scholarly journals Constructing C0-Semigroups via Picard Iterations and Generating Functions: An Application to a Black–Scholes Integro-Differential Operator

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 589
Author(s):  
Marianito R. Rodrigo
Keyword(s):  
Differential Operator ◽  
Generating Functions ◽  
Adomian Decomposition Method ◽  
Kernel Functions ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Diffusion Dynamics ◽  
Adomian Decomposition ◽  
Black Scholes ◽  
Picard Iterations

An alternative approach is proposed for constructing a strongly continuous semigroup based on the classical method of successive approximations, or Picard iterations, together with generating functions. An application to a Black–Scholes integro-differential operator which arises in the pricing of European options under jump-diffusion dynamics is provided. The semigroup is expressed as the Mellin convolution of time-inhomogeneous jump and Black–Scholes kernel functions. Other applications to the heat and transport equations are also given. The connection of the proposed approach to the Adomian decomposition method is explored.

scholarly journals Solving Some Derivative Equations Fractional Order Nonlinear Partials Using the Some Blaise Abbo Method

2021 ◽  
Vol 13 (2) ◽  
pp. 101
Author(s):  
Abdoul wassiha NEBIE ◽  
Frederic BERE ◽  
Bakari ABBO ◽  
Youssouf PARE
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Decomposition Method ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Adomian Decomposition ◽  
Diffusion Convection

In this paper, we propose the solution of some nonlinear partial differential equations of  fractional order that modeled diffusion, convection and reaction problems. For the solution of these equations we will use the SBA method which is a method based on the combination of the Adomian Decomposition Method (ADM), the Picard's principle  and the method of successive approximations.   

AN APPROPRIATE APPROACH TO PRICING EUROPEAN-STYLE OPTIONS WITH THE ADOMIAN DECOMPOSITION METHOD

2018 ◽  
Vol 59 (3) ◽  
pp. 349-369
Author(s):  
ZIWIE KE ◽  
JOANNA GOARD ◽  
SONG-PING ZHU
Keyword(s):  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
European Options ◽  
Black Scholes Model ◽  
Adomian Decomposition ◽  
Black Scholes ◽  
Interest Rate Model ◽  
Vasicek Interest Rate Model ◽  
Digital Options ◽  
Continuous Estimation

We study the numerical Adomian decomposition method for the pricing of European options under the well-known Black–Scholes model. However, because of the nondifferentiability of the pay-off function for such options, applying the Adomian decomposition method to the Black–Scholes model is not straightforward. Previous works on this assume that the pay-off function is differentiable or is approximated by a continuous estimation. Upon showing that these approximations lead to incorrect results, we provide a proper approach, in which the singular point is relocated to infinity through a coordinate transformation. Further, we show that our technique can be extended to pricing digital options and European options under the Vasicek interest rate model, in both of which the pay-off functions are singular. Numerical results show that our approach overcomes the difficulty of directly dealing with the singularity within the Adomian decomposition method and gives very accurate results.

scholarly journals On Comparative Analysis for the Black-Scholes Model in the Generalized Fractional Derivatives Sense via Jafari Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Saima Rashid ◽  
Sobia Sultana ◽  
Rehana Ashraf ◽  
Mohammed K. A. Kaabar
Keyword(s):  
Fractional Derivative ◽  
Adomian Decomposition Method ◽  
Capital Asset Pricing ◽  
Transform Method ◽  
Uniqueness Results ◽  
Black Scholes Model ◽  
Asset Pricing Models ◽  
Adomian Decomposition ◽  
Black Scholes ◽  
Generalized Fractional Derivatives

The Black-Scholes model is well known for determining the behavior of capital asset pricing models in the finance sector. The present article deals with the Black-Scholes model via the Caputo fractional derivative and Atangana-Baleanu fractional derivative operator in the Caputo sense, respectively. The Jafari transform is merged with the Adomian decomposition method and new iterative transform method. It is worth mentioning that the Jafari transform is the unification of several existing transforms. Besides that, the convergence and uniqueness results are carried out for the aforesaid model. In mathematical terms, the variety of equations and their solutions have been discovered and identified with various novel features of the projected model. To provide additional context for these ideas, numerous sorts of illustrations and tabulations are presented. The precision and efficacy of the proposed technique suggest its applicability for a variety of nonlinear evolutionary problems.

scholarly journals Application of the SBA Method to Solve the Nonlinear Biological Population Models

2019 ◽  
Vol 12 (3) ◽  
pp. 1096-1105
Author(s):  
Stevy Mikamona Mayembo ◽  
Joseph Bonazebi-Yindoula ◽  
Youssouf Pare ◽  
Gabriel Bissanga
Keyword(s):  
Parabolic Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Degenerate Parabolic Equations ◽  
Successive Approximations ◽  
Biological Population ◽  
Adomian Decomposition ◽  
Nonlinear Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Nonlinear Degenerate

This paper presents numerical solution of some nonlinear degenerate parabolic equations arising in the spatial di¤usion of biological populations. The SBA method based on combination of Adomian decomposition method, principle of Picard and successive approximations is used for solving these equations. The analytical obtained solutions show that the SBA method leads to more accurate results.

