An Hilbert space approach for a class of arbitrage free implied volatilities models

We present an Hilbert space formulation for a set of implied volatility models introduced in \cite{BraceGoldys01} in which the authors studied conditions for a family of European call options, varying the maturing time and the strike price $T$ an $K$, to be arbitrage free. The arbitrage free conditions give a system of stochastic PDEs for the evolution of the implied volatility surface ${\hat\sigma}_t(T,K)$. We will focus on the family obtained fixing a strike $K$ and varying $T$. In order to give conditions to prove an existence-and-uniqueness result for the solution of the system it is here expressed in terms of the square root of the forward implied volatility and rewritten in an Hilbert space setting. The existence and the uniqueness for the (arbitrage free) evolution of the forward implied volatility, and then of the the implied volatility, among a class of models, are proved. Specific examples are also given.Implied volatility; Option pricing; Stochastic SPDE; Hilbert space

Martingale solutions and Markov selections for stochastic partial differential equations

AbstractWe present a general framework for solving stochastic porous medium equations and stochastic Navier–Stokes equations in the sense of martingale solutions. Following Krylov [N.V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 691–708] and Flandoli–Romito [F. Flandoli, N. Romito, Markov selections for the 3D stochastic Navier–Stokes equations, Probab. Theory Related Fields 140 (2008) 407–458], we also study the existence of Markov selections for stochastic evolution equations in the absence of uniqueness

Martingale solutions and Markov selections for stochastic partial differential equations

We present a general framework for solving stochastic porous medium equations and stochastic Navier-Stokes equations in the sense of martingale solutions. Following Krylov [N.V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 691-708] and Flandoli-Romito [F. Flandoli, N. Romito, Markov selections for the 3D stochastic Navier-Stokes equations, Probab. Theory Related Fields 140 (2008) 407-458], we also study the existence of Markov selections for stochastic evolution equations in the absence of uniqueness.Markov selection Martingale solution Stochastic porous medium equation Stochastic Navier-Stokes equation

Label-free assessment of endothelial cell metabolic state using autofluorescent microscopy

To examine the process of endothelial cell aging we utilised hyperspectral imaging to collect broad autofluorescence emission at the individual cellular level and mathematically isolate the characteristic spectra of nicotinamide and flavin adenine dinucleotides (NADH and FAD, respectively). Quantitative analysis of this data provides the basis for a non-destructive spatial imaging method for cells and tissue. FAD and NADH are important factors in cellular metabolism and have been shown to be involved with the redox state of the cell; with the ratio between the two providing the basis for an ‘optical redox ratio’.2 page(s

Non-invasive monitoring of cytokine-based regenerative treatment of cartilage by hyperspectral unmixing

Extracting biochemical information from tissue autofluorescence is a promising approach to non-invasively monitor disease treatments at a cellular level, without using any external biomarkers. Our recently developed unsupervised hyperspectral unmixing by Dependent Component Analysis (DECA) provides robust and detailed metabolic information with proper account of intrinsic cellular heterogeneity. Moreover this method is compatible with established methods of fluorescent biomarker labelling. Recently adipose-derived stem cell (ADSC) – based therapies have been introduced for treating different diseases in animals and humans. ADSC have been shown promise in regenerative treatments for osteoarthritis and other bone and joint disorders. One of the mechanism of their action is their anti-inflammatory effects within osteoarthritic joints which aid the regeneration of cartilage. These therapeutic effects are known to be driven by secretions of different cytokines from the ADSCs. We have been using the hyperspectral unmixing techniques to study in-vitro the effects of ADSC-derived cytokine-rich secretions with the cartilage chip in both human and bovine samples. The study of metabolic effects of different cytokine treatment on different cartilage layers makes it possible to compare the merits of those treatments for repairing cartilage.1 page(s