We study smoothness of densities for the solutions of SDEs whose coefficients are smooth and nondegenerate only on an open domain D. We prove that a smooth density exists on D and give upper bounds for this density. Under some additional conditions (mainly dealing with the growth of the coefficients and their derivatives), we formulate upper bounds that are suitable to obtain asymptotic estimates of the density for large values of the state variable ("tail" estimates). These results specify and extend some results by Kusuoka and Stroock [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985) 1-76], but our approach is substantially different and based on a technique to estimate the Fourier transform inspired from Fournier [Electron. J. Probab. 13 (2008) 135-156] and Bally [Integration by parts formula for locally smooth laws and applications to equations with jumps I (2007)The Royal Swedish Academy of Sciences]. This study is motivated by existing models for financial securities which rely on SDEs with non-Lipschitz coefficients. Indeed, we apply our results to a square root-type diffusion (CIR or CEV) with coefficients depending on the state variable, that is, a situation where standard techniques for density estimation based on Malliavin calculus do not apply. We establish the existence of a smooth density, for which we give exponential estimates and study the behavior at the origin (the singular point).SDE are bounded C ∞ functions with bounded derivatives of any order and that, on the other hand, the Hormandër condition holds, then the solution of the equation is a smooth functional in Malliavin's sense, and it is nondegenerate at any fixed positive time. Then the general criterion given by Malliavin [15] allows one to say that the law of such a random variable is absolutely continuous with respect to the Lebesgue measure, and its density is a smooth function (see [16] for a general presentation of this topic).The aim of this paper is to relax the aforementioned conditions on the coefficients: roughly speaking, we assume that the coefficients are smooth only on an open domain D and have bounded partial derivatives therein. Moreover, we assume that the nondegeneracy condition on the diffusion coefficient holds true on D only. Under these assumptions, we prove that the law of a strong solution to the equation admits a smooth density on D (Theorem 2.1). Furthermore, when D is the complementary of a compact ball and the coefficients satisfy some additional assumptions on D (mainly dealing with their growth and the one of derivatives), we give upper bounds for the density for large values of the state variable (Theorem 2.2). We will occasionally refer to these aymptotic estimates of the density as "tail estimates" or estimates on the density's "tails."Local results have already been obtained by Kusuoka and Stroock in [13], Section 4. Here the authors work under local regularity and nondegeneracy hypotheses too, but the bounds they provide on the density are mostly significant on the diagonal (i.e., close to starting...
While all forms of Internet activity have an impact on the lives of Internet users, some are particularly beneficial and allow people to improve their daily lives. One of such Internet use is Digital Political Participation (DDP). In this paper we seek to understand how the influence of digital skills on the adoption of Digital Political Participation practices may form the basis of a second level of digital divide and of a set of political inequalities. We operationalize the digital skills construct in terms of users’ Internet competence and level of appropriation. We hypothesize that digital skills have a significant influence on the adoption of beneficial uses of the Internet, such as DPP. At the same time, we examine whether digital skill levels are stratified by socio-demographic background, thereby generating political and social inequality. By looking at the Spanish case, we first tested the adequateness of the items chosen to measure these two dimensions. Second, we looked into sequences of multiple influences between socio-demographic variables and digital skills and between digital skills and DPP. The results show that socio demographic variables have an influence on digital skills. At the same time, digital skills have a strong influence on DPP.
We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one cantheoretically-distinguish between mass at the origin and a heavy-left-tailed distribution. We numerically test our model-free results in stochastic models with absorption at the boundary, such as the CEV process, and in jump-to-default models. Note that while Lee's moment formula [25] tells that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as [3,17] do not apply in this setting-essentially due to the breakdown of Put-Call duality.
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