Invariant approach to optimal investment–consumption problem: the constant elasticity of variance (CEV) model

Bakkaloglu, Ahmet; Aziz, Taha; Fatima, Aeeman; Mahomed, F. M.; Khalique, Chaudry Masood

doi:10.1002/mma.4060

Cited by 12 publications

(7 citation statements)

References 28 publications

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“…e refined invariant criteria, for the reduction of (1 + 1) parabolic equation into different canonical forms, in terms of the coefficients of the parabolic equation were derived in [27]. Recently, the usefulness of the invariant criterion has been thoroughly demonstrated in studies [28][29][30].…”

Section: Theory and Backgroundmentioning

confidence: 99%

On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

Aziz

2021

Mathematical Problems in Engineering

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The classical Cox–Ingersoll–Ross (CIR) bond-pricing model is based on the evolution space-time dependent partial differential equation (PDE) which represents the standard European interest rate derivatives. In general, such class of evolution partial differential equations (PDEs) has generally been resolved by classical methods of PDEs and by ansatz-based techniques which have been previously applied in a similar context. The author here shows the application of an invariant approach, a systematic method based on deductive group-theoretical analysis. The invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. This method leads us to exactly solve the scalar linear space-time dependent parabolic PDE representing the CIR model. It was found that CIR PDE is transformed into the first canonical form, which is the heat equation. Under the proper choice of emerging parameters of the model, the CIR equation is also reduced to the second Lie canonical form. The equivalence transformations which map the CIR PDE into the different canonical forms are deduced. With the use of these equivalence transformations, the invariant solutions of the underlying model are found by using some well-known results of the heat equation and the second Lie canonical form. Furthermore, the Cauchy initial-value model of the CIR problem along with the terminal condition is discussed and closed-form solutions are deduced. Finally, the conservation laws associated with the CIR equation are derived by using the general conservation theorem.

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Section: Theory and Backgroundmentioning

confidence: 99%

On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

Aziz

2021

Mathematical Problems in Engineering

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“…Bakkaloğlu ve arkadaşları ise 2016'da [9] lie simetri değişmezlik koşulları ile optimal yatırım ve harcama probleminin çözümü üzerine makale yayınlamışlardır. Buna ek olarak, Bakkaloğlu ve arkadaşları 2017 yılında yayınladıkları makalelerinde [10] de Vasicek ve Cox-Ingersoll-Ross(CIR) rassal faiz oranları için değişmezlik kriterlerini uygulayarak bu modellerin sırasıyla birinci ve ikinci tip lie kanonik formuna indirgenebileceğini göstererek, temel çözümleri üzerine çalışmışlardır.…”

Section: Igi̇ri̇şunclassified

“…Daha sonra ise, Hull-White kısmi türevli diferansiyel denkleminin asıl çözümünü bulduğumuz bu dönüşümlerle ve ısı denkleminin literatürdeki özelliklerini kullanarak elde edeceğiz. literatüre baktığımızda bu konudaki öncü ilk çalışma Gazizov ve Ibragimov [2] tarafından 1998 yılında Merton-Black-Scholes [7] Bakkaloğlu ve arkadaşları ise 2016'da [9] lie simetri değişmezlik koşulları ile optimal yatırım ve harcama probleminin çözümü üzerine makale yayınlamışlardır. Buna ek olarak, Bakkaloğlu ve arkadaşları 2017 yılında yayınladıkları makalelerinde [10] de Vasicek ve Cox-Ingersoll-Ross(CIR) rassal faiz oranları için değişmezlik kriterlerini uygulayarak bu modellerin sırasıyla birinci ve ikinci tip lie kanonik formuna indirgenebileceğini göstererek, temel çözümleri üzerine çalışmışlardır.…”

Section: Igi̇ri̇şunclassified

Hull-White Stokastik Diferansiyel Denklemine Lie Simetri Analizi

İzgi

Bakkaloglu

2018

Marmara Fen Bilimleri Dergisi

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Bu makalede, stokastik diferansiyel denklemlere lie simetri analizinin bir uygulaması olarak asıl çözümün nasıl elde edileceğini göstereceğiz. Yapacağımız bu analizler stokastik faiz oranı modellerinden Hull-White modeli özelinde yapılacaktır. İlk olarak Hull-White stokastik modeline karşılık gelen Hull-White (1+1) lineer parabolik kısmi türevli denklemini elde edeceğiz. Daha sonra, elde ettiğimiz bu denklemin lie simetri analiz yöntemleriyle özellikle de değişmezlik kriterleri altında klasik anlamdaki ısı denklemine dönüşebileceğini göstereceğiz ve ilgili dönüşümleri bulacağız. Son olarak da, Hull-White kısmi türevli diferansiyel denkleminin asıl çözümünü, bulduğumuz bu dönüşümlerle ve ısı denkleminin literatürdeki özelliklerini kullanarak elde edeceğiz.

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“…Sophocleous et al provided solutions of the Stein-Stein model and Heston model for stochastic volatility based upon the Lie theory of infinitesimal transformations and its associated group theory [16,17]. Khalique et al solved the optimal investment-consumption problem under the constant elasticity of variance (CEV) model using the invariant approach [18]. Furthermore, the complete and enhanced group classifications of generalization of the Black-Scholes-Merton model and a semi-linear generalization of a general bond-pricing equation were carried out from the mathematical point by making use of the underlying equivalence and additional equivalence transformations [19,20].…”

Section: Introductionmentioning

confidence: 99%

Symmetry-based optimal portfolio for a DC pension plan under a CEV model with power utility

2021

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In this article, explicit representation of solution for the Hamilton-Jacobi-Bellman (HJB) equation associated with the portfolio optimization problem for an investor who seeks to maximize the expected power (CRRA) utility of the terminal wealth in a defined-contribution pension plan under a constant elasticity of variance model is derived based on the application of the Lie symmetry method to the partial differential equation and its associated terminal condition. Compared with the ingenious ansatz techniques used before, here we present a group theoretical analysis of the terminal value problem for the solution following the algorithmic procedure of the Lie symmetry analysis. It shows that the interesting properties of the group structures of the original HJB equation and its successive similarity reduced equations lead to an elegant resolution of the problem. Moreover, we identify the meaningful range of risk aversion coefficient which is ignored in the previous work. At last, the properties and sensitivity analysis of the derived optimal strategy are demonstrated by numerical simulations and several figures. The method used here is quite general and

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Invariant approach to optimal investment–consumption problem: the constant elasticity of variance (CEV) model

Cited by 12 publications

References 28 publications

On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

Hull-White Stokastik Diferansiyel Denklemine Lie Simetri Analizi

Symmetry-based optimal portfolio for a DC pension plan under a CEV model with power utility

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