Extremal Behavior of Diffusion Models in Finance

We investigate the extremal behavior of a diffusion X _t given by the SDE , where W is standard Brownian motion, μ is the drift term and σ is the diffusion coefficient. Under some appropriate conditions on X _t we prove that the point process of ε -upcrossings converges in distribution to a homogeneous Poisson process. As examples we study the extremal behavior of term structure models or asset price processes such as the Vasicek model, the Cox–Ingersoll–Ross model and the generalized hyperbolic diffusion. We also show how to construct a diffusion with pre-determined stationary density which captures any extremal behavior. As an example we introduce a new model, the generalized inverse Gaussian diffusion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Exterior Monge-Ampère solutions

We discuss the Siciak-Zaharjuta extremal function of a real convex body in Cn, a solution of the homogeneous complex Monge-Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadratic curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain. Finally, we construct extremal curves for two non-convex bodies in R².     

A mean-reverting SDE on correlation matrices

We introduce a mean-reverting SDE whose solution is naturally defined on the space of correlation matrices. This SDE can be seen as an extension of the well-known Wright–Fisher diffusion. We provide conditions that ensure weak and strong uniqueness of the SDE, and describe its ergodic limit. We also shed light on a useful connection with Wishart processes that makes understand how we get the full SDE. Then, we focus on the simulation of this diffusion and present discretization schemes that achieve a second-order weak convergence. Last, we give a possible application of these processes in finance and argue that they could easily replace and improve the standard assumption of a constant correlation.     

Definition of the boundary in the local Charzynski-Tammi conjecture

According to the Charzynski-Tammi conjecture, the symmetrized Pick function is extremal in the problem on the estimate for the nth Taylor coefficient in the class of holomorphic univalent functions close to the identical one. In this paper we find the exact value of M ₄ such that the symmetrized Pick function is locally extremal in the problem on the estimate for the 4th Taylor coefficient in the class of holomorphic normalized univalent functions, whose module is bounded byM ₄.     

Stability estimates of solutions to extremal problems for a nonlinear convection-diffusion-reaction equation

We consider an identification problem for a stationary nonlinear convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of the substance. This problem is reduced to an inverse extremal problem by an optimization method. The solvability of the boundary value problem and the extremal problem is proved. In the case that the reaction coefficient is quadratic, when the equation acquires cubic nonlinearity, we deduce an optimality system. Analyzing it, we establish some estimates of the local stability of solutions to the extremal problem under small perturbations both of the quality functional and the given velocity vector which occurs multiplicatively in the convection–diffusion–reaction equation.     

Stability of solutions of control problems for the convection–diffusion–reaction equation with a strong nonlinearity

We consider a boundary control problem for the stationary convection–diffusion–reaction equation in which the reaction constant depends on the concentration of matter in such a way that the equation has a fifth-order nonlinearity. We prove the solvability of the boundary value problem and an extremal problem, derive an optimality system, and analyze it to derive estimates for the local stability of the solution of the extremal problem under small perturbations of both the performance functional and one of the given functions.     

ON EXTREMALITY AND UNIQUE EXTREMALITY OF TEICHMULLER MAPPINGS

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Filtering for Non-Markovian SDEs Involving Nonlinear SPDEs and Backward Parabolic Equations

We study a filtering problem for non-Markovian SDE’s where the drift vector fields commute with diffusion vector fields. The evolution of the conditioned mean value will be decribed using a backward parabolic equation with parameters.     

Extremal functions as divisors for kernels of Toeplitz operators

In this paper, an extremal function of a Banach space of analytic functions in the unit disk (not all functions vanishing at 0) is a function solving the extremal problem for functions f of norm 1. We study extremal functions of kernels of Toeplitz operators on Hardy spaces H^p, 1<p<∞. Such kernels are special cases of so-called nearly invariant subspaces with respect to the backward shift, for which Hitt proved that when p=2, extremal functions act as isometric divisors. We show that the extremal function is still a contractive divisor when p<2 and an expansive divisor when p>2 (modulo p-dependent multiplicative constants). We give examples showing that the extremal function may fail to be a contractive divisor when p>2 and also fail to be an expansive divisor when p<2. We discuss to what extent these results characterize the Toeplitz operators via invariant subspaces for the backward shift.     

