A note on the nonlinear Volterra integral equation for the early exercise boundary

In this paper, we study the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the one-dimensional diffusion model for a stock price with constant interest rate and constant dividend yield and with a local volatility depending on the current time t and the current stock price S. In the classical Black–Sholes model for a stock price, Theorem 4.3 of [S. D. Jacka, Optimal stopping and the American put, Math. Finance 1 1991, 2, 1–14] states that if the family of integral equations (parametrized by the variable S) holds for all {S\leq b(t)} with a candidate function {b(t)}, then this {b(t)} must coincide with the American put early exercise boundary {c(t)}. We generalize Peskir’s result [G. Peskir, On the American option problem, Math. Finance 15 2005, 1, 169–181] to state that if the candidate function {b(t)} satisfies one particular integral equation (which corresponds to the upper limit {S=b(t)}), then all other integral equations (corresponding to S, {S\leq b(t)}) will be automatically satisfied by the same function {b(t)}.

A viscosity-type algorithm for an infinitely countable family of (f,g)-generalized k-strictly pseudononspreading mappings in CAT(0) spaces

In this paper, we propose a viscosity-type algorithm to approximate a common solution of a monotone inclusion problem, a minimization problem and a fixed point problem for an infinitely countable family of {(f,g)}-generalized k-strictly pseudononspreading mappings in a {\mathrm{CAT}(0)} space. We obtain a strong convergence of the proposed algorithm to the aforementioned problems in a complete {\mathrm{CAT}(0)} space. Furthermore, we give an application of our result to a nonlinear Volterra integral equation and a numerical example to support our main result. Our results complement and extend many recent results in literature.

The inflation of viscoelastic balloons and hollow viscera

For the first time, the problem of the inflation of a nonlinear viscoelastic thick-walled spherical shell is considered. Specifically, the wall has quasilinear viscoelastic constitutive behaviour, which is of fundamental importance in a wide range of applications, particularly in the context of biological systems such as hollow viscera, including the lungs and bladder. Experiments are performed to demonstrate the efficacy of the model in fitting relaxation tests associated with the volumetric inflation of murine bladders . While the associated nonlinear elastic problem of inflation of a balloon has been studied extensively, there is a paucity of studies considering the equivalent nonlinear viscoelastic case. We show that, in contrast to the elastic scenario, the peak pressure associated with the inflation of a neo-Hookean balloon is not independent of the shear modulus of the medium. Moreover, a novel numerical technique is described in order to solve the nonlinear Volterra integral equation in space and time originating from the fundamental problem of inflation and deflation of a thick-walled nonlinear viscoelastic shell under imposed pressure.

Non-polynomial spline functions and Quasi-linearization to approximate nonlinear Volterra integral equation

In this work, we want to use the Non-polynomial spline basis and Quasi-linearization method to solve the nonlinear Volterra integral equation. When the iterations of the Quasilinear technique employed in nonlinear integral equation we obtain a linear integral equation then by using the Non-polynomial spline functions and collocation method the solution of the integral equation can be approximated. Analysis of convergence is investigated. At the end, some numerical examples are presented to show the effectiveness of the method.