Papers by Enrique Villamor
Theory of Probability & Its Applications, 1977
In this short note, we present a new version of the Central Limit Theorem whose proof is based on... more In this short note, we present a new version of the Central Limit Theorem whose proof is based on Levy's characterization of Brownian motion. The method in the proof may allow to extend the result to a more general context, e.g. to averaged sums of properly compensated dependent random variables.
RePEc: Research Papers in Economics, 2020
In this paper we study the pricing of exchange options when underlying assets have stochastic vol... more In this paper we study the pricing of exchange options when underlying assets have stochastic volatility and stochastic correlation. An approximation using a closed-form approximation based on a Taylor expansion of the conditional price is proposed. Numerical results are illustrated for exchanges between WTI and Brent type oil prices.
arXiv (Cornell University), Jul 26, 2005
In this paper we establish results on the existence of nontangential limits for weighted A-harmon... more In this paper we establish results on the existence of nontangential limits for weighted A-harmonic functions in the weighted Sobolev space W 1,q w (B n), for some q > 1 and w in the Muckenhoupt A q class, where B n is the unit ball in R n. These results generalize the ones in section §3 of [KMV], where the weight was identically equal to one. Weighted A-harmonic functions are weak solutions of the partial differential equation div(A(x, ∇u)) = 0, where α w(x) |ξ| q ≤ A(x, ξ), ξ ≤ β w(x) |ξ| q for some fixed q ∈ (1, ∞), where 0 < α ≤ β < ∞, and w(x) is a q-admissible weight as in Chapter 1 in [HKM]. Later, we apply these results to improve on results of Koskela, Manfredi and Villamor [KMV] and Martio and Srebro [MS] on the existence of radial limits for bounded quasiregular mappings in the unit ball of R n with some growth restriction on their multiplicity function.
arXiv (Cornell University), Jul 27, 2005
There have been, over the last 8 years, a number of far reaching of the famous original F. and M.... more There have been, over the last 8 years, a number of far reaching of the famous original F. and M. Riesz's uniqueness theorem that states that if a bounded analytic function in the unit disc of the complex plane C has the same radial limit in a set of positive Lebesgue measure on its boundary, then the function has to be constant. First Beurling [B], considering the case of non-constant meromorphic functions mapping the unit disc on a Riemann surface of finite spherical area, was able to prove that if such a function showed an appropriate behavior in the neighborhood of the limit value where the function maps a set on the boundary of the unit disc, then those sets have capacity zero. Here the capacity considered is the logarithmic linear capacity. The author of the present note in [V], was able to weakened beurling condition on the limit value. Later Jenkins in [J], showed that in the presence of such a local condition on the limiting value, the global behavior of Riemann surface is irrelevant and at the same time he gave an improved and sharper condition. Those results where quite restrictive in a two folded way, namely, they were in dimension n = 2 and the regularity requirements on the treated functions were quite strong, analyticity and meromorphicity. Koskela in [K], was able to remove those two restrictions by proving a uniqueness result for functions in ACL p (B n) for values of p in the interval (1, n] and satisfying a condition on the limit value very similar in nature to the one of Jenkins in dimension 2. In particular, Koskela's result recovers Jenkins in the case p = n = 2. He proves that a continuous function in the Sobolev space W 1,p (B n) (here B n is the unit ball of R n and 1 < p ≤ n) vanishes identically provided u(x)−a <ǫ ∇u(x) p dx = O(ǫ p (log(1 ǫ)) p−1) as ǫ → 0 and there is a set E on ∂B n of positive p-capacity such that each x ∈ E is a terminal point of some rectifiable curve along which the function u tends to a. Koskela also shows in his paper that this result is sharp in the sense that (log(1 ǫ)) p−1 can not be replaced by (log(1 ǫ)) p−1+δ for any positive δ (even if u is assumed to be continuous in the closure of B n).
arXiv (Cornell University), Jul 27, 2005
Our approach will be different from the ones used in [MV] and [HK]. It is more geometrical in nat... more Our approach will be different from the ones used in [MV] and [HK]. It is more geometrical in nature and uses the method of extremal length.
Illinois Journal of Mathematics, Apr 1, 2001
Consider monotone functions u : B n → R in the weighted Sobolev space W 1,p (B n ; w), where n − ... more Consider monotone functions u : B n → R in the weighted Sobolev space W 1,p (B n ; w), where n − 1 < p ≤ n and w is a weight in the class Aq for some 1 ≤ q < p/(n − 1) which has a certain symmetry property with respect to ∂B n. We prove that u has nontangential limits at all points of ∂B n except possibly those on a set E of weighted (p, w)capacity zero. The proof is based on a new weighted oscillation estimate (Theorem 1) that may be of independent interest. In the special case w(x) = |1 − |x|| α , the weighted (p, w)-capacity of a ball can be easily estimated to conclude that the Hausdorff dimension of the set E is smaller than or equal to α + n − p, where 0 ≤ α < (p − (n − 1))/(n − 1).
