A new foundation for the mean–variance analysis

In this paper, I re-examine how the mean–variance analysis is consistent with its traditional theoretical foundations, namely, stochastic dominance and the expected utility theory. Then I propose a simplified version of the coarse utility theory as a new foundation. I prove that, by assuming risk aversion and the normality of asset variables, the simplified model is well behaved; indifference curves are convex and the opportunity set is concave. Therefore, there exist global optimal portfolios in the market. Finally, I prove that decision-making in accordance with the simplified model is consistent with the mean–variance analysis.     

On the mean convergence of Fourier–Jacobi series

The convergence of Fourier–Jacobi series in the spaces L _p,A,B is studied in the case where the Lebesgue constants are unbounded.     

Geometric representation of the mean–variance–skewness portfolio frontier based upon the shortage function

The literature suggests that investors prefer portfolios based on mean, variance and skewness rather than portfolios based on mean–variance (MV) criteria solely. Furthermore, a small variety of methods have been proposed to determine mean–variance–skewness (MVS) optimal portfolios. Recently, the shortage function has been introduced as a measure of efficiency, allowing to characterize MVS optimal portfolios using non-parametric mathematical programming tools. While tracing the MV portfolio frontier has become trivial, the geometric representation of the MVS frontier is an open challenge. A hitherto unnoticed advantage of the shortage function is that it allows to geometrically represent the MVS portfolio frontier. The purpose of this contribution is to systematically develop geometric representations of the MVS portfolio frontier using the shortage function and related approaches.     

Robust empirical optimization is almost the same as mean–variance optimization

We formulate a distributionally robust optimization problem where the deviation of the alternative distribution is controlled by a $?$-divergence penalty in the objective, and show that a large class of these problems are essentially equivalent to a mean–variance problem. We also show that while a “small amount of robustness” always reduces the in-sample expected reward, the reduction in the variance, which is a measure of sensitivity to model misspecification, is an order of magnitude larger.     

Optimal mean–variance selling strategies

Assuming that the stock price X follows a geometric Brownian motion with drift \(\mu \in \mathbb {R}\) and volatility \(\sigma >0\), and letting \(\mathsf {P}_{\!x}\) denote a probability measure under which X starts at \(x>0\), we study the dynamic version of the nonlinear mean–variance optimal stopping problem

$$\begin{aligned} \sup _\tau \Big [ \mathsf {E}\,\!_{X_t}(X_\tau ) - c\, \mathsf {V}ar\,\!_{\!X_t}(X_\tau ) \Big ] \end{aligned}$$

where t runs from 0 onwards, the supremum is taken over stopping times \(\tau \) of X, and \(c>0\) is a given and fixed constant. Using direct martingale arguments we first show that when \(\mu \le 0\) it is optimal to stop at once and when \(\mu \ge \sigma ^2\!/2\) it is optimal not to stop at all. By employing the method of Lagrange multipliers we then show that the nonlinear problem for \(0 < \mu < \sigma ^2\!/2\) can be reduced to a family of linear problems. Solving the latter using a free-boundary approach we find that the optimal stopping time is given by

$$\begin{aligned} \tau _* = \inf \,\! \left\{ \, t \ge 0\; \vert \; X_t \ge \tfrac{\gamma }{c(1-\gamma )}\, \right\} \end{aligned}$$

where \(\gamma = \mu /(\sigma ^2\!/2)\). The dynamic formulation of the problem and the method of solution are applied to the constrained problems of maximising/minimising the mean/variance subject to the upper/lower bound on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself.

     

Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin

We consider the problem of minimizing the probability of ruin by purchasing reinsurance whose premium is computed according to the mean–variance premium principle, a combination of the expected-value and variance premium principles. We derive closed-form expressions of the optimal reinsurance strategy and the corresponding minimum probability of ruin under the diffusion approximation of the classical Cramér–Lundberg risk process perturbed by a diffusion. We find an explicit expression for the reinsurance strategy that maximizes the adjustment coefficient for the classical risk process perturbed by a diffusion. Also, for this risk process, we use stochastic Perron’s method to prove that the minimum probability of ruin is the unique viscosity solution of its Hamilton–Jacobi–Bellman equation with appropriate boundary conditions. Finally, we prove that, under an appropriate scaling of the classical risk process, the minimum probability of ruin converges to the minimum probability of ruin under the diffusion approximation.     

