Mathematical models in finance

In this paper we illustrate the interplay between Mathematics and Finance, pointing out the relevance of stochastic calculus and mathematical modelling in some important aspects of modern finance. We present two types of mathematical models: the binomial asset pricing model and continuous-time models. We point out some sensitive points of research.info:eu-repo/semantics/publishedVersio

A fully nonlinear problem arising in financial modelling

We state existence and localisation results for a fully nonlinear boundary value problem using the upper and lower solutions method. With this study we aim to contribute to a better understanding of some analytical features of a problem arising in financial modelling related to the introduction of transaction costs in the classical Black-Scholes model. Our result concerns stationary solutions which become interesting in finance when the time does not play a relevant role such as, for instance, in perpetual options.info:eu-repo/semantics/publishedVersio

A note on a stationary problem for a Black-Scholes equation with transaction cost

In this paper, we consider the nonlinear Black-Scholes equation arising in certain option pricing models with transaction costs. Following the classical Leland approach and applying Itô’s Lemma, the stochastic model yields the nonlinear parabolic partial differential equation for the option price ….info:eu-repo/semantics/publishedVersio

A note on periodic solutions of some nonautonomous differential equations

We prove the existence of nontrivial periodic solutions of some nonlinear ordinary differential equations with time-dependent coefficients using variational methods.info:eu-repo/semantics/publishedVersio

On some non-linear boundary value problems related to a Black--Scholes model with transaction costs

We deal with some generalizations on a Black--Scholes model arising in financial mathematics. As novelty in this paper, we consider a variable volatility and abstract functional boundary conditions, which allow us to treat a very large class of problems involving Black--Scholes equation. Our main results involve the existence of extremal solutions in presence of lower and upper solutions. Some examples of application are provided too

A fully nonlinear problem arising in financial modelling

We state existence and localisation results for a fully nonlinear boundary value problem using the upper and lower solutions method. With this study we aim to contribute to a better understanding of some analytical features of a problem arising in financial modelling related to the introduction of transaction costs in the classical Black-Scholes model. Our result concerns stationary solutions which become interesting in finance when the time does not play a relevant role such as, for instance, in perpetual options.info:eu-repo/semantics/publishedVersio

note on the numerical approximation of parabolic equations in Holder spaces

We consider the initial-boundary value problem for a multidimensional linear parabolic PDE of second order. This problem is solvable in Holder spaces. The solution is numerically approximated, using finite differences, and the rate of convergence of the time-space finite difference scheme is estimated. Both explicit and implicit discrete operators are given.info:eu-repo/semantics/publishedVersio

The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation

In this paper we study the existence of solutions of the nonlinear fourth-order equation [1] under the asymmetric nonlinear boundary conditions [2], [3] where g is a strictly monotonous function that may have some “one-sided” discontinuities and f and h exhibit some singularities.info:eu-repo/semantics/publishedVersio

Periodic solutions of parabolic and telegraph equations with asymmetric nonlinearities

This paper is organized as follows: Introduction, In Section 2, we collect the notation and basic assumptions that we shall suppose fulfilled throughout this paper. Section 3 is devoted to second order nonlinear one-dimensional parabolic and (linearly) damped hyperbolic equations. We compare, in some sense, the nonlinearity g(x, u) with the Fuçik spectrum of the corresponding piecewise linear differential equations with homogeneous Dirichlet boundary conditions, and a resonance condition of Landesman-Lazer type with respect to the forcing term h(x, t). More specifically, we assume that (the asymptotic behavior of) u - ¹ g ( x , u) lies in a rectangle located in what we should call the Fucik -Landesman-Lazer "resolvent" set. In Section 4, we take up the case of second-order multi-dimensional equations, and we prove results on crossing at not necessarily simple (higher) eigenvalues. Finally, in Section 5 we indicate the conditions under which one can extend our results to higher-order multi-dimensional equations.info:eu-repo/semantics/publishedVersio