Convergence analysis of a global optimization algorithm using stochastic differential equations

We establish the convergence of a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility of the trajectory we introduce information from the Lagrange multipliers into the SDE. The analysis is performed in two steps. We first give a characterization of a probability measure (Π) that is defined on the set of global minima of the problem. We then study the transition density associated with the augmented diffusion process and show that its weak limit is given by Π.     

有限偏差函数与调和函数

分别借助解析函数与调和函数两类函数的Dirichlet积分,利用相关文献给定边界值的拟共形映射极值伸缩商的估计方法,通过有限偏差函数和拟共形映射的关系估计了具有给定边界值的有限偏差函数的极值伸缩商.得到了解析函数的Dirichlet积分在有限偏差函数下具有拟不变性,同时给出有限偏差函数极值伸缩商的下界估计.     

Univalent functions and holomorphic motions

We consider in this paper the coefficient problems for univalent functions slightly generalizing the Bieberbach and Zalcman conjectures and give their complete solution for the lower coefficients. Our approach is based on the properties of holomorphic motions and of extremal quasiconformal mappings on the complex plane. The proof possesses a geometric interpretation.     

A conformally invariant metric on Riemann surfaces associated with integrable holomorphic quadratic differentials

In this paper, we define a conformally invariant (pseudo-)metric on all Riemann surfaces in terms of integrable holomorphic quadratic differentials and analyze it. This metric is closely related to an extremal problem on the surface. As a result, we have a kind of reproducing formula for integrable quadratic differentials. Furthermore, we establish a new characterization of uniform thickness of hyperbolic Riemann surfaces in terms of invariant metrics.     

Moduli spaces of meromorphic connections and quiver varieties

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne–Simpson problem.     

Extremal values of multiple gamma and sine functions

We study the extremal values of multiple gamma and sine functions in the fundamental intervals. We show the number and locations of the extremal points, and prove that all the local maximum and minimum values are greater and less than one, respectively.     

Maximal continuants and the Fine-Wilf theorem

The following problem was posed by C.A. Nicol: given any finite sequence of positive integers, find the permutation for which the continuant (i.e. the continued fraction denominator) having these entries is maximal, resp. minimal. The extremal arrangements are known for the regular continued fraction expansion. For the singular expansion induced by the backward shift ⌈1/x⌉-1/x the problem is still open in the case of maximal continuants. We present the explicit solutions for sequences with pairwise different entries and for sequences made up of any pair of digits occurring with any given (fixed) multiplicities. Here the arrangements are uniquely described by a certain generalized continued fraction. We derive this from a purely combinatorial result concerning the partial order structure of the set of permutations of a linearly ordered vector. This set has unique extremal elements which provide the desired extremal arrangements. We also prove that the palindromic maximal continuants are in a simple one-to-one correspondence with the Fine and Wilf words with two coprime periods which gives a new analytic and combinatorial characterization of this class of words.     

Łojasiewicz–type inequalities in complex analysis

We show how some variants of the ?ojasiewicz inequality, which is a powerful tool in real analysis, can also be used to study certain problems in complex analysis and approximation theory. In particular, we discuss whether the so-called ?ojasiewicz–Siciak condition of the Siciak extremal function is preserved when taking holomorphic preimages or images.     

On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds

We study the second best constant problem for logarithmic Sobolev inequalities on complete Riemannian manifolds and investigate its relationship with optimal heat kernel bounds and the existence of extremal functions.     

On holomorphic harmonic morphisms

We study holomorphic harmonic morphisms from K?hler manifolds to almost Hermitian manifolds. When the codomain is also K?hler we get restrictions on such maps in the case of constant holomorphic curvature. We also prove a Bochner-type formula for holomorphic harmonic morphisms which, under certain curvature conditions of the domain, gives insight to the structure of the vertical distribution. We thus prove that when the domain is compact and non-negatively curved, the vertical distribution is totally geodesic. Received: 28 May 2001     

Extremal holomorphic curves for defect relations

Drasin’s theorem describing meromorphic functions of finite order with maximal sum of deficiencies is extended to holomorphic curves in projective space. A conjecture about holomorphic curves extremal for Cartan’s defect relation is discussed. Supported by NSF grant DMS-950036. This paper was written at the Norwegian Technology and Science University (NTNU, Trondheim), which the author thanks for its hospitality.