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Lie Symmetry Method for Partial Differential Equations with Application in finance

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Lie Symmetry Method for Partial Differential Equations with Application in finance Università degli Studi di Trento Dipartimento di Matematica Corso di Laurea magistrale in Matematica Lie symmetry method for partial differential equations with application in finance FINAL THESIS Supervisor: Graduant: Prof. Luciano Tubaro Francesco Giuseppe Cordoni Co-supervisor: Prof. Luca Di Persio Co-supervisor: Prof. Andrea Pascucci Accademic Year 2012-2013 i "Per cui, quella sera, nell’ansa che il fiume descrive costeggiando la zona degli orti per anziani, alcuni pensionati avevano fiduciosamente messo a mollo le lenze." Preface Preface Partial differential equations (PDE) arise in many areas of scientific research and they have been classically used in describing models spanning from quantum mechanics to biological systems. More recently the theory of PDE have attracted an increasing interest by mathematicians and practitioners working in those areas related to banks and insurances. When we study a particular differential equation, we have to face with two fundamental questions: " Is there (at least) a solution ?" and (just in case) "Is it possible to write it (them) explicitly ?" Unfortunately it does not exist a standard procedure to follow to reach these goals. Even if we are able to show that a given differential problem has solutions, we often cannot show them analytically. This makes the derivation of exact solutions for PDE a corner stone in studying differential equations theory. In fact, although numerical analysis has made impressive strides in recent years, still the quest of find analytic solution is the main goal in studying PDE. It follows that a method able to recursively produce solutions of a given PDE may be very useful. In this thesis we deeply analyse one of the most fruitful technique for finding exact solutions of a given differential equations. Such a method, first developed by the Norwegian mathematician Sophus Lie (1842-1899) in the second half of the XIX century, unifies and extends ad hoc techniques in order to produce exact solutions for PDE by exploiting the fact that many phenomena in nature has internal symmetries. Lie developed a systematic way for finding analytic solution for a wide class of PDEs through continuous transformation groups. These groups are usually called Lie groups. These groups are both a "weird" and startling merger of algebraic group, topological structures and elements of analysis. The core idea is to use internal symmetries admitted by a given PDE in order to reduce the number of independent variables. Thus a solution of the given PDE can be found solving a different differential equations with fewer independent variables. The Lie work was inspired by Galois’s theory for polynomial equations. In particular Lie tried to create a unified theory of integration for ordinary differential equations similar to the Abelian theory developed to solve algebraic equations. For almost 100 years nobody has further studied the theory. It was in the mid 90’s that Ovsiannikov, in the ex URSS, and Bluman, in the West,have brought back the theory to light. Since then a number of works have appeared. This is mainly due to the high algorithmic complexity characterizing the procedure which gives the internal symmetries of a given PDE and, thus, its solutions. A great improvement in concrete applications of the Lie approach have iii iv been realized by the use of symbolic calculus and fast calculators. Modern computers can be easily programmed to find the desired symmetries, making the theory even more appealing and effective. In recent years, particularly after the worldwide financial crisis of 1987, the field of mathematical finance has seen a massive development leading it to became one of the most studied field in applied mathematics. Particular attention have been attracted by problems related to the fair pricing procedure for a huge pletora of structured financial instruments such in the case of financial derivatives. Financial derivatives are particular type of contracts whose value depends on an underlying asset. The time behaviour of the price of such a contract is usually modeled by a partial differential equations, eventually driven by stochastic terms. In such a scenario is clearly of particular relevance the existence of a (possibly) unique fair price. Moreover, when such a fair price is shown to exist, is of great importance to explicitly write it as an analytical function of its (eventually stochastic) parameters .This is why many mathematicians have applied the Lie’s theory to obtain efficient ways to price a wide class of financial instruments. We will see further (see Sec. [] and []) how such techniques can be exploited to derive not only analytical solutions but also fundamental solutions and transition density functions associated to particular financial models. Since the huge realm of applications of the Lie approach, we are sadly not able to