Guy Merlin MBAKOP

OPTIMAL DYNAMIC PORTFOLIOS UNDER A TAIL CONDITIONAL EXPECTATION CONSTRAINT

Daniel Akume, G. M. Mbakop

Abstract. We consider a portfolio problem when a tail conditional ex-pectation constraint is imposed. The financial market is composed of n risky assets driven by geometric Brownian motion and one risk-free asset. The tail conditional expectation is derived, re-calculated at short intervals of time and imposed continuously. The method of Lagrange multipliers is combined with the Hamilton-Jacobi-Bellman equation to insert the con-straint into the resolution framework. A numerical method is applied to obtain an approximate solution to the problem. We find that the imposi-tion of the tail conditional expectation constraint when risky assets evolve following a log-normal distribution, curbs investment in the risky assets and increases consumption. 1.

• #stochastic processes and financial applications
• #risk and portfolio optimization
• #financial markets and investment strategies
• #constraint (computer-aided design)

Polar varieties and efficient real elimination

B. Bank, M Giusti, Joos Heintz, G. M. Mbakop

Mathematische Zeitschrift

• #polynomial and algebraic computation
• #algebraic geometry and number theory
• #tensor decomposition and applications
• #mathematics

Equations for Polar Varieties and Efficient Real Elimination

Bernd Bank, M Giusti, Joos Heintz, G. M. Mbakop

edoc Publication server (Humboldt University of Berlin)

Let $V_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1,..., f_p$. This paper is devoted to the algorithmic problem of finding efficiently for each connected component of $V_0$ a representative point. For this purpose we exhibit explicit polynomial equations which describe for generic variables the polar varieties of $V_0$ of all dimensions. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations $f_1,...,f_p$ and in a suitably introduced geometric (extrinsic) parameter, called the degree of the real interpretation of the given equation system $f_1,...,f_p$.

• #advanced control systems optimization
• #polar
• #mathematics
• #computer science

Polar Varieties and Efficient Real Elimination

B. Bank, M Giusti, Joos Heintz, G. M. Mbakop

arXiv (Cornell University)

Let $S_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1, ..., f_p$. This paper is devoted to the algorithmic problem of finding {\em efficiently} a representative point for each connected component of $S_0$ . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of $S_0$. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations $f_1, >..., f_p$ and in a suitably introduced, intrinsic geometric parameter, called the {\em degree} of the real interpretation of the given equation system $f_1, >..., f_p$.

• #polynomial and algebraic computation
• #algebraic geometry and number theory
• #commutative algebra and its applications
• #variety (cybernetics)

Effiziente Lösung reeller Polynomialer Gleichungssysteme

G. M. Mbakop

edoc Publication server (Humboldt University of Berlin)

Diese Arbeit beinhaltet {\it geometrische Algorithmen} zur L\"osung reeller polynomialer Gleichungssysteme mit rationalen Koeffizienten, wobei die Polynome eine reduzierte regul\"are Folge bilden (vgl. Abschnitt \ref{abschgeo}). Unter reellem L\"osen verstehen wir hier die Bestimmung eines Punktes in jeder Zusammenhangskomponente einer kompakten glatten reellen Variet\"at $V:=W \cap \R^n$.\\ Im Mittelpunkt steht die Anwendung des f\"ur den algebraisch abgeschlossenen Fall ver\"offentlichten symbolischen geometrischen Algorithmus nach \cite{gh2} und \cite{gh3}. Die Berechenungsmodelle sind {\em Straight--Line Programme} und {arithmetische Netzwerke} mit Parametern in $\; \Q$. Die Input--Polynome sind durch ein Straight--Line Programm der Gr\"o{\ss}e $L$ gegeben. Eine geometrische L\"osung des Input--Glei\-chungs\-sys\-tems besteht aus einem primitiven Element der Ringerweiterung, welche durch die Nullstellen des Systems beschrieben ist, aus einem mininalen Polynom dieses primitiven Elements, und aus den Parametrisierungen der Koordinaten. Diese Darstellung der L\"osung hat eine lange Geschichte und geht mindestens auf Leopold Kronecker \cite{kron} zur\"uck. Die Komplexit\"at des in dieser Arbeit begr\"undeten Algorithmus erweist sich als linear in $L$ und polynomial bez\"uglich $n, d, \delta$ bzw. $\delta \;'$, wobei $n$ die Anzahl der Variablen und $d$ eine Gradschranke der Polynome im System ist. Die Gr\"o{\ss}en $\delta$ und $\delta \; '$ sind geometrische Invarianten, die das Maximum der {\em Grade des Inputsystems} und geeigneter {\em polarer Variet\"aten} repr\"asentieren (bzgl. des ({\em geometrischen}) Grades vgl. \cite{he}). Die Anwendung eines Algorithmus \"uber den komplexen Zahlen auf das L\"osen von polynomialen Gleichungen im Reellen wird durch die Einf\"urung polarer Variet\"aten m\"oglich (vgl. \cite{bank}). Die polaren Variet\"aten sind das Kernst\"uck und das vorbereitende Werzeug zur effizienten Nutzung des oben erw\"ahnten geometrischen Algorithmus. Es wird ein inkrementelles Verfahren zur Auffindung reeller Punkte in jeder Zusammenhangskomponente der Nullstellenmenge des Inputsystems abgeleitet, welches einen beschr\"ankten glatten (lokalen) vollst\"andigen Durchschnitt in $\R^n$ beschreibt. Das Inkrement des Algorithmus ist die Kodimension der polaren Variet\"aten. Die Haupts\"atze sind Satz $\ref{theorem12}$ auf Seite $\pageref{theorem12}$ f\"ur den Hyperfl\"achenfall, und Satz $\ref{theoresult}$ auf Seite $\pageref{theoresult}$, sowie die Aussage in der Einf\"uhrung dieser Arbeit, Seite $\pageref{vollres}$ f\"ur den vollst\"andigen Durchschnitt.

• #mathematical control systems and analysis
• #humanities
• #philosophy

Polar Varieties, Real Equation Solving, and Data Structures: The Hypersurface Case

B. Bank, M Giusti, Joos Heintz, G. M. Mbakop

Journal of Complexity

• #polynomial and algebraic computation
• #commutative algebra and its applications
• #algebraic geometry and number theory
• #hypersurface

Polar Varieties and Efficient Real Equation Solving

Bernd Bank, M Giusti, Joos Heintz, Reinhard Mandel, G. M. Mbakop

edoc Publication server (Humboldt University of Berlin)

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in yields a method for symbolically solving a zero-dimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined affine degree of the equation system. Replacing the affine degree of the equation system by a suitably defined real degree of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straight-line program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.

• #matrix theory and algorithms
• #distributed and parallel computing systems
• #polar
• #mathematics

Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

B. Bank, M Giusti, Joos Heintz, Rainer Mandel, G. M. Mbakop

arXiv (Cornell University)

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for symbolically solving a zero-dimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined {\em affine degree} of the equation system. Replacing the affine degree of the equation system by a suitably defined {\em real degree} of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straight-line program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.

• #polynomial and algebraic computation
• #commutative algebra and its applications
• #algebraic geometry and number theory
• #hypersurface