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Capital Requirements and Optimal Investment with Solvency Probability Constraints
Quantifying economic capital and optimally allocating it into portfolios of financial instruments are two key topics in the asset–liability management of an insurance company. In general, these problems are studied in the literature by minimizing standard risk measures such as the value at risk and the conditional VaR. Motivated by Solvency II regulations, we introduce a novel optimization problem to solve for the optimal required capital and the portfolio structure simultaneously, when the ruin probability is used as an insurance solvency constraint. Besides the generic optimal required capital and portfolio problem formulation, we propose a two-model hierarchy of optimization models, where both models admit the so-called second-order conic reformulation, in turn making them particularly well suited for numerics. The first model, albeit naively asserting the normality of the returns on assets and liabilities, under minor further simplifications admits a closed-form solution—a set of formulas, which may be used as simple decision-making guidelines in the analysis of more complex scenarios. A potentially more realistic second model aims to represent the ‘heavy-tailed’ nature of an insurer's liabilities more accurately, while also allowing arbitrary distributions of asset returns via a semi-parametric approach. Extensive numerical simulations illustrate the sensitivity and robustness of the proposed approach relative to the model's parameters. In addition, we explore the potential of insurance risk diversification and discuss if combining several liabilities into a single insurance portfolio may always be beneficial for the insurer. Finally, we propose an extension of the model with an expected return on capital constraint added
On the cohomology of pseudoeffective line bundles
The goal of this survey is to present various results concerning the
cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and
related properties of their multiplier ideal sheaves. In case the curvature is
strictly positive, the prototype is the well known Nadel vanishing theorem,
which is itself a generalized analytic version of the fundamental
Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested
here in the case where the curvature is merely semipositive in the sense of
currents, and the base manifold is not necessarily projective. In this
situation, one can still obtain interesting information on cohomology, e.g. a
Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a
surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his
PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing
theorem that depends on the concept of numerical dimension of a given
pseudoeffective line bundle. The proof of these results depends in a crucial
way on a general approximation result for closed (1,1)-currents, based on the
use of Bergman kernels, and the related intersection theory of currents.
Another important ingredient is the recent proof by Guan and Zhou of the strong
openness conjecture. As an application, we discuss a structure theorem for
compact K{\"a}hler threefolds without nontrivial subvarieties, following a
joint work with F.Campana and M.Verbitsky. We hope that these notes will serve
as a useful guide to the more detailed and more technical papers in the
literature; in some cases, we provide here substantially simplified proofs and
unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the
Abel Symposium, Trondheim, July 201
Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles
The construction of sections of bundles with prescribed jet values plays a
fundamental role in problems of algebraic and complex geometry. When the jet
values are prescribed on a positive dimensional subvariety, it is handled by
theorems of Ohsawa-Takegoshi type which give extension of line bundle valued
square-integrable top-degree holomorphic forms from the fiber at the origin of
a family of complex manifolds over the open unit 1-disk when the curvature of
the metric of line bundle is semipositive. We prove here an extension result
when the curvature of the line bundle is only semipositive on each fiber with
negativity on the total space assumed bounded from below and the connection of
the metric locally bounded, if a square-integrable extension is known to be
possible over a double point at the origin. It is a Hensel-lemma-type result
analogous to Artin's application of the generalized implicit function theorem
to the theory of obstruction in deformation theory. The motivation is the need
in the abundance conjecture to construct pluricanonical sections from flatly
twisted pluricanonical sections. We also give here a new approach to the
original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the
punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi
to a simple application of the standard method of constructing holomorphic
functions by solving the d-bar equation with cut-off functions and additional
blowup weight functions
Strigolactone synthesis is ancestral in land plants, but canonical strigolactone signalling is a flowering plant innovation
Background
Strigolactones (SLs) are an important class of carotenoid-derived signalling molecule in plants, which function both as exogenous signals in the rhizosphere and as endogenous plant hormones. In flowering plants, SLs are synthesized by a core pathway of four enzymes and are perceived by the DWARF14 (D14) receptor, leading to degradation of SMAX1-LIKE7 (SMXL7) target proteins in a manner dependent on the SCFMAX2 ubiquitin ligase. The evolutionary history of SLs is poorly understood, and it is not clear whether SL synthesis and signalling are present in all land plant lineages, nor when these traits evolved.
Results
We have utilized recently-generated genomic and transcriptomic sequences from across the land plant clade to resolve the origin of each known component of SL synthesis and signalling. We show that all enzymes in the core SL synthesis pathway originated at or before the base of land plants, consistent with the previously observed distribution of SLs themselves in land plant lineages. We also show that the late-acting enzyme LATERAL BRANCHING OXIDOREDUCTASE (LBO) may be considerably more ancient than previously thought. We perform a detailed phylogenetic analysis of SMXL proteins and show that specific SL target proteins only arose in flowering plants. We also assess diversity and protein structure in the SMXL family, identifying several previously unknown clades.
