اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدApplications of the quadratic covariation differentiation theory: variants of the Clark-Ocone and Stroocks formulas
177
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In a 2006 article (cite{A1}), Allouba gave his quadratic covariation differentiation theory for It^os integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic differentiation theory containing a fundamental theorem of stochastic calculus relating this derivative to It^os integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. Here, we use this differentiation theory to obtain variants of the Clark-Ocone and Stroock formulas, with and without change of measure. We prove our variants of the Clark-Ocone formula under $L^{2}$-type conditions; with no Malliavin calculus, without the use of weak distributional or Radon-Nikodym type derivatives, and without the significant machinery of the Hida-Malliavin calculus. Unlike Malliavin or Hida-Malliavin calculi, the form of our variant of the Clark-Ocone formula under change of measure is as simple as it is under no change of measure, and without requiring any further differentiability conditions on the Girsanov transform integrand beyond Novikovs condition. This is due to the invariance under change of measure of the first authors derivative in cite{A1}. The formulations and proofs are natural applications of the differentiation theory in cite{A1} and standard It^o integral calculus. Iterating our Clark-Ocone formula, we obtain variants of Stroocks formula. We illustrate the applicability of these formulas by easily, and without Hida-Malliavin methods, obtaining the representation of the Brownian indicator $F=mathbb{I}_{[K,infty)}(W_{T})$, which is not standard Malliavin differentiable, and by applying them to digital options in finance. We then identify the chaos expansion of the Brownian indicator.
قيم البحث
اقرأ أيضاً
In this paper, we construct a Malliavin derivative for functionals of square-integrable Levy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian
motion and Poisson random measure. As an illustration, we compute the explicit martingale representation for the maximum of a Levy process.
A peculiar feature of It^os calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimarti
ngales with respect to Brownian motion that leads to a differentiation theory counterpart to It^os integral calculus? From It^os definition of his integral, such a derivative must be based on the quadratic covariation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolles theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications some of which we address in an upcoming article.
We use Rices formulas in order to compute the moments of some level functionals which are linked to problems in oceanography and optics. For instance, we consider the number of specular points in one or two dimensions, the number of twinkles, the dis
tribution of normal angle of level curves and the number or the length of dislocations in random wavefronts. We compute expectations and in some cases, also second moments of such functionals. Moments of order greater than one are more involved, but one needs them whenever one wants to perform statistical inference on some parameters in the model or to test the model itself. In some case we are able to use these computations to obtain a Central Limit Theorem.
An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malli
avin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Itos formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Levy models. Since the payoff of digital options is described by an indicator function, we also discuss the Malliavin differentiability of indicator functions with respect to Levy processes.
We revisit two classical formulas for the Bessel function of the first kind, due to von Lommel and Weber-Schafheitlin, in a probabilistic setting. The von Lommel formula exhibits a family of solutions to the van Dantzig problem involving the generali
zed semi-circular distributions and the first hitting times of a Bessel process with positive parameter, whereas the Weber-Schafheitlin formula allows one to construct non-trivial moments of Gamma type having a signed spectral measure. Along the way, we observe that the Weber-Schafheitlin formula is a simple consequence of the von Lommel formula, the Fresnel integral and the Selberg integral.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
