Ralf Wunderlich

Dynamic Portfolio Optimization with Bounded Shortfall Risks

Abstract We address a dynamic portfolio optimization problem where the expected utility from terminal wealth has to be maximized. The special feature of this paper is an additional constraint on the portfolio strategy modeling bounded shortfall risks, which are measured by value at risk or expected loss. Using a continuous-time model of a complete financial market and applying martingale methods, analytic expressions for the optimal terminal wealth and the optimal portfolio strategies are given. Finally, some numerical results are presented.

PORTFOLIO OPTIMIZATION UNDER PARTIAL INFORMATION WITH EXPERT OPINIONS

This paper investigates optimal portfolio strategies in a market with partial information on the drift. The drift is modelled as a function of a continuous-time Markov chain with finitely many states which is not directly observable. Information on the drift is obtained from the observation of stock prices. Moreover, expert opinions in the form of signals at random discrete time points are included in the analysis. We derive the filtering equation for the return process and incorporate the filter into the state variables of the optimization problem. This problem is studied with dynamic programming methods. In particular, we propose a policy improvement method to obtain computable approximations of the optimal strategy. Numerical results are presented at the end.

Random eigenvalue problems for bending vibrations of beams

The paper deals with the determination of statistical characteristics of eigenvalues for a class of ordinary differential operators with random coefficients. This problem arises from the computation of eigenfrequencies for the bending vibrations of beams possessing random geometry and material properties. Representations of eigenvalues are found by applying the Ritz method and perturbation results for matrix eigenvalue problems. Approximations of the probability density function and the moments of the random eigenvalues are given by means of expansions in powers of the correlation length of weakly correlated random functions which are used for modelling the random terms. The eigenvalue statistics determined analytically are compared favourably with Monte-Carlo simulations.

Utility Maximization Under Bounded Expected Loss

We consider optimal selection of portfolios for utility maximizing investors under joint budget and shortfall risk constraints. The shortfall risk is measured in terms of the expected loss. Depending on the parameters of the risk constraint, we show the existence of an optimal solution and uniqueness of the corresponding Lagrange multipliers. Using Malliavin calculus we also provide the optimal trading strategy.

Optimal portfolio policies under bounded expected loss and partial information

In a market with partial information we consider the optimal selection of portfolios for utility maximizing investors under joint budget and shortfall risk constraints. The shortfall risk is measured in terms of expected loss. Stock returns satisfy a stochastic differential equation. Under general conditions on the corresponding drift process we provide the optimal trading strategy using Malliavin calculus. We give extensive numerical results in the case that the drift is modeled as a continuous-time Markov chain with finitely many states. To deal with the problem of time-discretization when applying the results to market data, we propose a method to detect and correct possible tracking errors.

Optimal portfolio strategies benchmarking the stock market

The paper investigates the impact of adding a shortfall risk constraint to the problem of a portfolio manager who wishes to maximize his utility from the portfolios terminal wealth. Since portfolio managers are often evaluated relative to benchmarks which depend on the stock market we capture risk management considerations by allowing a prespecified risk of falling short such a benchmark. This risk is measured by the expected loss in utility. Using the Black–Scholes model of a complete financial market and applying martingale methods, explicit analytic expressions for the optimal terminal wealth and the optimal portfolio strategies are given. Numerical examples illustrate the analytic results.

Random transverse vibrations of a one-sided fixed beam and model reduction

This paper is devoted to the computation of second-order moment functions of the solution of large-scale systems of linear random ODEs. Such systems result from the semi-discretization of PDEs describing continuous vibration systems with random excitation. Model reduction techniques are applied to find approximations of the desired moment functions of the large-scale system by computing them for a suitable low-dimensional system. Numerical results concerning transverse vibrations of a one-sided fixed beam subjected to a random excitation modeled by a random field are presented. Special attention is paid to e-correlated excitations characterized by a short correlation length.

Partially observable stochastic optimal control problems for an energy storage

In this paper we study the valuation of an energy storage facility in the presence of stochastic energy prices as it arises in the case of a hydro-electric pump station. The valuation problem is related to the problem of determining the optimal charging/discharging strategy that maximizes the expected value of the resulting discounted cash flows over the lifetime of the storage. Adding to the literature we use a regime switching model for the energy price which allows for a changing economic environment described by a non-observable Markov chain. We formulate the problem as a stochastic control problem under partial information in continuous time. Applying filtering theory we find an alternative state process containing the filter of the Markov chain, which is adapted to the observable filtration. For this alternative control problem we derive the associated Hamilton-Jacobi-Bellman (HJB) equation which is not strictly elliptic. Therefore we study the HJB equation using regularization arguments. Finally, we solve the optimization problem numerically.

Remarks on randomly excited oscillators

In the paper linear oscillators with stochastic excitation are considered. Explicit formulae for second-order characteristics of random solution processes for input processes which are wide-sense stationary or possess wide-sense stationary increments are given. Further, asymptotic expansions in the case of weakly correlated excitation processes are derived.

Asymptotic Expansions of Integral Functionals of Weakly Correlated Random Processes

In the paper asymptotic expansions for second-order moments of integral functionals of a family of random processes are considered. The random processes are assumed to be wide-sense stationary and c-correlated, i.e. the values are not correlated excluding an c-neighbourhood of each point. The asymptotic expansions are derived for e 0. Using a special weak assumption there are found easier expansions as in the case of general weakly correlated random processes. Expansions are given for integral functionals of real-valued as well as of complex vector-valued processes.