An Invariance Principle of G-Brownian Motion for the Law of the Iterated Logarithm under G-expectation

The classical law of the iterated logarithm (LIL for short)as fundamental limit theorems in probability theory play an important role in the development of probability theory and its applications. Strassen (1964) extended LIL to large classes of functional random variables, it is well known as the invariance principle for LIL which provide an extremely powerful tool in probability and statistical inference. But recently many phenomena show that the linearity of probability is a limit for applications, for example in finance, statistics. As while a nonlinear expectation--- G-expectation has attracted extensive attentions of mathematicians and economists, more and more people began to study the nature of the G-expectation space. A natural question is: Can the classical invariance principle for LIL be generalized under G-expectation space? This paper gives a positive answer. We present the invariance principle of G-Brownian motion for the law of the iterated logarithm under G-expectation

Consistency of the kernel density estimator - a survey

Various consistency proofs for the kernel density estimator have been developed over the last few decades. Important milestones are the pointwise consistency and almost sure uniform convergence with a fixed bandwidth on the one hand and the rate of convergence with a fixed or even a variable bandwidth on the other hand. While considering global properties of the empirical distribution functions is sufficient for strong consistency, proofs of exact convergence rates use deeper information about the underlying empirical processes. A unifying character, however, is that earlier and more recent proofs use bounds on the probability that a sum of random variables deviates from its mean

Pitfalls and Opportunities in the Use of Extreme Value Theory in Risk Management

Recent literature has trumpeted the claim that extreme value theory (EVT) holds promise for accurate estimation of extreme quantiles and tail probabilities of financial asset returns, and hence hold promise for advances in the management of extreme financial risks. Our view, based on a disinterested assessment of EVT from the vantage point of financial risk management, is that the recent optimism is partly appropriate but also partly exaggerated, and that at any rate much of the potential of EVT remains latent. We substantiate this claim by sketching a number of pitfalls associate with use of EVT techniques. More constructively, we show how certain of the pitfalls can be avoided, and we sketch a number of explicit research directions that will help the potential of EVT to be realized

A new poly(ortho ester)-based drug delivery system as an adjunct treatment in filtering surgery.

PURPOSE: Pharmacologic modulation of wound healing after glaucoma filtering surgery remains a major clinical challenge in ophthalmology. Poly(ortho ester) (POE) is a bioerodible and biocompatible viscous polymer potentially useful as a sustained drug delivery system that allows the frequency of intraocular injections to be reduced. The purpose of this study was to determine the efficacy of POE containing a precise amount of 5-fluorouracil (5-FU) in an experimental model of filtering surgery in the rabbit. METHODS: Trabeculectomy was performed in pigmented rabbit eyes. An ointmentlike formulation of POE containing 1% wt/wt 5-FU was injected subconjunctivally at the site of surgery, during the procedure. Intraocular pressure (IOP), bleb persistence, and ocular inflammatory reaction were monitored until postoperative day 30. Quantitative analysis of 5-FU was performed in the anterior chamber. Histologic analysis was used to assess the appearance of the filtering fistula and the polymer's biocompatibility. RESULTS: The decrease in IOP from baseline and the persistence of the filtering bleb were significantly more marked in the 5-FU-treated eyes during postoperative days 9 through 28. Corneal toxicity triggered by 5-FU was significantly lower in the group that received 5-FU in POE compared with a 5-FU tamponade. Histopathologic evaluation showed that POE was well tolerated, and no fibrosis occurred in eyes treated with POE containing 5-FU. CONCLUSIONS: In this rabbit model of trabeculectomy, the formulation based on POE and containing a precise amount of 5-FU reduced IOP and prolonged bleb persistence in a way similar to the conventional method of a 5-FU tamponade, while significantly reducing 5-FU toxicity

Evaluation of a novel biomaterial in the suprachoroidal space of the rabbit eye.

PURPOSE: Drug delivery to treat diseases of the posterior segment of the eye, such as choroidal neovascularization and its complications, is hampered by poor intraocular penetration and rapid elimination of the drug from the eye. The purpose of this study was to investigate the feasibility and tolerance of suprachoroidal injections of poly(ortho ester) (POE), a bioerodible and biocompatible polymer, as a biomaterial potentially useful for development of sustained drug delivery systems. METHODS: After tunnelization of the sclera, different formulations based on POE were injected (100 microL) into the suprachoroidal space of pigmented rabbits and compared with 1% sodium hyaluronate. Follow-up consisted of fundus observations, echography, fluorescein angiography, and histologic analysis over 3 weeks. RESULTS: After injection, POE spread in the suprachoroidal space at the posterior pole. It was well tolerated and progressively disappeared from the site of injection without sequelae. No bleeding or retinal detachment occurred. Echographic pictures showed that the material was present in the suprachoroidal space for 3 weeks. Angiography revealed minor pigment irregularities at the site of injection, but no retinal edema or necrosis. Histology showed that POE was well tolerated in the choroid. CONCLUSIONS: POE suprachoroidal injections, an easy, controllable, and reproducible procedure, were well tolerated in the rabbit eye. POE appears to be a promising biomaterial to deliver drugs focally to the choroid and the retina

Therapeutic applications of viscous and injectable poly(ortho esters).

Poly(ortho esters) (POE) are hydrophobic and bioerodible polymers that have been investigated for pharmaceutical use since the early 1970s. Among the four described generations of POE, the third (POE III) and fourth (POE IV) are promising viscous and injectable materials which have been investigated in numerous biomedical applications. POE III has been extensively studied for ophthalmic drug delivery, it presents an excellent biocompatibility and is currently being investigated as a vehicle for sustained drug delivery to treat diseases of the posterior segment of the eye. POE IV is distinguishable by a highly reproducible and controlled synthesis, a higher hydrophobicity, and an excellent biocompatibility. It is currently under development for a variety of applications, such as ocular delivery, periodontal disease treatment and applications in veterinary medicine. This review will also focus on new perspectives for this promising family of polymers, such as guided tissue regeneration, treatment of osteoarthritis, as well as peptide and protein delivery