×

Geometrical properties of differential equations. Applications of the Lie group analysis in financial mathematics. (English) Zbl 1393.22001

Hackensack, NJ: World Scientific (ISBN 978-981-4667-24-1/hbk). xi, 328 p. (2015).
This textbook is an introduction to Lie group analysis of ordinary and partial differential equations with applications to financial mathematics. The book consists of nine chapters. Chapter 1 is an introduction and contains some historical remarks on Lie group analysis and financial mathematics. Chapter 2 deals with point transformations on \(\mathbb R^2\). The author discusses point transformations on the plane, groups, group representations, infinitesimal actions of a group, and the group orbit. Definitions are provided of the direction field, an integral curve to a direction field and the orbit of a point transformation group. The Lie equations which are the key relation between the infinitesimal and the closed form of a point transformation group are introduced. In Chapter 3 the ideas are introduced of invariant points, invariant curves, and invariant families of curves in the plane. The author discusses the connection between the closed form and the infinitesimal form of point transformations. Notions are presented of infinitesimal generator, canonical variables and the normal form of an infinitesimal generator. The author analyzes the geometrical meaning of canonical coordinates. Chapter 4 deals with first order ordinary differential equations. The author examines in great detail the geometrical side of the ordinary differential equations theory and the connection to Lie group theory. Almost all practical tips and tricks given by the Lie group methods in application to differential equations are demonstrated. The geometrical image of the first order ordinary differential equations is considered. A linear first order differential operator is introduced and the relation between this operator and the infinitesimal generator is studied. Chapter 5 is devoted to the prolongation procedure. It is discussed how someone can prolong the action of a one-parameter group of point transformations on the first and higher derivatives of a dependent variable. The author defines the prolongation of an infinitesimal generator and obtains a linear operator in higher dimensional space. The invariance is discussed of differential expressions under the action of a one-parameter group of point transformations, and the determining equations are defined which play a crucial role in the applications of Lie group theory to differential equations. A short overview of Lie algebra properties is presented in Chapter 6. The linear space of differential operators is introduced and some typical examples of linear spaces are presented. Then the author defines a bilinear operation with certain properties on a linear space. This operation defines the Lie algebra structure on the linear space. Classifications are provided of two-, three-, and four-dimensional real solvable Lie algebras often arising in applications. In Chapter 7 \(n\)-th order differential equations are considered. The author introduces the differential operator for a high order ordinary differential equation and shows that the way of using admitted symmetry groups depends on their structure. Second order ordinary differential equations with two symmetries are considered, the case of \(n\)-th order ordinary differential equations which admits a high-dimensional solvable Lie algebra, the case of \(n\)-th order ordinary differential equations which are invariant under a non-solvable Lie algebra with a special property, and the case of \(n\)-th order ordinary differential equations with a general type of the admitted symmetry algebra. In Chapter 8 the author provides a detailed Lie group analysis of the heat equation and Black-Scholes model. They can be transformed to each other. It is explained that this transformation provides a similarity transformation of the admitted symmetry algebras. Since the admitted symmetry algebras are six-dimensional the author finds a lot of interesting reductions and corresponding families of invariant solutions. The famous solution for the call option from the Black-Scholes model is presented. Problems with solutions and exercises are presented at the end of each chapter. Chapter 9 is devoted to the study of new models in financial mathematics. The author considers an ideal financial market and the Black-Scholes model for option pricing. Then pricing option is considered in illiquid markets. The author studies the Frey, the Frey-Patie, the Frey-Stremme models of risk management for derivatives and the Sircar-Papanicolaou model. Then the equilibrium or the reaction-function models are investigated. The Schönbucher-Wilmott model for self-financing trading strategies is studied, pricing and hedging in incomplete markets are considered. The Musiela-Zariphopoulou model, problems of optimal consumption with random income, transaction costs models are also studied. The bibliography consists of 80 items.

MSC:

22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
91G80 Financial applications of other theories
91-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
91G20 Derivative securities (option pricing, hedging, etc.)
34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
54H15 Transformation groups and semigroups (topological aspects)
PDFBibTeX XMLCite
Full Text: DOI