Differential Galois Theory and Lie Symmetries

Blázquez-Sanz, David; Ruiz, Juan José Morales; Weil, Jacques-Arthur

doi:10.3842/sigma.2015.092

Cited by 3 publications

(5 citation statements)

References 23 publications

(38 reference statements)

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“…. Correspondingly, the analytic class of this normal form system is the one that has the largest possible Lie algebra of analytic infinitesimal symmetries (see [5]) of all the systems within the formal class. It turns out that the same holds also for non-resonant irregular LDEs (1.6): they are analytically equivalent to their formal normal form (1.20) if and only their Lie algebra of linear analytic point symmetries is the largest possible (Theorem 1.13 below).…”

Section: Lie Symmetriesmentioning

confidence: 99%

On equivalence of singularities of second order linear differential equations by point transformations

Klimes¹

2021

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Section: Lie Symmetriesmentioning

confidence: 99%

On equivalence of singularities of second order linear differential equations by point transformations

Klimes¹

2021

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…It came as a surprise when Ibragimov found a bridge [52] between Lie symmetries and Galois groups: He constructed the Galois groups for several simple algebraic equations by first calculating their Lie symmetries and then restricting the symmetry group to the roots of the equation in question. Thereafter more papers have appeared [53] which study the interplay and connections between the differential Galois group and Lie symmetries of linear homogeneous ODEs of n th order.…”

Section: Galois Theory Lie Theory Picard−vessiot Theorymentioning

confidence: 99%

“…We note that Equation (52) implies that C is Liouvillian if and only if C is Liouvillian. Thus it suffices to apply Kovacic's algorithm to Equation (53). Two remarks are now in order regarding Equation (53): 1.…”

Section: Application Of Kovacic's Algorithmmentioning

confidence: 99%

“…The order of ν at ∞, o(∞), is defined as (Equation ( 50)) o(∞) = max(0, 4 + d o s − d o t) where d o s and d o t denote the leading degree of s and t. This is related to the order of ∞ as a zero of ν. The cases to be considered in order to decide if Equation (53) admits Liouvillian solutions depend crucially on the orders associated to the poles of ν and to ∞.…”

Section: Application Of Kovacic's Algorithmmentioning

confidence: 99%

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Classes of elementary function solutions to the CEV model I

Melas

2019

Journal of Computational and Applied Mathematics

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In the equity markets the stock price volatility increases as the stock price declines. The classical Black−Scholes−Merton (BSM) option pricing model does not reconcile with this association. Cox introduced the constant elasticity of variance (CEV) model in 1975, in order to capture this inverse relationship between the stock price and its volatility. An important parameter in the model is the parameter β, the elasticity of volatility. The CEV model subsumes some of the previous option pricing models. For β = 0, β = −1/2, and β = −1 the CEV model reduces respectively to the BSM model, the square−root model of Cox and Ross, and the Bachelier model. Both in the case of the BSM model and in the case of the CEV model it has become traditional to begin a discussion of option pricing by starting with the vanilla European calls and puts. The pricing formulas for these financial instruments give concrete information about the pricing of options only after the employment of some intermediate approximation scheme. However, there are simpler solutions to both models than those pertaining to the standard calls and puts. Mathematically, it makes sense to investigate the simpler cases first. Furthermore we do not allow ourselves to be drawn into any rash generalizations or inferences from the vanilla European case by prematurely focusing on those cases and we obtain concrete information for the pricing of options without needing to introduce any intermediate approximation schemes. In the case of BSM model simpler solutions are the log and power solutions. These contracts, despite the simplicity of their mathematical description, are attracting increasing attention as a trading instrument. Similar simple solutions have not been studied so far in a systematic fashion for the CEV model. We use Kovacic's algorithm to derive, for all half−integer values of β, all solutions "in quadratures" of the CEV ordinary differential equation. These solutions give rise, by separation of variables, 1 emelas@econ.uoa.gr to simple solutions to the CEV partial differential equation. In particular, when β = ..., − 5 2 , −2, − 3 2 , −1, 1, 3 2 , 2, 5 2 , ..., we obtain four classes of of denumerably infinite elementary function solutions, when β = − 1 2 and β = 1 2 we obtain two classes of of denumerably infinite elementary function solutions, whereas, when β = 0 we find two elementary function solutions. In the derived solutions we have also dispensed with the unnecessary assumption made in the the BSM model asserting that the underlying asset pays no dividends during the life of the option.

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“…. Correspondingly, the analytic class of this normal form system is the one that has the largest possible Lie algebra of analytic infinitesimal symmetries (see [BMW15]) of all the systems within the formal class. It turns out that the same holds also for non-resonant irregular LDEs (6): they are analytically equivalent to their formal normal form (20) if and only their Lie algebra of linear analytic point symmetries is the largest possible (Theorem 12 below).…”

Section: Lie Symmetriesmentioning

confidence: 99%