that the rescaled resolvent operator associated to the Ornstein-Uhlenbeck operator with homogeneous Dirichlet boundary conditions on $O$ is gradient contractive in $L^p(X,\gamma)$ for every $p\in(1,\infty)$. This is the Gaussian counterpart of an analogous result for the rescaled resolvent operator associated to the Laplace operator $\Delta$ in $L^p$ with respect to the Lebesgue measure, $p\in[1,\infty)$, with homogeneous Dirichlet boundary conditions on a bounded convex open set $O\subseteq \mathbb R^n$.

We present a new high-precision strong lensing model of PLCK G287.0$+$32.9, a massive lens galaxy cluster at $z=0.383$, with the aim to get an accurate estimation of its effective Einstein radius and total mass distribution. We also present a spectroscopic catalog containing accurate redshift measurements for 490 objects, including multiply-lensed sources and cluster member galaxies. We exploit high-quality spectroscopic data from three pointings of the VLT Multi Unit Spectroscopic Explorer, covering a central $3~\rm{arcmin}^2$ region of the cluster. We complete the spectroscopic catalog by including redshift measurements from VLT-VIMOS and KECK-DEIMOS. We identify 129 spectroscopic cluster member galaxies, with redshift values $0.360 \leq z \leq 0.405$ and $m_{\rm{F160W}} \leq 21$, and 24 photometric ones identified with a Convolutional Neural Network from ancillary HST imaging. We also identify 114 multiple images from 28 background sources, of which 84 images from 16 sources are new and the remaining ones were identified in previous work. The best-fitting lens model shows a root mean square separation value between the predicted and observed positions of the multiple images of $0.75''$, corresponding to an improvement in reconstructing the observed positions of the multiple images of a factor of $2.5$ with respect to previous models. Using the predictive power of our new lens model we find 3 new multiple images and we confirm the configuration of three systems of multiple images that were not used for the optimization of the model. The derived total mass distribution confirms this cluster to be a very prominent gravitational lens with an effective Einstein $\theta_{E} = 43.4'' \pm 0.1''$, that is in agreement with previous estimates and corresponds to a total mass enclosed in the critical curve of $M_E = {3.33}_{-0.07}^{+0.02} \times{ 10^{14} M_\odot}$., Comment: submitted to "Astronomy & Astrophysics" and accepted the on January 29th 2024

In this paper we consider the vector-valued Schr\"{o}dinger operator $-\Delta + V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb{R}^d)$ and, for every $x\in\mathbb{R}^d$, $V(x)$ is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in $L^1(\mathbb{R}^d,\mathbb{R}^m).$ Assuming further that the minimal eigenvalue of $V$ belongs to some reverse H\"older class of order $q\in(1,\infty)\cup\{\infty\}$, we obtain maximal inequality in $L^p(\mathbb{R}^d,\mathbb{R}^m)$, for $p$ in between $1$ and some $q$.

In this paper, we prove pathwise uniqueness for stochastic differential equations in infinite dimension. Under our assumptions, we are able to consider the stochastic heat equation up to dimension $3$, the stochastic damped wave equation in dimension $1$ and the stochastic Euler-Bernoulli damped beam equation up to dimension $3$. We do not require that the so-called {\it structure condition} holds true., Comment: Added a section about weak existence, which combined with pathwise uniqueness, implies strong existence

In this paper we prove existence and uniqueness of a mild solution to the Young equation $dy(t)=Ay(t)dt+\sigma(y(t))dx(t)$, $t\in[0,T]$, $y(0)=\psi$. Here, $A$ is an unbounded operator which generates a semigroup of bounded linear operators $(S(t))_{t\geq 0}$ on a Banach space $X$, $x$ is a real-valued $\eta$-H\"older continuous. Our aim is to reduce, in comparison to [4] and [1] (see also [2,5]) in the bibliography, the regularity requirement on the initial datum $\psi$ eventually dropping it. The main tool is the definition of a sewing map for a new class of increments which allows the construction of a Young convolution integral in a general interval $[a,b]\subset \mathbb R$ when the $X_\alpha$-norm of the function under the integral sign blows up approaching $a$ and $X_{\alpha}$ is an intermediate space between $X$ and $D(A)$.

