We consider a two-dimensional steady motion of an inviscid incompressible fluid described by the equation &UDelta; u(x,y) = F(u(x,y)), where u(x,y) is the streamfunction, &UDelta; is the Laplace operator, and F((.)) an arbitrary function measuring the flow vorticity. Apparently, until now, the only way to treat an equation of the above type with nontrivial function F analytically is to use the algebro-geometric method for integrable equations. In particular, we investigate the Cosh-Laplace equation (ChL) &UDelta; u(x,y) = ± 4cosh(u(x,y)) by means of the special technique of finite-gap integration, which allows us to obtain real solutions of the ChL equation by using a Riemann surface with appropriate symmetry. We study the first nontrivial case corresponding to a Riemann surface of genus g = 3. The hydrodynamical interpretation of finite-gap solutions is meaningful, and we try to understand the fluid processes described by these solutions. To this end, we take a Riemann surface with additional symmetry properties. We present four five-parameter families of exact solutions. These solutions are given in terms of Jacobi elliptic functions, which enables us to directly investigate the relevant properties. We also find explicit formulas for the lines of singularity. It is of interest from the point of view of algebraic geometry that the structure of the theta divisor can be described.

The present model describes a perfect hedging strategy for a large trader. In this case the hedging strategy affects the price of the underlying security. The feedback-effect leads to a nonlinear version of the Black-Scholes partial differential equation. Using Lie group theory we reduce in special cases the partial differential equation to some ordinary differential equations. The Lie group found for the model equation gives rise to invariant solutions. Families of exact invariant solutions for special values of parameters are described. © 2008 Springer-Verlag Berlin Heidelberg.

We study a class of nonlinear pricing models which involves the feedback effect from the dynamic hedging strategies on the price of asset introduced by  Sircar and Papanicolaou. We are first to study the case of a nonlinear demand function involved in the model. Using a Lie group analysis we investigate the symmetry properties of these nonlinear diffusion equations. We provide the optimal systems of subalgebras and the complete set of non-equivalent reductions of studied PDEs to ODEs. In most cases we obtain families of exact solutions or derive particular solutions to the equations.

Families of exact solutions are found to a nonlinear modification of the Black-Scholes equation. This risk-adjusted pricing methodology model (RAPM) incorporates both transaction costs and the risk from a volatile portfolio. Using the Lie group analysis we obtain the Lie algebra admitted by the RAPM equation. It gives us the possibility to describe an optimal system of subalgebras and the corresponding set of invariant solutions to the model. In this way we can describe the complete set of possible reductions of the nonlinear RAPM model. Reductions are given in the form of different second order ordinary differential equations. In all cases we provide exact solutions to these equations in an explicit or parametric form. Each of these solutions contains a reasonable set of parameters which allows one to approximate a wide class of boundary conditions. We discuss the properties of these reductions and the corresponding invariant solutions.

The symmetry properties of nonlinear diffusion equations are studied using a Lie group analysis. Reductions and families of exact solutions are found for some of these equations.

© 2009 World Scientific Publishing Company.

We analyze the stability of a Taylor-Couette flow under the imposition of a weak axial flow in the case of a very short cylinder with a narrow annulus gap. We consider an incompressible viscous fluid contained in the narrow gap between two concentric short cylinders, in which the inner cylinder rotates with constant angular velocity. The caps of the cylinders have narrow tubes conically tapering to very narrow slits, allowing an axial flow along the surface of the inner cylinder. The approximated solution for the Taylor-Couette flow for short cylinders was found and used for the stability analysis instead of the precise but bulky solution. The sensitivity of the Taylor-Couette flow to small perturbations and to weak axial flow was studied. We demonstrate that perturbations coming from the axial flow cause the propagation of dispersive waves in the Taylor-Couette flow. While in long cylinders the presence of an axial flow leads to the breaking of axial symmetry, in small cylinders it leads to the breaking of mirror symmetry. The coexistence of a rotation and an axial flow requires that, in addition to the energy and the angular momentum of the flow, the helicity must also be studied. The approximated form for the helicity formula in the case of short cylinders was derived. We found that the axial flow stabilizes the Taylor-Couette flow. The supercritical flow includes a rich variety of vortical structures, including a symmetric pair of Taylor vortices, an anomalous single vortex and quasiperiodic oscillating vortices. Pattern formation was studied at large for rated ranges of azimuthal and axial Reynolds numbers. A region where three branches of different states occur was localized. Numerical simulations in 3-D and in the axisymmetrical case of the model flow are presented, which illustrate the instabilities analyzed.

Families of explicit solutions are found to a nonlinear Black-Scholes equation which incorporates the feedback-effect of a large trader in case of market illiquidity. The typical solution of these families will have a payoff which approximates a strangle. These solutions were used to test numerical schemes for solving a nonlinear Black-Scholes equation. © World Scientific Publishing Company.

The present paper is concerned with nonlinear Black Scholes equations arising in certain option pricing models with a large trader and/or transaction costs. In the first part we give an overview of existing option pricing models with frictions. While the financial setup differs between models, it turns out that in many of these models derivative prices can be characterized by fully nonlinear versions of the standard parabolic Black-ScholesPDE. In the second part of the paper we study a typical nonlinear Black-Scholes equation using methods from Lie group analysis. The equation possesses a rich symmetry group. By introducing invariant variables,  invariant solutions can therefore be characterized in terms of solutions to ordinary differential equations. Finally we discuss properties and applications of these solutions.

We study the general model of self-financing trading strategies inilliquid markets introduced by Schoenbucher and Wilmott, 2000.A hedging strategy in the framework of this model satisfies anonlinear partial differential equation (PDE) which contains somefunction g(alpha). This function is deep connected to anutility function.

We describe the Lie symmetry algebra of this PDE and provide acomplete set of reductions of the PDE to ordinary differentialequations (ODEs). In addition we are able to describe all types offunctions g(alpha) for which the PDE admits an extended Liegroup. Two of three special type functions lead to modelsintroduced before by different authors, one is new. We clarify theconnection between these three special models and the generalmodel for trading strategies in illiquid markets. We study withthe Lie group analysis the new special case of the PDE describingthe self-financing strategies. In both, the general model and thenew special model, we provide the optimal systems of subalgebrasand study the complete set of reductions of the PDEs to differentODEs. In all cases we are able to provide explicit solutions tothe new special model. In one of the cases the solutions describepower derivative products.