Related papers: Elementary functions solutions to the Bachelier mo…
We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options and we compute their non-trivial scaling limit for a vanishing price impact which is inversely…
In this paper we will discuss the Dirichlet problem of nonlinear second order partial differential equations resolved with any derivatives. First, we transform it into generalized integral equations. Next, we discuss the existence of the…
The problem of finding a martingale on a manifold with a fixed random terminal value can be solved by considering BSDEs with a generator with quadratic growth. We study here a generalization of these equations and we give uniqueness and…
This study investigates enhancing option pricing by extending the Black-Scholes model to include stochastic volatility and interest rate variability within the Partial Differential Equation (PDE). The PDE is solved using the finite…
The so-called unified method expresses the solution of an initial-boundary value problem (IBVP) for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple IBVP, which will be referred…
Expectiles are statistical parameters which also provide a class of sublinear risk measures in finance. They are solutions of continuous optimization problems. The corresponding first order condition provides two different fixed point…
We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that…
Employing a limiting case of a conjecture for constructing piecewise separable-variables functions, the elements of the Pseudoanalytic Function Theory are used for numerically approaching solutions of the forward Dirichlet boundary value…
In this paper, we consider a class of nonlinear fractional differential equations involving Hilfer derivative with boundary conditions. First, we obtain an equivalent integral for the given boundary value problem in weighted space of…
Lie symmetry analysis is applied to study the nonlinear rotating shallow water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow water equations are related…
In this didactic note, we describe a procedure to derive successive approximations of $\pi$ using Euler Beta functions. It is an interesting exercise for undergraduate students, since it involves polynomial roots, integral calculations,…
We analyze the polynomial solutions of the linear differential equation $p_2(x)y''+p_1(x)y'+p_0(x)y=0$ where $p_j(x)$ is a $j^{\rm th}$-degree polynomial. We discuss all the possible polynomial solutions and their dependence on the…
We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Ito and point processes, such PDEs are…
In this paper we consider a boundary value problem for fully fourth order nonlinear functional differential equation which contains all lower derivatives of proportional delay arguments. By the reduction of the problem to operator equation…
We consider a general nonsymmetric second-order linear elliptic PDE in the framework of the Lax-Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive…
We analyze properties of apportionment functions in context of the problem of allocating seats in the European Parliament. Necessary and sufficient conditions for apportionment functions are investigated. Some exemplary families of…
This paper introduces the study of occurrence of symmetries in binary differential equations (BDEs). These are implicit differential equations given by the zeros of a quadratic 1-form, $a(x,y)dy^2 + b(x,y)dxdy + c(x,y)dx^2 = 0,$ for $a, b,…
The Bank-Laine conjecture concerning the oscillation of solutions of second order homogeneous linear differential equations has recently been disproved by Bergweiler and Eremenko. It is shown here, however, that the conjecture is true if…
Motivated by the integral representation of the Euler Beta function, we introduce its Cauchy siblings and investigate some of their properties. Two of these newly introduced functions happen to coincide with some classical means, such as…
We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost…