Solving a nonlinear pde that prices real options using utility based pricing methods

We provide group invariant solutions to two nonlinear differential equations associated with the valuing of real options with utility pricing theory. We achieve these through the use of the Lie theory of continuous groups, namely, the classical Lie point symmetries. These group invariant solutions, constructed through the use of the symmetries that also leave the boundary conditions invariant, are consistent with the results in the literature. Thus it may be shown that Lie symmetry algorithms underlie many ad hoc methods that are utilised to solve differential equations in finance.     

Existence and multiplicity of wave trains in 2D lattices

We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov–Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and p:q$p:q$ resonance, respectively. We show the impact of the direction θ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of traveling waves of the original two-dimensional lattice in the direction θ of propagation satisfying tan θ is rational.     

Nagumo type existence results of Sturm-Liouville BVP for impulsive differential equations

In this paper, the method of upper and lower solutions and the Schauder degree theory are employed in the study of Sturm-Liouville boundary value problems for second order impulsive differential equations. We obtain the existence of at least three solutions to the problem under the assumption that the nonlinear term f satisfies a Nagumo condition with respect to the first order derivative.     

Invariant solutions as internal singularities of nonlinear differential equations and their use for qualitative analysis of implicit and numerical solutions

Lie group analysis of nonlinear differential equations reveals existence of singularities provided by invariant solutions and invisible from the form of the equation in question. We call them internal singularities in contrast with external singularities manifested by the form of the equation. It is illustrated by way of examples that internal singularities are useful for analyzing a behaviour of solutions of nonlinear differential equations near external singularities.     

Doubly stochastic Yule cascades (Part I): The explosion problem in the time-reversible case

Motivated by the probabilistic methods for nonlinear differential equations introduced by McKean (1975) for the Kolmogorov-Petrovski-Piskunov (KPP) equation, and by Le Jan and Sznitman (1997) for the incompressible Navier-Stokes equations (NSE), we identify a new class of stochastic cascade models, referred to as doubly stochastic Yule cascades. We establish non-explosion criteria under the assumption that the randomization of Yule intensities from generation to generation is by an ergodic time-reversible Markov process. In addition to the cascade models that arise in the analysis of certain deterministic nonlinear differential equations, this model includes the multiplicative branching random walks, the branching Markov processes, and the stochastic generalizations of the percolation and/or cell ageing models introduced by Aldous and Shields (1988) and independently by Athreya (1985).     

Periodic solutions for nonlinear nth order differential equations with delays

By applying the continuation theorem of coincidence degree theory, we establish the existence of 2π-periodic solutions for a class of nonlinear nth order differential equations with delays.     

Existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter

In this paper, we study the existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter. By using the properties of the Green’s function, u ₀-positive function and the fixed point index theory, we obtain some existence results of positive solution under some conditions concerning the first eigenvalue with respect to the relevant linear operator. The method of this paper is a unified method for establishing the existence of multiple positive solutions for a large number of nonlinear differential equations of arbitrary order with any allowed number of non-local boundary conditions.     

A note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations

Following the lead of [Carrillo, Arch. Ration. Mech. Anal. 147 (1999) 269-361], recently several authors have used Kru?kov's device of “doubling the variables” to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order differential operator is not allowed to depend explicitly on the spatial variable, which certainly restricts the range of applications of entropy solution theory. The purpose of this paper is to extend a version of Carrillo's uniqueness result to a class of degenerate parabolic equations with spatially dependent second order differential operator. The class is large enough to encompass several interesting nonlinear partial differential equations coming from the theory of porous media flow and the phenomenological theory of sedimentation-consolidation processes.     

Oscillation of second-order nonlinear differential equations with damping

We study oscillatory properties of solutions to a class of nonlinear second-order differential equations with a nonlinear damping. New oscillation criteria extend those reported in [ROGOVCHENKO, Yu. V.—TUNCAY, F.: Oscillation criteria for second-order nonlinear differential equations with damping, Nonlinear Anal. 69 (2008), 208–221] and improve a number of related results.     

Branching Near Plane Waves in Perturbed Dispersive Systems

A Poincaré-Lindstedt type technique for partial differential equations is used to study branching phenomena in perturbed dispersive systems arising in hydrodynamic stability theory. Multi-periodic waves with two frequencies which branch from a family of neutrally stable nonlinear periodic plane waves are constructed, the second frequency as a power series expansion in ε. The branching is compared with that of the unperturbed equations described in an earlier paper for the purpose of understanding how higher order perturbation terms effect the properties of the lower order amplitude equations. We find that in general the perturbation terms alter the leading order frequency shifts, thus changing the bifurcation from pitchfork to transverse type. The method is used to study the perturbed nonlinear Schrödinger equation and the perturbed MKdV equation.     

Matrix polynomials satisfying first order differential equations and three term recurrence relations

We describe families of matrix valued polynomials satisfying simultaneously a first order differential equation and a three term recurrence relation. Our goal is to address the classification of the matrix valued polynomials satisfying first order differential equations through the solutions of the so-called bispectral problem. At the heart of this lies the need to solve some complicated nonlinear equations with matrix coefficients called ad-conditions. The solutions of these equations are studied under a variety of sufficient conditions on its coefficients.     

Positive solutions of m-point boundary value problems for a class of nonlinear fractional differential equations

This paper deals with the existence and multiplicity of positive solutions for a class of nonlinear fractional differential equations with m-point boundary value problems. We obtain some existence results of positive solution by using the properties of the Green’s function, u ₀-bounded function and the fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator.     

Ermakov–Pinney and Emden–Fowler Equations: New Solutions from Novel Bäcklund Transformations

We study the class of nonlinear ordinary differential equations y″ y = F(z, y²), where F is a smooth function. Various ordinary differential equations with a well-known importance for applications belong to this class of nonlinear ordinary differential equations. Indeed, the Emden–Fowler equation, the Ermakov–Pinney equation, and the generalized Ermakov equations are among them. We construct Bäcklund transformations and auto-Bäcklund transformations: starting from a trivial solution, these last transformations induce the construction of a ladder of new solutions admitted by the given differential equations. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficult to apply.     

Solutions of nonlinear impulsive Itô functional-differential equations: Stability by the linear approximation

We study the p-stability (2 ≤ p < ∞) of solutions of nonlinear impulsive Itô functional-differential equations. To this end, we use the stability theory developed for deterministic functional-differential equations. The moment stability of solutions of nonlinear impulsive Itô functional-differential equations is studied with the use of the problem on the admissibility of a pair of spaces for linear impulsive Itô functional-differential equations. We prove assertions similar to traditional theorems on stability by the first approximation.     

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.     

Some problems on super-diffusions and one class of nonlinear differential equations

The historical superprocesses are considered on bounded regular domains with a complete branching form, as a probabilistic argument, the limit property of superprocesses is studied when the domains enlarge to the whole space. As an important application of superprocess, the representation of solutions of involved differential equations is used in term of historical superprocesses. The differential equations including the existence of nonnegative solution, the closeness of solutions and probabilistic representations to the maximal and minimal solutions are discussed, which helps develop the well-known results on nonlinear differential equations. Project supported by the National Natural Science Foundation of China (Grant No. 19631060) and the Postdoctoral Foundation of China.     

Stochastic partial differential equations with unbounded and degenerate coefficients

In this article, using DiPerna-Lions theory (DiPerna and Lions, 1989) [1], we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and uniqueness. Moreover, we also prove the L¹-integrability and a general maximal principle for generalized solutions of SPDEs. As applications, we study nonlinear filtering problem and also obtain the existence and uniqueness of generalized solutions for a degenerate nonlinear SPDE.     

Inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation

We consider the inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

Random nonlinear Hammerstein equations

We prove existence theorems for random differential equations defined in a separable reflexive Banach space. These theorems are proved through the use of theory of random analysis established in [X. Z. Yuan, Random nonlinear mappings of monotone type, J. Math. Anal. Appl. 19] which differs from the other means, for example in [R. Kannan and H. Salehi, Random nonlinear equations and monotonic nonlinearities, J. Math. Anal. Appl. 57 (1977), 234–256; D. Kravvaritis, Existence theorems for nonlinear random equations and inequalities, J. Math. Anal. Appl. 86 (1982), 61–73; D. A. Kandilakis and N. S. Papageorgious, On the existence of solutions for random differential inclusions in a Banach space, J. Math. Anal. Appl. 126 (1987), 11–23].     

The representation of meromorphic solutions for a class of odd order algebraic differential equations and its applications

In this paper, we employ the Nevanlinna's value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations. We obtain the representations of all meromorphic solutions for a class of odd order algebraic differential equations with the weak ?p,q?and dominant conditions. Moreover, we give the complex method to find all traveling wave exact solutions of corresponding partial differential equations. As an example, we obtain all meromorphic solutions of the Kuramoto–Sivashinsky equation by using our complex method. Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd.