J.-C. Cortés

A NEW DIRECT METHOD FOR SOLVING THE BLACK-SCHOLES EQUATION

Abstract Using the Mellin transform a new method for solving the Black–Scholes equation is proposed. Our approach does not require either variable transformations or solving diffusion equations.

Random differential operational calculus: Theory and applications

In this article, we obtain a product rule and a chain rule for mean square derivatives. An application of the chain rule to the mean square solution of random differential equations is shown. However, to achieve such mean square differentiation rules, fourth order properties were needed and, therefore, we first studied a mean fourth order differential and integral calculus. Results are applied to solve random linear variable coefficient differential problems.

Random linear-quadratic mathematical models: Computing explicit solutions and applications

This paper deals with the study of linear random population models and with a random logistic model (where parameters are random variables). Assuming appropriate conditions, the stochastic processes solutions are obtained under closed form using mean square calculus. Expectation and variance expressions for the stochastic processes solutions are given and illustrative examples are included.

Computing mean square approximations of random diffusion models with source term

This paper presents a discrete numerical method for computing mean square approximations of random diffusion models. Mean square consistency of the random difference scheme is established. Sufficient conditions for the mean square stability of the proposed numerical solution are given.

Random Hermite differential equations: Mean square power series solutions and statistical properties

Abstract This paper deals with the construction of random power series solution of second order linear differential equations of Hermite containing uncertainty through its coefficients and initial conditions. Under appropriate hypotheses on the data, we establish that the constructed random power series solution is mean square convergent. We provide conditions in order to obtain random polynomial solutions and, as a consequence, random Hermite polynomial are introduced. Also, the main statistical functions of the approximate stochastic process solution generated by truncation of the exact power series solution are given. Finally, we apply the proposed technique to several illustrative examples comparing the numerical results with respect to those provided by other available approaches including Monte Carlo simulation.

Random Airy type differential equations: Mean square exact and numerical solutions

This paper deals with the construction of power series solutions of random Airy type differential equations containing uncertainty through the coefficients as well as the initial conditions. Under appropriate hypotheses on the data, we establish that the constructed random power series solution is mean square convergent over the whole real line. In addition, the main statistical functions, such as the mean and the variance, of the approximate solution stochastic process generated by truncation of the exact power series solution are given. Finally, we apply the proposed technique to several illustrative examples which show substantial speed-up and improvement in accuracy compared to other approaches such as Monte Carlo simulations.

Analysing the effect of public health campaigns on reducing excess weight: A modelling approach for the Spanish Autonomous Region of the Community of Valencia

Excess weight is fast becoming a serious health concern in the developed and developing world. The concern of the public health sector has lead to the development of public health campaigns, focusing on two-fold goals: to inform the public as to the health risks inherent in being overweight, and the benefits of a change in nutritional behaviour. Recent studies indicate that the effects of the average public health campaign on the target community is around 5%. In this study we aim to quantify the effect of different public health campaigns on lifestyle behaviour in the target populations in order to bring about weightloss in a significant number of people over the next few years. This study is based on recent works that consider excess weight as a consequence of the transmission of unhealthy lifestyles from one individual to another. Following this point of view, first a mathematical model is presented. Then, policies based on public health campaigns addressed to stop people gaining weight (prevention; this type of policy acts on individuals in order to maintain their weight and to stop an increase in weight) and, policies addressed to overweight individuals to reduce their weight (treatment; these campaigns act on overweight and/or obese individuals in order to reduce their weight) are simulated in order to evaluate their effectiveness. The study concludes that combination of preventive plus treatment campaigns are more effective than considering them separately.

Mean square convergent numerical methods for nonlinear random differential equations

This paper deals with the construction of numerical solution of nonlinear random matrix initial value problems by means of a random Euler scheme. Conditions for the mean square convergence of the method are established avoiding the use of pathwise information. Finally, one includes several illustrative examples where the main statistics properties of the stochastic approximation processes are given.

Determining the First Probability Density Function of Linear Random Initial Value Problems by the Random Variable Transformation (RVT) Technique: A Comprehensive Study

Deterministic differential equations are useful tools for mathematical modelling. The consideration of uncertainty into their formulation leads to random differential equations. Solving a random differential equation means computing not only its solution stochastic process but also its main statistical functions such as the expectation and standard deviation. The determination of its first probability density function provides a more complete probabilistic description of the solution stochastic process in each time instant. In this paper, one presents a comprehensive study to determinate the first probability density function to the solution of linear random initial value problems taking advantage of the so-called random variable transformation method. For the sake of clarity, the study has been split into thirteen cases depending on the way that randomness enters into the linear model. In most cases, the analysis includes the specification of the domain of the first probability density function of the solution stochastic process whose determination is a delicate issue. A strong point of the study is the presentation of a wide range of examples, at least one of each of the thirteen casuistries, where both standard and nonstandard probabilistic distributions are considered.

Mean square power series solution of random linear differential equations

This paper deals with the construction of random power series solutions of linear differential equations containing uncertainty through the diffusion coefficient, the source term as well as the initial condition. Under appropriate hypotheses on the data, we establish that the constructed random power series solution is mean square convergent in a certain interval whose length depends on the mean square norm of the random variable coefficient. Also, the main statistical functions of the approximating stochastic process solution generated by truncation of the exact power series solution are given. Finally, we apply the proposed technique to several illustrative examples.