We consider a singularly perturbed boundary value problem with two small parameters. The problem is numerically treated by a quadratic spline collocation method. The suitable choice of collocation points provides the discrete minimum principle. Error bounds for the numerical approximations are...

This paper studies the probability distribution and option pricing for drawdown in a stochastic volatility environment. Their analytical approximation formulas are derived by the application of a singular perturbation method (Fouque et al., 2000). The mathematical validity of the approximation...

This paper considers an elliptic PDE with a small parameter multiplied with one of the derivatives. Recognizing the equation as an ill-posed equation, Tikhonov’s regularization is used to cast a related well-posed problem. A-priori estimates of the regularized solution and of the difference...

In this paper, we develop numerical methods based on a non-polynomial spline function with uniform grid for solving certain class of singularly perturbed boundary value problems. The proposed methods are second-order and fourth-order accurate. Numerical examples are provided to demonstrate the...

A population model including diffusion, chemotaxis and growth is studied. Assuming that the diffusion rate and the chemotactic rate are both very small compared with the growth rate, we derive a new equation to describe the time-evolution of the aggregating region of biological individuals and...

We apply a singular perturbation analysis to some option pricing models. To illustrate the technique we first consider the European put option under the standard Black-Scholes model, with or without barriers. Then we consider the same option under the constant elasticity of variance (CEV)...

In this paper, a study of a stochastic volatility model for asset pricing is described. Originally presented by J. Da Fonseca, M. Grasselli and C. Tebaldi, the Wishart volatility model identifies the volatility of the asset as the trace of a Wishart process. Contrary to a classic multifactor...

New methods for nonlinear model reduction of dynamic models, described by nonlinear differential equations have been developed. The methods are based on singular perturbations, weak coupling and cost functionals for computing and analyzing the reduced order model. By introducing a formally...

The evolution of renewable resources is characterized in many cases by different time scales where some state variables such as biomass, may evolve relatively faster than other state variables such as carrying capacity. Ignoring this time scale separation means that a slowly changing variable is...