Given an arbitrary choice of two sets of nonzero Boltzmann weights for $n$-color lattice models, we provide explicit algebraic conditions on these Boltzmann weights which guarantee a solution (i.e., a third set of weights) to the Yang-Baxter equation. Furthermore we provide an explicit one-dimensional parametrization of all solutions in this case. These $n$-color lattice models are so named because their admissible vertices have adjacent edges labeled by one of $n$ colors with additional restrictions. The two-colored case specializes to the six-vertex model, in which case our results recover the familiar quadric condition of Baxter for solvability. The general $n$-color case includes important solutions to the Yang-Baxter equation like the evaluation modules for the quantum affine Lie algebra $U_q(\hat{\mathfrak{sl}}_n)$. Finally, we demonstrate the invariance of this class of solutions under natural transformations, including those associated with Drinfeld twisting., Comment: 39 pages, 8 figures

In this paper we consider an abstract Wiener space $(X,\gamma,H)$ and an open subset $O\subseteq X$ which satisfies suitable assumptions. For every $p\in(1,+\infty)$ we define the Sobolev space $W_{0}^{1,p}(O,\gamma)$ as the closure of Lipschitz continuous functions which support with positive distance from $\partial O$ with respect to the natural Sobolev norm, and we show that under the assumptions on $O$ the space $W_{0}^{1,p}(O,\gamma)$ can be characterized as the space of functions in $W^{1,p}(O,\gamma)$ which have null trace at the boundary $\partial O$, or, equivalently, as the space of functions defined on $O$ whose trivial extension belongs to $W^{1,p}(X,\gamma)$.

We consider approximate pricing formulas for European options based on approximating the logarithmic return's density of the underlying by a linear combination of rescaled Hermite polynomials. The resulting models, that can be seen as perturbations of the classical Black-Scholes one, are nonpararametric in the sense that the distribution of logarithmic returns at fixed times to maturity is only assumed to have a square-integrable density. We extensively investigate the empirical performance, defined in terms of out-of-sample relative pricing error, of this class of approximating models, depending on their order (that is, roughly speaking, the degree of the polynomial expansion) as well as on several ways to calibrate them to observed data. Empirical results suggest that such approximate pricing formulas, when compared with simple nonparametric estimates based on interpolation and extrapolation on the implied volatility curve, perform reasonably well only for options with strike price not too far apart from the strike prices of the observed sample., Comment: 44 pages. Final version

We prove maximal Schauder regularity for solutions to elliptic systems and Cauchy problems, in the space $C_b(\mathbb{R}^d;\mathbb{R}^m)$ of bounded and continuous functions, associated to a class of nonautonomous weakly coupled second-order elliptic operators $\bf{\mathcal A}$, with possibly unbounded coefficients and diffusion and drift terms which vary from equation to equation. We also provide estimates of the spatial derivatives up to the third-order and continuity properties both of the evolution operator ${\bf G}(t,s)$ associated to the Cauchy problem $D_t{\bf u}=\bf{\mathcal A}(t){\bf u}$ in $C_b(\mathbb{R}^d;\mathbb{R}^m)$, and, for fixed $\overline t$, of the semigroup ${\bf T}_{\overline t}(\tau)$ associated to the autonomous Cauchy problem $D_{\tau}{\bf u}={\bf{\mathcal A}}(\overline t){\bf u}$ in $C_b(\mathbb{R}^d;\mathbb{R}^m)$. These results allow us to deal with elliptic problems whose coefficients also depend on time.

In this paper we provide sufficient conditions which ensure that the non-linear equation $dy(t)=Ay(t)dt+\sigma(y(t))dx(t)$, $t\in(0,T]$, with $y(0)=\psi$ and $A$ being an unbounded operator, admits a unique mild solution which is classical, i.e., $y(t)\in D(A)$ for any $t\in (0,T]$, and we compute the blow-up rate of the norm of $y(t)$ as $t\rightarrow 0^+$. We stress that the regularity of $y$ is independent on the smoothness of the initial datum $\psi$, which in general does not belong to $D(A)$. As a consequence we get an integral representation of the mild solution $y$ which allows us to prove a chain rule formula for smooth functions of $y$ and necessary conditions for the invariance of hyperplanes with respect to the non-linear evolution equation.