Constant mean curvature tori as stationary solutions to the Davey–Stewartson equation

A well known result of Da Rios and Levi-Civita says that a closed planar curve is elastic if and only if it is stationary under the localized induction (or smoke ring) equation, where stationary means that the evolution under the localized induction equation is by rigid motions. We prove an analogous result for surfaces: an immersion of a torus into the conformal 3-sphere has constant mean curvature with respect to a space form subgeometry if and only if it is stationary under the Davey–Stewartson flow.     

The Radon–Kipriyanov transform of the generalized spherical mean of a function

A formula relating the Radon transform of functions of spherical symmetries to the Radon–Kipriyanov transform K_γ for a naturalmulti-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the K_γ-transform of a radial function as fractional integrals of Erdelyi–Kober integral type and of Riemann–Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon–Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables.     

The influence of perceived stock value price histories in the mean–variance-instability model

The model introduced by H. Talpaz, A. Harpaz and J.B. Penson (1984. European Journal of Operational Research 14, 262–269) extends the mean–variance model introducing the concept of instability. In this way it is possible to see an investor's attitude towards predicted instability. In this paper we show how optimisation procedures based on penalty (or preferred) weighted instability matrices can be interpreted in terms of real time utility functions which depend on an `actual' and a `remembered' time series due to fading memory. This approach justifies some bounded normalised functions used to represent the investors preference between the irregular frequency fluctuations.     

Michael–Simon inequalities for k-th mean curvatures

This paper continues the study of Alexandrov–Fenchel inequalities for quermassintegrals for \(k\) -convex domains. It focuses on the application to the Michael–Simon type inequalities for \(k\) -curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a domain.     

Hyperelliptic integrals and generalized arithmetic–geometric mean

We show how certain determinants of hyperelliptic periods can be computed using a generalized arithmetic-geometric mean iteration, whose initialisation parameters depend only on the position of the ramification points. Special attention is paid to the explicit form of this dependence and the signs occurring in the real domain.     

Global bifurcation of solutions of the mean curvature spacelike equation in certain Friedmann–Lemaître–Robertson–Walker spacetimes

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied.     

Stochastic goal programming: A mean–variance approach

We propose a stochastic goal programming (GP) model leading to a structure of mean–variance minimisation. The solution to the stochastic problem is obtained from a linkage between the standard expected utility theory and a strictly linear, weighted GP model under uncertainty. The approach essentially consists in specifying the expected utility equation corresponding to every goal. Arrow's absolute risk aversion coefficients play their role in the calculation process. Once the model is defined and justified, an illustrative example is developed.     

The Hasegawa–Petz mean: Properties and inequalities

We study a family of means introduced by H. Hasegawa and D. Petz (1996) [13]. Properties with respect to the parameter, such as monotonicity and logarithmic concavity, further, monotonicity and concavity in the mean variables are shown. Besides, the comparison between the Hasegawa–Petz mean and the geometric mean is completely solved. The connection to earlier results on operator monotonicity and some applications are also discussed.     

Optimal multi-period mean–variance policy under no-shorting constraint

We consider in this paper the mean–variance formulation in multi-period portfolio selection under no-shorting constraint. Recognizing the structure of a piecewise quadratic value function, we prove that the optimal portfolio policy is piecewise linear with respect to the current wealth level, and derive the semi-analytical expression of the piecewise quadratic value function. One prominent feature of our findings is the identification of a deterministic time-varying threshold for the wealth process and its implications for market settings. We also generalize our results in the mean–variance formulation to utility maximization with no-shorting constraint.