treat all possible applications even restricting ourselves to the financial applications. In particular we have chosen to focus our attention on linear PDE’s problems with one single dependant variable. Particular attention have been given to local symmetries with emphasis on point transformations, i.e. symmetries defined by infinitesimal transformations whose infinitesimals depend on independent variable, dependent variable and its derivatives. We would like to underline that one of the main reason for the huge success encountered by Lie theory during recent years, relies on the fact that it provides a unified approach to treat basically. any kind of PDE. It follows that in this thesis we are able to merely scratch the surface of a wider theory. In fact symmetries more general than point symmetries, e.g. contact symmetries, Lie- Backlund symmetries, etc., have been developed. Further non-local symmetries i.e. symmetries whose infinitesimals at any point x depend on the global behaviour of u(x), are largely used and perhaps they are the area where Lie groups are actually an improvement with respect to standard approaches to PDE’s. In addition system of PDE’s depending on more than just one variable, can be successfully studied and non-linear differential equations are the perfect application of such a method. Outline of thesis Actually mainly three (slightly) different approaches to the Lie groups theory do exist which differ each one from another by emphasising a different characteristic of the theory. In this work we aim at both unifying different notations coming from such approaches and show how Lie theory of group invariants can be applied for widely used financial models described by stochastic differential equations. TThe first approach is due to Ovsiannikov, see Ovsiannikov [Ovs82]. It gives a rigorous math- ematical foundation of the theory wich is based on both representation theory and transformation groups analysis. During recent years a former student of him, Ibragimov, improved Ovsiannikov v work in a series of works, see e.g. Ibragimov [Ibr09], Ibragimov et al. [Ibr+94; Ibr+95], Ibragi- mov et al. [Ibr+96], and Ibragimov [Ibr94], related to both the mathematical theory of continuous transformations and some of their application to mathematical physics. Independetly and simultaneously to Ovsiannikov, Bluman studied Lie’s theory in the USA. His original work, see Bluman and Cole [BC74], (together with the one made by Ovsiannikov,see Ovsiannikov [Ovs82] ) has been the first that brought back methods associated to Lie groups theory to the attention of the mathematical community. Works by Bluman, see e.g. Bluman and Kumei [BK89], Bluman, Cheviakov, and Anco [BCA10], and Bluman and Anco [BA02], are focused on similarity reduction providing a highly efficient approach to find invariant solutions and mapping of a wide set of differential equations This thesis follows anyway a different approach mainly based on a geometric point of view, with a emphasis on differential geometry techniques. This approach can be found in the works of Olver, particularly in his book Olver [Olv93]. A part from the intuitive approach and examples helping understand properly the theory, this book presents one of the best section exploiting the algorithm for finding admitted symmetries. A good presentation of the topic can be found on Olver’s home page http://www.math.umn.edu/~olver/sm.html. Furthermore it has to be said that, as it was easily understandable, such a theory has an algebraic foundation. The topic is broad and entire books are devoted to it. We do not treat any of such a theory in this thesis since it is beyond our purposes. For the interested reader we refer anyway to Hall [Hal03] and Varadarajan [Var84]. Chapter§1 is devoted to the theory developed by Lie. At first (almost) all preliminary notions fundamental to the understanding of the method are stated. In section 1.1 we will introduce the fundamental tools such as Lie groups, flows, vector fields, infinitesimal generator and Lie algebras.A complementary part has been introduced in appendix A.1.5. The appendix is not necessary to the understanding of the theory but for an interested reader it better explains the mathematical foundation of the method. Section 1.2 treat the algorithm for finding admitted symmetries. It is the main reason why such a method has seen such a huge development in the past years. The notion of prolongation, Jet-space and total derivative will be introduced. Then the algorithm itself is exploited. Eventually the main results to find any admitted symmetric group of a given PDE are stated. Section 1.3 will focus on the concept of invariant (or self-similar) solutions. It is widely explained how to construct invariant solutions for a PDE. The notions of invariant surface, invariant curve and invariant point will introduced. Section 1.4 treat how to construct a mapping from a given DE to a target DE. In particular such a method it is largely used in finance. For instance the well known Black&Scholes equation can be mapped into the heat equation.
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