Application of Artificial Viscosity in Establishing Supercritical Solutions to the Transonic Integra

The nonlinear singular integral equation of transonic flow is examined in the free-stream Mach number range where only solutions with shocks are known to exist. It is shown that, by the addition of an artificial viscosity term to the integral equation, even the direct iterative scheme, with the linear solution as the initial iterate, leads to convergence. Detailed tables indicating how the solution varies with changes in the parameters of the artificial viscosity term are also given. In the best cases (when the artificial viscosity is smallest), the solutions compare well with known results, their characteristic feature being the representation of the shock by steep gradients rather than by abrupt discontinuities. However, 'sharp-shock solutions' have also been obtained by the implementation of a quadratic iterative scheme with the 'artificial viscosity solution' as the initial iterate; the converged solution with a sharp shock is obtained with only a few more iterates. Finally, a review is given of various shock-capturing and shock-fitting schemes for the transonic flow equations in general, and for the transonic integral equation in particular, frequent comparisons being made with the approach of this paper.

Dynamic programming for stochastic target problems, Viscosity solutions and hedging in markets with Portfolio constraints and large investors

The purpose of this expository arti le is to present a self- ontained overview of some results on the hara terization of the optimal value fun tion of a sto hasti target problem as (dis ontinuous) vis osity solution of a ertain dynami programming PDE and its appli ation to the problem of hedging ontingent laims in the presen e of portfolio onstraints and large investors

Differential Games of Mixed Type with Control and Stopping Times

We study a zero sum differential game of mixed type where each player uses both control and stopping times. Under certain conditions we show that the value function for this problem exists and is the unique viscosity solution of the corresponding variational inequalities. We also show the existence of saddle point equilibrium for a special case of differential game.

Existence of Value in Stochastic Differential Games of Mixed Type

In this article, we address stochastic differential games of mixed type with both control and stopping times. Under standard assumptions, we show that the value of the game can be characterized as the unique viscosity solution of corresponding Hamilton-Jacobi-Isaacs (HJI) variational inequalities.

Les processus additifs markoviens et leurs applications en finance mathématique

Cette thèse porte sur les questions d'évaluation et de couverture des options dans un modèle exponentiel-Lévy avec changements de régime. Un tel modèle est construit sur un processus additif markovien un peu comme le modèle de Black- Scholes est basé sur un mouvement Brownien. Du fait de l'existence de plusieurs sources d'aléa, nous sommes en présence d'un marché incomplet et ce fait rend inopérant les développements théoriques initiés par Black et Scholes et Merton dans le cadre d'un marché complet. Nous montrons dans cette thèse que l'utilisation de certains résultats de la théorie des processus additifs markoviens permet d'apporter des solutions aux problèmes d'évaluation et de couverture des options. Notamment, nous arrivons à caracté- riser la mesure martingale qui minimise l'entropie relative à la mesure de probabilit é historique ; aussi nous dérivons explicitement sous certaines conditions, le portefeuille optimal qui permet à un agent de minimiser localement le risque quadratique associé. Par ailleurs, dans une perspective plus pratique nous caract érisons le prix d'une option Européenne comme l'unique solution de viscosité d'un système d'équations intégro-di érentielles non-linéaires. Il s'agit là d'un premier pas pour la construction des schémas numériques pour approcher ledit prix.

Ecuaciones diferenciales estocásticas con condición final y soluciones de viscosidad de EDPS semilineales de segundo orden

El objetivo de este documento es recopilar algunos resultados clasicos sobre existencia y unicidad ´ de soluciones de ecuaciones diferenciales estocasticas (EDEs) con condici ´ on final (en ingl ´ es´ Backward stochastic differential equations) con particular enfasis en el caso de coeficientes mon ´ otonos, y su cone- ´ xion con soluciones de viscosidad de sistemas de ecuaciones diferenciales parciales (EDPs) parab ´ olicas ´ y el´ıpticas semilineales de segundo orden.

Portifolio selection with random transaction costs

Transaction costs have a random component in the bid-ask spread. Facing a high bid-ask spread, the consumer has the option to wait for better terms oI' trade, but only by carrying an undesirable portfolio balance. We present the best policy in this case. We pose the control problem and show that the value function is the uni que viscosity solution of the relevant variational inequality. Next, a numerical procedure for the problem is presented.

Black-scholes type equations

In this work we are concerned with the analysis and numerical solution of Black-Scholes type equations arising in the modeling of incomplete financial markets and an inverse problem of determining the local volatility function in a generalized Black-Scholes model from observed option prices. In the first chapter a fully nonlinear Black-Scholes equation which models transaction costs arising in option pricing is discretized by a new high order compact scheme. The compact scheme is proved to be unconditionally stable and non-oscillatory and is very efficient compared to classical schemes. Moreover, it is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. In the next chapter we turn to the calibration problem of computing local volatility functions from market data in a generalized Black-Scholes setting. We follow an optimal control approach in a Lagrangian framework. We show the existence of a global solution and study first- and second-order optimality conditions. Furthermore, we propose an algorithm that is based on a globalized sequential quadratic programming method and a primal-dual active set strategy, and present numerical results. In the last chapter we consider a quasilinear parabolic equation with quadratic gradient terms, which arises in the modeling of an optimal portfolio in incomplete markets. The existence of weak solutions is shown by considering a sequence of approximate solutions. The main difficulty of the proof is to infer the strong convergence of the sequence. Furthermore, we prove the uniqueness of weak solutions under a smallness condition on the derivatives of the covariance matrices with respect to the solution, but without additional regularity assumptions on the solution. The results are illustrated by a numerical example.