We prove strong well-posedness for a class of stochastic evolution equations in Hilbert spaces H when the drift term is Holder continuous. This class includes examples of semilinear stochastic damped wave equations which describe elastic systems with structural damping (for such equations even existence of solutions in the linear case is a delicate issue) and semilinear stochastic 3D heat equations. In the deterministic case, there are examples of non-uniqueness in our framework. Strong (or pathwise) uniqueness is restored by means of a suitable additive Wiener noise. The proof of uniqueness relies on the study of related systems of infinite dimensional forward-backward SDEs (FBSDEs). This is a different approach with respect to the well-known method based on the Ito formula and the associated Kolmogorov equation (the so-called Zvonkin transformation or Ito-Tanaka trick). We deal with approximating FBSDEs in which the linear part generates a group of bounded linear operators in H; such approximations depend on the type of SPDEs we are considering. We also prove Lipschitz dependence of solutions from their initial conditions.

We investigate the problem of the equivalence of $L^q$-Sobolev norms in Malliavin spaces for $q\in [1,\infty)$, focusing on the graph norm of the $k$-th Malliavin derivative operator and the full Sobolev norm involving all derivatives up to order $k$, where $k$ is any positive integer. The case $q=1$ in the infinite-dimensional setting is challenging, since at such extreme the standard approach involving Meyer's inequalities fails. In this direction, we are able to establish the mentioned equivalence for $q=1$ and $k=2$ relying on a vector-valued Poincar\'e inequality that we prove independently and that turns out to be new at this level of generality, while for $q=1$ and $k>2$ the equivalence issue remains open, even if we obtain some functional estimates of independent interest. With our argument (that also resorts to the Wiener chaos) we are able to recover the case $q\in (1,\infty)$ in the infinite-dimensional setting; the latter is known since the eighties, however our proof is more direct than those existing in the literature, and allows to give explicit bounds on all the multiplying constants involved in the functional inequalities. Finally, we also deal with the finite-dimensional case for all $q\in [1,\infty)$ (where the equivalence, without explicit constants, follows from standard compactness arguments): our proof in such setting is much simpler, relying on Gaussian integration-by-parts formulas and an adaptation of Sobolev inequalities in Euclidean spaces, and it provides again quantitative bounds on the multiplying constants, which however blow up when the dimension diverges to $\infty$ (whence the need for a different approach in the infinite-dimensional setting).

Let $X$ be a separable Banach space and let $Q:X^*\rightarrow X$ be a linear, bounded, non-negative and symmetric operator and let $A:D(A)\subseteq X\rightarrow X$ be the infinitesimal generator of a strongly continuous semigroup of contractions on $X$. We consider the abstract Wiener space $(X,\mu_\infty,H_\infty)$ where $\mu_\infty$ is a centred non-degenerate Gaussian measure on $X$ with covariance operator defined, at least formally, as \begin{align*} Q_\infty=\int_0^{+\infty} e^{sA}Qe^{sA^*}ds, \end{align*} and $H_\infty$ is the Cameron--Martin space associated to $\mu_\infty$. Let $H$ be the reproducing kernel Hilbert space associated with $Q$ with inner product $[\cdot,\cdot]_H$. We assume that the operator $Q_\infty A^*:D(A^*)\subseteq X^*\rightarrow X$ extends to a bounded linear operator $B\in \mathcal L(H)$ which satisfies $B+B^*=-{\rm Id}_H$, where ${\rm Id}_H$ denotes the identity operator on $H$. Let $D$ and $D^2$ be the first and second order Fr\'echet derivative operators, we denote by $D_H$ and $D^2_H$ the closure in $L^2(X,\mu_\infty)$ of the operators $QD$ and $QD^2$ and by $W^{1,2}_H(X,\mu_\infty)$ and and $W^{2,2}_H(X,\mu_\infty)$ their domains in $L^2(X,\mu_\infty)$, respectively,. Furthermore, we denote by $D_{A_\infty}$ the closure of the operator $Q_\infty A^*D$ and by $W^{1,2}_{A_\infty}(X,\mu_\infty)$ its domain in $L^2(X,\mu_\infty)$. We characterize the domain of the operator $L$, associated to the bilinear form \begin{align*} (u,v)\mapsto-\int_{X}[BD_Hu,D_Hv]_Hd\mu_\infty, \qquad u,v\in W^{1,2}_H(X,\mu_\infty), \end{align*} in $L^2(X,\mu_\infty)$. More precisely, we prove that $D(L)$ coincides, up to an equivalent remorming, with a subspace of $W^{2,2}_H(X,\mu_\infty)\cap W^{1,2}_{A_\infty}(X,\mu_\infty)$. We stress that we are able to treat the case when $L$ is degenerate and non-symmetric.

