Ljudmila A. Bordag

Zittau/Görlitz University of Applied Sciences | HSZG · Fakultät für Mathematik/Naturwissenschaften

Management of the portfolios containing low liquidity assets is a tedious problem. The buyer proposes the price that can differ greatly from the paper value estimated by the seller, the seller, on the other hand, can not liquidate his portfolio instantly and waits for a more favorable offer. To minimize losses in this case we need to develop new me...

We study an optimization problem for a portfolio with a risk-free, a liquid , and an illiquid risky asset. The illiquid risky asset is sold in an exogenous random moment with a prescribed liquidation time distribution. We assume that the investor chooses an exponential utility function. Study of optimization problems with three assets including an...

We study an optimization problem for a portfolio with a risk-free, a liquid risky, and an illiquid asset which is sold in an exogenous random moment of time with a prescribed liquidation time distribution. Problems of such type lead to three dimensional nonlinear partial differential equations (PDEs) on the value function. We study the optimization...

In this chapter we discuss the problem of financial illiquidity and give an overview of different modeling approaches to this problem. We focus on one of the approaches to an optimization problem of a portfolio with an illiquid asset sold in an exogenous random moment of time. We formulate the problem in a mathematically rigorous way and apply it t...

Management of a portfolio that includes an illiquid asset is an important problem of modern mathematical finance. One of the ways to model illiquidity among others is to build an optimization problem and assume that one of the assets in a portfolio can not be sold until a certain finite, infinite or random moment of time. This approach arises a cer...

During financial crises investors manage portfolios with low liquidity, where the paper value of an asset differs from the price proposed by the buyer. We consider an optimization problem for a portfolio with an illiquid, a risky and a riskfree asset. We work in the Merton's optimal consumption framework with continuous time. The liquid part of the...

Families of exact solutions are found to a nonlinear modification of the Black–Scholes equation. This risk-adjusted pricing methodology (RAPM) model incorporates both transaction costs and 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...

We study the general model of self-financing trading strategies in illiquid markets introduced by Schoenbucher and Wilmott, 2000. A hedging strategy in the framework of this model satisfies a nonlinear partial differential equation (PDE) which contains some function g(alpha). This function is deep connected to an utility function. We describe the L...

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 dif...

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 dif...

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...

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.

This chapter 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...

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 or...

Several models for the pricing of derivative securities in illiquid markets are discussed. A typical type of nonlinear partial differential equations arising from these investigation is studied. The scaling properties of these equations are discussed. Explicit solutions for one of the models are obtained and studied.

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 equa...

The studied model was suggested to design a perfect hedging strategy for a large trader. In this case the implementation of a 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 the Lie group theory we reduce the partial differentia...

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 equa...

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 th...

We consider a two-dimensional steady motion of an inviscid incompressible fluid described by the equation Du(x,y)=F(u(x,y)),where u(x,y) is the streamfunction, D is the Laplace operator, and F(·) is an arbitrary function measuring the flow vorticity. Apparently, until now, the only way to treat an equation of the above type with nontrivial function...

The equivalence classes for Emden equations were investigated. The radial solutions of the Emden equation, satisfying ordinary differential equations, was found to have a significant role in the diffusion process. The equivalence of the Emden equation to the first and second Painleve equations was also examined.

This paper is devoted to the investigation of Emden-Fowler equations of the type y '' (x)+e(x)y p (x)=0,x∈ℝ· The motivation to study such equations systematically came from astro-, atomic physics and Riemannian geometry. The author studies the properties of Emden-Fowler equations which are invariant under general point transformations. The underlyi...

The necessary and sufficient conditions that an equation of the form y″=f(x, y, y′) can be reduced to one of the Painlevé equations under a general point transformation are obtained. A procedure to check these conditions is found. The theory of invariants plays a leading role in this investigation. The reduction of all six Painlevé equations to the...

Real three-phase solutions of the sine-Laplace equation are constructed. All smooth and singular real doubly periodic solutions are found. The corresponding three-dimensional theta functions are reduced to the elliptic Jacobi functions. Some classes of solutions with symmetries giving possibilities for physical applications are determined. Bibliogr...

New Lax pairs for the chiral O(3)-field equations and for the Landau-Lifshitz equation have been found. In contrast to the already-known pairs, these pairs are polynomial in the spectral parameter lambda . We also find a new 4*4 Lax pair in the case of cnoidal waves for the generalized Landau-Lifshitz equation.

N-phase solutions of the Kadomtsev-Petviashvili (KP) equation (1970), that are periodic in space variables x and y, were obtained and effectively investigated using the Schottky uniformisation, of which a short description is given. Many wave patterns are represented graphically as contour plots and as isometric projections for different parameter...

We investigate new polynomial hierarchies of Lax pairs which contain the polynomial pairs for the system of O(3) chiral field equations and Landau - Lifshitz equation introduced recently and give an algorithmic construction of the corresponding hierarchies of soliton equations. We compare the Landau - Lifshitz equation hierarchy obtained via a poly...

The investigation of nonlinear dynamical systems of the type $\dot{x}=P(x,y,z),\dot{y}=Q(x,y,z),\dot{z}=R(x,y,z)$ by means of reduction to some ordinary differential equations of the second order in the form $y''+a_1(x,y)y'^3+3a_2(x,y)y'^2+3a_3(x,y)y'+a_4(x,y)=0$ is done. The main backbone of this investigation was provided by the theory of invaria...

By the automorphic approach and the use of computer graphics, we carry out a qualitative analysis of finite-gap periodic solutions of the KgΦ equation. Graphs are constructed for multiphase solutions.

The connection between Painleve equations P2 and P34 is concidered and the Baecklund transformation (BT) for P34 which helps us to find interrelation between general solution of the P2 equation and the Toda and Volterra lattice equations are obtained. We present solutions of these lattice equations expressed through Airy functions. For degenerated...

One considers the one-dimensional Dirac operator with a slowly oscillating potential$$H = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right)\frac{d}{{dx}} + q\left( {\begin{array}{*{20}c} {\cos z(x)} & {\sin z(x)} \\ {\sin z(x)} & { - \cos z(x)} \\ \end{array} } \right)_, x \in ( - \infty ,\infty ),q - const,$$ (1) where . T...

Explicit analytic formulae for two-dimensional solitons are given. It is proved that, unlike one-dimensional solitons, two-dimensional ones do not interact at all.