Conclusions
Overall, our results suggest that SL synthesis is much more ancient than canonical SL signalling, consistent with the idea that SLs first evolved as rhizosphere signals and were only recruited much later as hormonal signals
Lagrangian Floer superpotentials and crepant resolutions for toric orbifolds
We investigate the relationship between the Lagrangian Floer superpotentials
for a toric orbifold and its toric crepant resolutions. More specifically, we
study an open string version of the crepant resolution conjecture (CRC) which
states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold
and that of its toric crepant resolution coincide after
analytic continuation of quantum parameters and a change of variables. Relating
this conjecture with the closed CRC, we find that the change of variable
formula which appears in closed CRC can be explained by relations between open
(orbifold) Gromov-Witten invariants. We also discover a geometric explanation
(in terms of virtual counting of stable orbi-discs) for the specialization of
quantum parameters to roots of unity which appears in Y. Ruan's original CRC
["The cohomology ring of crepant resolutions of orbifolds", Gromov-Witten
theory of spin curves and orbifolds, 117-126, Contemp. Math., 403, Amer. Math.
Soc., Providence, RI, 2006]. We prove the open CRC for the weighted projective
spaces using an equality between open
and closed orbifold Gromov-Witten invariants. Along the way, we also prove an
open mirror theorem for these toric orbifolds.Comment: 48 pages, 1 figure; v2: references added and updated, final version,
to appear in CM
Cohomology of bundles on homological Hopf manifold
We discuss the properties of complex manifolds having rational homology of
including those constructed by Hopf, Kodaira and
Brieskorn-van de Ven. We extend certain previously known vanishing properties
of cohomology of bundles on such manifolds.As an application we consider
degeneration of Hodge-deRham spectral sequence in this non Kahler setting.Comment: To appear in Proceedings of 2007 conference on Several complex
variables and Complex Geometry. Xiamen. Chin
Exponent Bounds for a Family of Abelian Difference Sets
Which groups G contain difference sets with the parameters (v, k, λ)= (q3 + 2q2 , q2 + q, q), where q is a power of a prime p? Constructions of K. Takeuchi, R.L. McFarland, and J.F. Dillon together yield difference sets with these parameters if G contains an elementary abelian group of order q2 in its center. A result of R.J. Turyn implies that if G is abelian and p is self-conjugate modulo the exponent of G, then a necessary condition for existence is that the exponent of the Sylow p-subgroup of G be at most 2q when p = 2 and at most q if p is an odd prime. In this paper we lower these exponent bounds when q ≠p by showing that a difference set cannot exist for the bounding exponent values of 2q and q. Thus if there exists an abelian (96, 20, 4)-difference set, then the exponent of the Sylow 2-subgroup is at most 4. We also obtain some nonexistence results for a more general family of (v, k, λ)-parameter values
Extension of holomorphic functions and cohomology classes from non reduced analytic subvarieties
The goal of this survey is to describe some recent results concerning the L 2
extension of holomorphic sections or cohomology classes with values in vector
bundles satisfying weak semi-positivity properties. The results presented here
are generalized versions of the Ohsawa-Takegoshi extension theorem, and borrow
many techniques from the long series of papers by T. Ohsawa. The recent
achievement that we want to point out is that the surjectivity property holds
true for restriction morphisms to non necessarily reduced subvarieties,
provided these are defined as zero varieties of multiplier ideal sheaves. The
new idea involved to approach the existence problem is to make use of L 2
approximation in the Bochner-Kodaira technique. The extension results hold
under curvature conditions that look pretty optimal. However, a major unsolved
problem is to obtain natural (and hopefully best possible) L 2 estimates for
the extension in the case of non reduced subvarieties -- the case when Y has
singularities or several irreducible components is also a substantial issue.Comment: arXiv admin note: text overlap with arXiv:1703.00292,
arXiv:1510.0523
Positivity of relative canonical bundles and applications
Given a family of canonically polarized manifolds, the
unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the
relative canonical bundle . We use a global elliptic
equation to show that this metric is strictly positive on , unless
the family is infinitesimally trivial.
For degenerating families we show that the curvature form on the total space
can be extended as a (semi-)positive closed current. By fiber integration it
follows that the generalized Weil-Petersson form on the base possesses an
extension as a positive current. We prove an extension theorem for hermitian
line bundles, whose curvature forms have this property. This theorem can be
applied to a determinant line bundle associated to the relative canonical
bundle on the total space. As an application the quasi-projectivity of the
moduli space of canonically polarized varieties
follows.
The direct images , , carry natural hermitian metrics. We prove an
explicit formula for the curvature tensor of these direct images. We apply it
to the morphisms that are induced by the Kodaira-Spencer map and obtain a differential
geometric proof for hyperbolicity properties of .Comment: Supercedes arXiv:0808.3259v4 and arXiv:1002.4858v2. To appear in
Invent. mat
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