International Journal of Financial Studies, Mar 27, 2023
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Uniqueness Theorems for Mappings of Finite Distortion
Journal of The Australian Mathematical Society, Nov 7, 2017
In this paper we prove uniqueness theorems for mappings $F\in W_{\text{loc}}^{1,n}(\mathbb{B}^{n}... more In this paper we prove uniqueness theorems for mappings $F\in W_{\text{loc}}^{1,n}(\mathbb{B}^{n};\mathbb{R}^{n})$ of finite distortion $1\leq K(x)=\Vert \mathit{DF}(x)\Vert ^{n}/J_{F}(x)$ satisfying some integrability conditions. These types of theorems fundamentally state that if a mapping defined in $\mathbb{B}^{n}$ has the same boundary limit $a$ on a ‘relatively large’ set $E\subset \unicode[STIX]{x2202}\mathbb{B}^{n}$ , then the mapping is constant. Here the size of the set $E$ is measured in terms of its $p$ -capacity or equivalently its Hausdorff dimension.
Israel Journal of Mathematics, Dec 1, 2001
A necessary and sufficient condition is given for a discrete multiplicity variety in the unit bal... more A necessary and sufficient condition is given for a discrete multiplicity variety in the unit ball Bn of C n to be an interpolating variety for weighted spaces of holomorphic functions in Bn.
Analytic properties of monotone sobolev functions
Complex Variables, Nov 1, 2001
ABSTRACT We prove that if is a monotone function in the weighted Sobolev space with n; 1 &lt;... more ABSTRACT We prove that if is a monotone function in the weighted Sobolev space with n; 1 &lt; ρ ≤ n, and w; a weight in the Muckenhoupt class for 1≦q&gt;p/(n−1), then u has to be constant. This constitutes a Liouviile type theorem for this class of functions. We will also prove a quasi-uniform continuity result for the same class of functions
arXiv (Cornell University), Mar 31, 1995
Let F ∈ W 1 , n loc (Ω ; R n) be a mapping with nonnegative Jacobian J F (x) = det DF (x) ≥ 0 for... more Let F ∈ W 1 , n loc (Ω ; R n) be a mapping with nonnegative Jacobian J F (x) = det DF (x) ≥ 0 for a.e. x in a domain Ω ⊂ R n. The dilatation of F is defined (almost everywhere in Ω) by the formula K(x) = |DF (x)| n J F (x) • Iwaniec andŠverák [IS] have conjectured that if p ≥ n − 1 and K ∈ L p loc (Ω) then F must be continuous, discrete and open. Moreover, they have confirmed this conjecture in the two-dimensional case n = 2. In this article, we verify it in the higher-dimensional case n ≥ 2 whenever p > n − 1 .
International Journal of Financial Studies
In this paper we study the pricing of exchange options between two underlying assets whose dynami... more In this paper we study the pricing of exchange options between two underlying assets whose dynamic show a stochastic correlation with random jumps. In particular, we consider a Ornstein-Uhlenbeck covariance model, with Levy Background Noise Processes driven by Inverse Gaussian subordinators. We use expansions in terms of Taylor polynomials and cubic splines to approximately compute the price of the derivative contract. Our findings show that the later approach provides an efficient way to compute the price when compared with a Monte Carlo method, while maintaining an equivalent degree of accuracy.
Quantitative Structural Models to Assess Credit Risk on Individuals
Journal of Applied Mathematics and Physics
In this paper we establish results on the existence of nontangential limits for weighted A-harmon... more In this paper we establish results on the existence of nontangential limits for weighted A-harmonic functions in the weighted Sobolev space W 1,q w (B ), for some q > 1 and w in the Muckenhoupt Aq class, where B is the unit ball in R. These results generalize the ones in section §3 of [KMV], where the weight was identically equal to one. Weighted A-harmonic functions are weak solutions of the partial differential equation div(A(x,∇u)) = 0, where α w(x) |ξ| ≤ 〈A(x, ξ), ξ〉 ≤ β w(x) |ξ| for some fixed q ∈ (1,∞), where 0 < α ≤ β < ∞, and w(x) is a q-admissible weight as in Chapter 1 in [HKM]. Later, we apply these results to improve on results of Koskela, Manfredi and Villamor [KMV] and Martio and Srebro [MS] on the existence of radial limits for bounded quasiregular mappings in the unit ball of R with some growth restriction on their multiplicity function. 1991 Mathematics Subject Classification. Primary: 30C65, Secondary: 46E35.