In this paper we consider the symmetric Kolmogorov operator $L=\Delta +\frac{\nabla \mu}{\mu}\cdot \nabla$ on $L^2(\mathbb R^N,d\mu)$, where $\mu$ is the density of a probability measure on $\mathbb R^N$. Under general conditions on $\mu$ we prove first weighted Rellich's inequalities with optimal constants and deduce that the operators $L$ and $-L^2$ with domain $H^2(\mathbb R^N,d\mu)$ and $H^4(\mathbb R^N,d\mu)$ respectively, generate analytic semigroups of contractions on $L^2(\mathbb R^N,d\mu)$. We observe that $d\mu$ is the unique invariant measure for the semigroup generated by $-L^2$ and as a consequence we describe the asymptotic behaviour of such semigroup and obtain some local positivity properties. As an application we study the bi-Ornstein-Uhlenbeck operator and its semigroup on $L^2(\mathbb R^N,d\mu)$.

We consider the spaces $L^p(X,\nu;V)$, where $X$ is a separable Banach space, $\mu$ is a centred non-degenerate Gaussian measure, $\nu:=Ke^{-U}\mu$ with normalizing factor $K$ and $V$ is a separable Hilbert space. In this paper we prove a vector-valued Poincar\'e inequality for functions $F\in W^{1,p}(X,\nu;V)$, which allows us to show that for every $p\in(1,+\infty)$ and every $k\in\mathbb N$ the norm in $W^{k,p}(X,\nu)$ is equivalent to the graph norm of $D_H^k$ (the $k$-th Malliavin derivative) in $L^p(X,\nu)$. To conclude, we show exponential decay estimates for $(T^V(t))_{t\geq0}$ as $t\rightarrow+\infty$. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck $(T(t))_{t\geq0}$, and pointwise estimates for $|D_HT(t)f|_H^p$ by means both of $T(t)|D_Hf|^p_H$ and of $T(t)|f|^p$., Comment: The paper contains many mistakes and it will be completely changed

The Probability Density Function (PDF) provides an estimate of the photometric redshift (zphot) prediction error. It is crucial for current and future sky surveys, characterized by strict requirements on the zphot precision, reliability and completeness. The present work stands on the assumption that properly defined rejection criteria, capable of identifying and rejecting potential outliers, can increase the precision of zphot estimates and of their cumulative PDF, without sacrificing much in terms of completeness of the sample. We provide a way to assess rejection through proper cuts on the shape descriptors of a PDF, such as the width and the height of the maximum PDF's peak. In this work we tested these rejection criteria to galaxies with photometry extracted from the Kilo Degree Survey (KiDS) ESO Data Release 4, proving that such approach could lead to significant improvements to the zphot quality: e.g., for the clipped sample showing the best trade-off between precision and completeness, we achieve a reduction in outliers fraction of $\simeq 75\%$ and an improvement of $\simeq 6\%$ for NMAD, with respect to the original data set, preserving the $\simeq 93\%$ of its content., Comment: Preprint version of the manuscript to appear in the Volume "Intelligent Astrophysics" of the series "Emergence, Complexity and Computation", Book eds. I. Zelinka, D. Baron, M. Brescia, Springer Nature Switzerland, ISSN: 2194-7287

Every field of Science is undergoing unprecedented changes in the discovery process, and Astronomy has been a main player in this transition since the beginning. The ongoing and future large and complex multi-messenger sky surveys impose a wide exploiting of robust and efficient automated methods to classify the observed structures and to detect and characterize peculiar and unexpected sources. We performed a preliminary experiment on KiDS DR4 data, by applying to the problem of anomaly detection two different unsupervised machine learning algorithms, considered as potentially promising methods to detect peculiar sources, a Disentangled Convolutional Autoencoder and an Unsupervised Random Forest. The former method, working directly on images, is considered potentially able to identify peculiar objects like interacting galaxies and gravitational lenses. The latter instead, working on catalogue data, could identify objects with unusual values of magnitudes and colours, which in turn could indicate the presence of singularities., Comment: Preprint version of the manuscript to appear in the Volume "Intelligent Astrophysics" of the series "Emergence, Complexity and Computation", Book eds. I. Zelinka, D. Baron, M. Brescia, Springer Nature Switzerland, ISSN: 2194-7287