A different approach to the European option pricing model with new fractional operator

2018 ◽  
Vol 13 (1) ◽  
pp. 12 ◽  
Author(s):  
M. Yavuz ◽  
N. Özdemir
Keyword(s):  
Option Pricing ◽  
Homotopy Perturbation Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
European Option Pricing ◽  
Adomian Decomposition ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Pricing Problems ◽  
Fractional Differentiation Operator

In this work, we have derived an approximate solution of the fractional Black-Scholes models using an iterative method. The fractional differentiation operator used in this paper is the well-known conformable derivative. Firstly, we redefine the fractional Black-Scholes equation, conformable fractional Adomian decomposition method (CFADM) and conformable fractional modified homotopy perturbation method (CFMHPM). Then, we have solved the fractional Black-Scholes (FBS) and generalized fractional Black-Scholes (GFBS) equations by using the proposed methods, which can analytically solve the fractional partial differential equations (FPDE). In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of these two option pricing problems by using in pricing the actual market data. Also, we have found out that the proposed models are very efficient and powerful techniques in finding approximate solutions of the fractional Black-Scholes models which are considered in conformable sense.

scholarly journals Solution of the Black-Scholes Equation: LDM and SDM

2021 ◽  
Vol 23 (07) ◽  
pp. 1111-1115
Author(s):  
R. K. Pavan Kumar. Pannala ◽  
Keyword(s):  
Differential Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Decomposition Methods ◽  
Financial Mathematics ◽  
Discretization Method ◽  
Discretization Methods ◽  
Adomian Decomposition ◽  
Black Scholes ◽  
Laplace Decomposition Method

The main aim of this paper is to discuss a new way of a non-discretization method for the solution of the Black-Scholes equation. Black-Scholes is a mathematical model based on a partial differential equation. The solution of the model is of utmost importance in financial mathematics to estimate option pricing. Several analytical, numerical, and non-discretization methods are existing in the literature to solve the model. Two decomposition methods namely the Laplace decomposition method (LDM) and Sumudu decomposition method (SDM) are adopted for the present study. The results of the present techniques have closed an agreement with an approximate solution which has been obtained with the help of the Adomian Decomposition Method (ADM).

scholarly journals Resolution of Nonlinear Convection - Diffusion - Reaction Equations of Cauchy Kind by the Laplace SBA Method

2019 ◽  
Vol 12 (3) ◽  
pp. 771-789
Author(s):  
Youssouf Pare ◽  
Minoungou Youssouf ◽  
Abdoul Wassiha Nebie
Keyword(s):  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Diffusion Reaction ◽  
Successive Approximations ◽  
Cauchy Type ◽  
Nonlinear Convection ◽  
Convection Diffusion ◽  
S Transform ◽  
Adomian Decomposition ◽  
Reaction Equations

In this paper, we propose the solution of a few non linear partial di¤erential equa-tions modelling di¤usion, convection and reaction problems Cauchy type. The Laplace SBA method based on combination of Laplace’s transform, Adomian Decomposition Method(ADM), Picard prnciple and successive approximations is used for solving these equations.

scholarly journals An Effcient Method to Find Approximate Solutions for Emden-Fowler Equations of nth Order

2020 ◽  
pp. 9-27
Author(s):  
Nuha Mohammed Dabwan ◽  
Yahya Qaid Hasan
Keyword(s):  
Exact Solution ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Higher Order ◽  
New Method ◽  
Adomian Decomposition

In this research, Emden-Fowler equations of higher order with boundary conditions are considered and solved using Modied Adomian Decomposition Method (MADM). We dened a new differential operator under two conditions: rst condition when m ≤ 0 and second condition when m ≥ 0. From this operator, we got three types of Emden-Fowler equations of higher order. The new method is evaluated by using many examples, the results obtained through this method reveal the effectiveness of this method for these type of equations, especially when comparisons are made with the exact solution.

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