Let F∈ W_loc^1,n(Ω; R^n) be a mapping with non-negative Jacobian J_F(x)=det DF(x)> 0 a.e. in a... more Let F∈ W_loc^1,n(Ω; R^n) be a mapping with non-negative Jacobian J_F(x)=det DF(x)> 0 a.e. in a domain Ω∈ R^n. The dilatation of the mapping F is defined, almost everywhere in Ω, by the formula K(x)=|DF(x)|^nJ_F(x). If K(x) is bounded a.e., the mapping is said to be quasiregular. Quasiregular mappings are a generalization to higher dimensions of holomorphic mappings. The theory of higher dimensional quasiregular mappings began with Rešhetnyak's theorem, stating that non constant quasiregular mappings are continuous, discrete and open. In some problems appearing in the theory of non-linear elasticity, the boundedness condition on K(x) is too restrictive. Tipically we only know that F has finite dilatation, that is, K(x) is finite a.e. and K(x)^p is integrable for some value p. In two dimensions, Iwaniec and Šverak [IS] have shown that K(x)∈ L^1_loc is sufficient to guarantee the conclusion of Rešhetnyak's theorem. For n> 3, Heinonen and Koskela [HK], showed that if the m...
There have been, over the last 8 years, a number of far reaching extensions of the famous origina... more There have been, over the last 8 years, a number of far reaching extensions of the famous original F. and M. Riesz's uniqueness theorem that states that if a bounded analytic function in the unit disc of the complex plane C has the same radial limit in a set of positive Lebesgue measure on its boundary, then the function has to be constant. First Beurling [B], considering the case of non-constant meromorphic functions mapping the unit disc on a Riemann surface of finite spherical area, was able to prove that if such a function showed an appropriate behavior in the neighborhood of the limit value where the function maps a set on the boundary of the unit disc, then those sets have logarithmic capacity zero. The author of the present note, in [V], was able to weaken Beurling's condition on the limit value. Those results where quite restrictive in a two folded way, namely, they were in dimension n=2 and the regularity requirements on the treated functions were quite strong. Kosk...
Let F ∈ W 1,n loc (Ω; R n) be a mapping with non-negative Jacobian J F (x) = detDF (x) ≥ 0 a.e. i... more Let F ∈ W 1,n loc (Ω; R n) be a mapping with non-negative Jacobian J F (x) = detDF (x) ≥ 0 a.e. in a domain Ω ∈ R n. The dilatation of the mapping F is defined, almost everywhere in Ω, by the formula K(x) = |DF (x)| n J F (x) .
Proceedings of the American Mathematical Society, 1994
We study the sets of uniqueness of areally mean p-valent functions in the unit disc. Namely, if f... more We study the sets of uniqueness of areally mean p-valent functions in the unit disc. Namely, if f{z) is in this class and has the same angular limit in a set E on the boundary of the unit disc, we prove that if p is small compared to the size of E then f(z) is constant. We then construct an areally mean p-valent function which shows that some condition on the size of the set E must be imposed. From now on we are going to denote this class of functions by AMP. This class has been studied by several authors; good references are Hayman [5] and Eke [4]. Let us consider now the following class of functions.
Abstract. There have been, over the last 8 years, a number of far reaching of the famous original... more Abstract. There have been, over the last 8 years, a number of far reaching of the famous original F. and M. Riesz’s uniqueness theorem that states that if a bounded analytic function in the unit disc of the complex plane C has the same radial limit in a set of positive Lebesgue measure on its boundary, then the function has to be constant. First Beurling [B], considering the case of non-constant meromorphic functions mapping the unit disc on a Riemann surface of finite spherical area, was able to prove that if such a function showed an appropriate behavior in the neighborhood of the limit value where the function maps a set on the boundary of the unit disc, then those sets have capacity zero. Here the capacity considered is the logarithmic linear capacity. The author of the present note in [V], was able to weakened beurling condition on the limit value. Later Jenkins in [J], showed that in the presence of such a local condition on the limiting value, the global behavior of Riemann s...
Mappings with integrable dilatation in higher dimensions, preprint
Abstract. Let F ∈ Wloc (Ω; Rn) be a mapping with nonnegative Jacobian JF(x) = det DF(x) ≥ 0 for a... more Abstract. Let F ∈ Wloc (Ω; Rn) be a mapping with nonnegative Jacobian JF(x) = det DF(x) ≥ 0 for a.e. x in a domain Ω ⊂ Rn. The dilatation of F is defined (almost everywhere in Ω) by the formula K(x) = |DF(x)|n JF(x) Iwaniec and ˇ Sverák [IS] have conjectured that if p ≥ n − 1 and K ∈ L p loc (Ω) then F must be continuous, discrete and open. Moreover, they have confirmed this conjecture in the two-dimensional case n = 2. In this article, we verify it in the higher-dimensional case n ≥ 2 whenever p> n − 1. 1.
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Papers by Enrique Villamor