arXiv subjects
Publications and source records attributed to Enrique Villamor.
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.
Let $F\in W_{loc}^{1,n}(Ω;\Bbb R^n)$ be a mapping with non-negative Jacobian $J_F(x)=\text{det} DF(x)\ge 0$ a.e. in a domain $Ω\in \Bbb R^n$. The dilatation of the mapping $F$ is defined, almost everywhere in $Ω$, by the formula $$K(x)={{|DF(x)|^n}\over {J_F(x)}}.$$ 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 $K(x)\in L^1_{loc}$ is sufficient to guarantee the conclusion of Rešhetnyak's theorem. For $n\ge 3$, Heinonen and Koskela [HK], showed that if the mapping is quasi-light and $K(x)\in L^p_{loc}$ for $p>n-1$, then the mapping $F(x)$ is continuous, discrete and open. Manfredi and Villamor [MV] proved a similar result without assuming that the mapping $f(x)$ was quasi-light. The result is known to be false, see [Ball], when $p<n-1$. In this paper we attempt to improve in those results. In particular, we will deal with the case $p=n-1$ for $n\ge 3$, and will assume that our mapping $F(x)$ is quasi-light, that is, the inverse image of any point is compact in $Ω$. 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.
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 $\Bbb C$ 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. Koskela in [K], was able to remove those two restrictions by proving a uniqueness result for functions in $ACL^p(\Bbb B^n)$ for values of $p$ in the interval $(1,n]$. Koskela also shows in his paper that his result is sharp..
In this paper we establish results on the existence of nontangential limits for weighted $\Cal A$-harmonic functions in the weighted Sobolev space $W_w^{1,q}(\Bbb B^n)$, for some $q>1$ and $w$ in the Muckenhoupt $A_q$ class, where $\Bbb B^n$ is the unit ball in $\Bbb R^n$. These results generalize the ones in section \S3 of [KMV], where the weight was identically equal to one. Weighted $\Cal A$-harmonic functions are weak solutions of the partial differential equation $$\text{div}(\Cal A(x,\nabla u))=0,$$ where $αw(x) |ξ|^{q} \le < \Cal A(x,ξ),ξ>\le βw(x) |ξ|^{q}$ for some fixed $q\in (1,\infty)$, where $0<α\leq β<\infty$, 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 $\Bbb R^n$ with some growth restriction on their multiplicity function.
Let $F\in W^{1,n}_{\text{loc}}(Ω; \Bbb R^n)$ be a mapping with nonnegative Jacobian $J_F(x)=\det DF(x)\ge 0$ for a.e. $x$ in a domain $Ω\subset\Bbb R^n$. The {\it dilatation} of $F$ is defined (almost everywhere in $Ω$) by the formula $$K(x)=\frac{|DF(x)|^n}{J_F(x)}\cdot$$ Iwaniec and \v Sver\' ak \ncite{IS} have conjectured that if $p\ge n-1$ and $K\in L^{p}_{\text{loc}}(Ω)$ 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 $n\ge 2$ whenever $p>n-1$.