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<p>We consider one transaction costs model which was suggested by Cetin, Jarrow and Protter (2004) for an illiquid market. In this case the hedging strategy of programming traders can affect the assets prises. We study the corresponding partial differential equation (PDE) which is a non-linear Black-Scholes equation for illiquid markets. We use the Lie group analysis to determine the symmetry group of this equations. We present the Lie algebra of the Lie point transformations, the complete symmetry group and invariants. For different subgroups of the main symmetry group we describe the corresponding invariants. We use these invariants to reduce the PDE under investigation to ordinary differential equations (ODE). Solutions of these ODE's are subgroup-invariant solutions of the non-linear Black-Scholes equation. For some classes of those ODE's we find exact solutions and studied their properties.</p><p>% reduce non-linear PDE to ODE's. To some ODE's we find exact solutions.</p><p>%In this work we are studying one model for pricing derivatives in illiquid market. We discuss it structure and properties. Make a symmetry reduction for the PDE corresponding our model.</p>

We consider one transaction costs model which was suggested by Cetin, Jarrow and Protter (2004) for an illiquid market. In this case the hedging strategy of programming traders can affect the assets prises. We study the corresponding partial differential equation (PDE) which is a non-linear Black-Scholes equation for illiquid markets. We use the Lie group analysis to determine the symmetry group of this equations. We present the Lie algebra of the Lie point transformations, the complete symmetry group and invariants. For different subgroups of the main symmetry group we describe the corresponding invariants. We use these invariants to reduce the PDE under investigation to ordinary differential equations (ODE). Solutions of these ODE's are subgroup-invariant solutions of the non-linear Black-Scholes equation. For some classes of those ODE's we find exact solutions and studied their properties. % reduce non-linear PDE to ODE's. To some ODE's we find exact solutions. %In this work we are studying one model for pricing derivatives in illiquid market. We discuss it structure and properties. Make a symmetry reduction for the PDE corresponding our model.

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We define the Lie group G2 and show several equivalent ways to view G2. We do the same with its Lie algebra g2. We identify a new basis for g2 using Bryant’s view of g2 and geometric considerations we develop. We then show how to construct a matrix representation of G2 given our particular basis for g2. We examine the geometry of 1 and 2-parameter subgroups of G2. Finally, we suggest an area of further research using the new geometric characterization we developed for g2.

This thesis investigates the set stabilization problem for systems with Lie group symmetry. Initially, we examine left-invariant systems on Lie groups where the target set is a left or right coset of a closed subgroup. We broaden the scope to systems defined on smooth manifolds that are invariant under a Lie group action. Inspired by the solution of this problem for linear time-invariant systems, we show its equivalence to an equilibrium stabilization problem for a suitable quotient control system. We provide necessary and sufficient conditions for the existence of the quotient control system and analyze various properties of such a system. This theory is applied to the formation stabilization of three kinematic unicycles, the path stabilization of a particle in a gravitational field, and the conversion and temperature control of a continuously stirred tank reactor.

This thesis investigates the set stabilization problem for systems with Lie group symmetry. Initially, we examine left-invariant systems on Lie groups where the target set is a left or right coset of a closed subgroup. We broaden the scope to systems defined on smooth manifolds that are invariant under a Lie group action. Inspired by the solution of this problem for linear time-invariant systems, we show its equivalence to an equilibrium stabilization problem for a suitable quotient control system. We provide necessary and sufficient conditions for the existence of the quotient control system and analyze various properties of such a system. This theory is applied to the formation stabilization of three kinematic unicycles, the path stabilization of a particle in a gravitational field, and the conversion and temperature control of a continuously stirred tank reactor.

Cette thèse est consacrée à une classe de groupes, appelés groupes de Kac-Moody, qui généralise de façon naturelle les groupes de Lie semi-simples, ou plus précisément, les groupes algébriques réductifs, dans un contexte infini-dimensionnel. On s'intéresse plus particulièrement au problème d'isomorphismes pour ces groupes, en vue d'obtenir un analogue infini-dimensionnel de la célèbre théorie des homomorphismes 'abstraits' de groupes algébriques simples, due à Armand Borel et Jacques Tits. Le problème d'isomorphismes qu'on étudie s'avère être un cas particulier d'un problème plus général, qui consiste à caractériser les homomorphismes de groupes algébriques vers les groupes de Kac-Moody, dont l'image est bornée. Ce problème peut à son tour s'énoncer comme un problème de rigidité pour les actions de groupes algébriques sur les immeubles, via l'action naturelle d'un groupe de Kac-Moody sur une paire d'immeubles jumelés. Les résultats partiels, relatifs à ce problème de rigidité, que nous obtenons, nous permettent d'apporter une solution complète au problème d'isomorphismes pour les groupes de Kac-Moody déployés. En particulier, on obtient un résultat de dévissage pour les automorphismes de ces objets. Celui-ci fournit à son tour une description complète de la structure du groupe d'automorphismes d'un groupe de Kac-Moody déployé sur un corps de caractéristique~$0$. Nos arguments permettent également de traiter de façon analogue certaines formes anisotropes de groupes de Kac-Moody complexes, appelées formes unitaires. On montre en particulier que la topologie Hausdorff naturelle que portent ces formes est un invariant de leur structure de groupe abstrait. Ceci généralise un résultat bien connu de H. Freudenthal pour les groupes de Lie compacts. Enfin, l'on s'intéresse aux homomorphismes de groupes de Kac-Moody à image fini-dimensionnelle, et l'on démontre la non-existence de tels homomorphismes à noyau central, lorsque le domaine est un groupe de Kac-Moody de type indéfini sur un corps infini. Ceci réduit un problème ouvert, dit problème de linéarité pour les groupes de Kac-Moody, au cas de corps de base finis.

This paper presents the Lie-Backlund symmetry method to give the equivalence between differential equations and describe the equivalent transformation procedure of nonlinear control systems of partial differential equations. The equivalent linear systems found by solving the infinitesimal generator of one-parameter Lie groups with prolongations and the infinitesimal generator are used to construct the parameters of invertible mapping u. And the equivalence linear form of the nonlinear system is constructed via u. Some necessary conditions for mapping a nonlinear control system of PDE¡¦s to a linear control system of PDE¡¦s are discussed, and application of Lie-Backlund symmetries and invertible mapping u constructed linear time-invariant control system of partial differential equations.