Erik Taflin

A theory of bond portfolios

We introduce a bond portfolio management theory based on foundations similar to those of stock portfolio management. A general continuous-time zero-coupon market is considered. The problem of optimal portfolios of zero-coupon bonds is solved for general utility functions, under a condition of no-arbitrage in the zero-coupon market. A mutual fund theorem is proved, in the case of deterministic volatilities. Explicit expressions are given for the optimal solutions for several utility functions.

Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations

Introduction The nonlinear representation

The cauchy problem for non-linear Klein-Gordon equations

T

On global solutions of the Maxwell-Dirac equations

and spaces of differentiable vectors The asymptotic nonlinear representation Construction of the approximate solution Energy estimates and

Wave operators and analytic solutions for systems of nonlinear Klein-Gordon equations and of nonlinear Schrödinger equations

L^2-L^\infty

In which financial markets do mutual fund theorems hold true

estimates for the Dirac field Construction of the modified wave operator and its inverse Appendix.

Bond market completeness and attainable contingent claims

We consider in ℝn+1,n≧2, the non-linear Klein-Gordon equation. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincaré covariant then the non-linear representation of the Poincaré Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincaré group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincaré group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra.

No-arbitrage of second kind in countable markets with proportional transaction costs

We prove, for the Maxwell-Dirac equations in 1+3 dimensions, that modified wave operators exist on a domain of small entire test functions of exponential type and that the Cauchy problem, inR+×R3, has a unique solution for each initial condition (att=0) which is in the image of the wave operator. The modification of the wave operator, which eliminates infrared divergences, is given by approximate solutions of the Hamilton-Jacobi equation, for a relativistic electron in an electromagnetic potential. The modified wave operator linearizes the Maxwell-Dirac equations to their linear part.

Formal linearization of nonlinear massive representations of the connected Poincaré group

We consider, in a 1+3 space time, arbitrary (finite) systems of nonlinear Klein-Gordon equations (respectively Schrödinger equations) with an arbitrary local and analytic non-linearity in the unknown and its first and second order space-time (respectively first order space) derivatives, having no constant or linear terms. No restriction is given on the frequency sign of the initial data. In the case of non-linear Klein-Gordon equations all masses are supposed to be different from zero.We prove, for such systems, that the wave operator (fromt=∞ tot=0) exists on a domain of small entire test functions of exponential type and that the analytic Cauchy problem, in ℝ+×ℝ3, has a unique solution for each initial condition (att=0) being in the image of the wave operator. The decay properties of such solutions are discussed in detail.

Generalized integrands and bond portfolios: Pitfalls and counter examples.

The mutual fund theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are: (i)Let N be the wealth process of the numéraire portfolio (i.e., the optimal portfolio for the log utility). If any path-independent option with maturity T written on the numéraire portfolio can be replicated by trading only in N and the risk-free asset, then the MFT holds true for general utility functions, and the numéraire portfolio may serve as mutual fund. This generalizes Merton’s classical result on Black–Merton–Scholes markets as well as the work of Chamberlain in the framework of Brownian filtrations (Chamberlain in Econometrica 56:1283–1300, 1988).Conversely, under a supplementary weak completeness assumption, we show that the validity of the MFT for general utility functions implies the replicability property for options on the numéraire portfolio described above.(ii)If for a given class of utility functions (i.e., investors) the MFT holds true in all complete Brownian financial markets S, then all investors use the same utility function U, which must be of HARA type. This is a result in the spirit of the classical work by Cass and Stiglitz.