The Effects of Credit Risk on Dynamic Portfolio Management: A New Computational Approach

The study investigates the role of credit risk in a continuous time stochastic asset allocation model, since the traditional dynamic framework does not provide credit risk flexibility. The general model of the study extends the traditional dynamic efficiency framework by explicitly deriving the optimal value function for the infinite horizon stochastic control problem via a weighted volatility measure of market and credit risk. The model's optimal strategy was then compared to that obtained from a benchmark Markowitz-type dynamic optimization framework to determine which specification adequately reflects the optimal terminal investment returns and strategy under credit and market risks. The paper shows that an investor's optimal terminal return is lower than typically indicated under the traditional mean-variance framework during periods of elevated credit risk. Hence I conclude that, while the traditional dynamic mean-variance approach may indicate the ideal, in the presence of credit-risk it does not accurately reflect the observed optimal returns, terminal wealth and portfolio selection strategies.

Effects of solution viscosity on heterogeneous electron transfer across a liquid/liquid interface

Scanning electrochemical microscopy (SECM) is employed to investigate the effect of solution viscosity on the rate constants of electron transfer (ET) reaction between potassium ferricyanide in water and 7,7,8,8-tetracyanoquinodimethane (TCNQ) in 1,2-dichloroethane. Either tetrabutylammonium (TBA(+)) or ClO4- is chosen as the common ion in both phases to control the interfacial potential drop. The rate constant of heterogeneous ET reaction between TCNQ and ferrocyanide produced in-situ, k(12), is evaluated by SECM and is inversely proportional to the viscosity of the aqueous solution and directly proportional to the diffusion coefficient of K4Fe(CN)(6) in water when the concentration of TCNQ in the DCE phase is in excess. The k(12) dependence on viscosity is explained in terms of the longitudinal relaxation time of the solution. The rate constant of the heterogeneous ET reaction between TCNQ and ferricyanide, k(21), is also obtained by SECM and these results cannot be explained by the same manner.

THE VISCOSITY BEHAVIOUR OF SEP BLOCK COPOLYMER SOLUTION DURING ITS MICELLIZATION

The viscosities of polystyrene-b-poly (ethylene/propylene) diblock copolymer in mixed solvent of n-octane and benzene were measured. The influences of the constitution of the mixed solvent, temperature and concentration were on the viscosity investigated. During the micellization the solution viscosity increases rapidly. The results are consistent with the study on the micellization by light scattering. The average mass of micelleswas measured and the hydrodynamic radius of gyrations were calculated.

Influence of solution viscosity and injection protocol on distribution patterns of jet injectors: Application to photodynamic tumour targeting

Photodynamic therapy of deep or nodular skin tumours is currently limited by the poor tissue penetration of the porphyrin precursor 5-aminolevulinic acid (ALA) and preformed photosensitisers. In this study, we investigated the potential of jet injection to deliver both ALA and a preformed photosensitiser (meso-tetra (N-methyl-4-pyridyl) porphine tetra tosylate, TMP) into a defined volume of skin. Initial studies using a model hydrogel showed that as standoff distance is increased, injection depth decreases. As the ejected volume is increased, injection depth increases. It was also shown, for the first time, that, as injection solution viscosity was increased, for a given injection setting and standoff distance, both total depth of jet penetration, L-t, and depth at which the maximum width of the penetration pattern occurred, L-m, decreased progressively. For a standoff distance of zero, the maximum width of the penetration pattern, L-w, increased progressively with increasing viscosity at each of the injection settings. Conversely, when the standoff distance was 2.5 mm, L-w decreased progressively with increasing viscosity. Studies with neonate porcine skin revealed that an injection protocol comprising an 8.98 mPas solution, an arbitrary injection setting of 8 and a standoff distance of zero was capable of delivering photosensitisers to a volume of tissue (L-t of 2.91 mm, L-m of 2.14 mm, L-w of 5. 10 mm) comparable to that occupied by a typical nodular basal cell carcinoma. Both ALA and TMP were successfully delivered using jet injection, with peak tissue concentrations (67.3 mg cm(-3) and 5.6 mg cm(-3), respectively) achieved at a depth of around 1.0 mm and substantial reductions in drug concentration seen at depths below 3.0 mm. Consequently, jet injection may be suitable for selective targeting of ALA or preformed photosensitisers to skin tumours. (c) 2007 Elsevier B.V. All rights reserved.

Influence of the concentration of Sb2O3 and the viscosity of the precursor solution on the electrical and optical properties of SnO2 thin films produced by the Pechini method

SnO2:Sb multi-layer coatings were prepared by the Pechini method. An investigation was made of the influence of the concentration of Sb2O3 and the viscosity of the precursor solution on the electrical and optical properties of SnO2 thin films. The use of a multi-layer system as an alternative form of increasing the packing and. thus. decreasing porosity proved to be efficient, decreasing the system's resistivity without altering its optical properties. (C) 2002 Elsevier B.V. B.V. All rights reserved.