Qiji Zhu

• PhD
• Professor (Full) at Western Michigan University

In this chapter, we carefully develop a general framework of portfolio theory involving a scalar risk. This will establish a foundation for our central result on a portfolio theory involving multiple risks (or a vector risk) in the next chapter.

The theory that we have developed in the previous chapters can be applied to several management problems, coming from corporations, individuals, or governments. The manager has to balance profit with more than one risk. We present a few situations where multiple risks arise.

As we are approaching the end of the description of this stage of our interdisciplinary collaboration, a look back may help the reader to obtain a more global view of the material provided in this monograph and furthermore have a glimpse of the road ahead.

In the last chapter, we presented a thorough outline of the general framework of portfolio theory for abstract, scalar-valued risk and utility functions.

We formulate the optimal balance sheet management problem as a linear program and study it using a duality approach. In addition to helping determine the optimal balance sheet, the dual problem also provides us the interest rate risk and credit risk pricing. We deploy our methodology to determine premia on credit risk and interest rate risk for com...

The Growth-Optimal Portfolio (GOP) theory determines the path of bet sizes that maximize long-term wealth. This multi-horizon goal makes it more appealing among practitioners than myopic approaches, like Markowitz’s mean-variance or risk parity. The GOP literature typically considers risk-neutral investors with an infinite investment horizon. In th...

This is Part III of a series of papers which focus on a general framework for portfolio theory. Here, we extend a general framework for portfolio theory in a one-period financial market as introduced in Part I [Maier-Paape and Zhu, Risks 2018, 6(2), 53] to multi-period markets. This extension is reasonable for applications. More importantly, we tak...

Banks make profits from the difference between short-term and long-term loan interest rates. To issue loans, banks raise funds from capital markets. Since the long-term loan rate is relatively stable, but short-term interest is usually variable, there is an interest rate risk. Therefore, banks need information about the optimal leverage strategies...

We turn to discuss continuous financial models. These models in general involve infinite dimensional spaces and are more complex. Our focus here is to use relatively simple models to illustrate the convex duality between the price of a contingent claim and the process of cash borrowed in delta hedging. This reveals the root of the convexity in cont...

We present a concise description of the convex duality theory in this chapter. The goal is to lay a foundation for later application in various financial problems rather than to be comprehensive. We emphasize the role of the subdifferential of the value function of a convex programming problem. It is both the set of Lagrange multiplier and the set...

We now expand our discussion to a multi-period economy with finite status. This setting models trading in the real world quite well, where we always only deal with finite number of transactions and finite number of possible scenarios. On the technical side, both payoffs and trading strategies are still belonging to finite dimensional vector spaces....

This chapter focuses on financial models in a one period economy with a finite sample space. Mathematically, these models involve only finite dimensional spaces yet they still illustrate the main patterns. In modeling the behavior of agents in a financial market, we usually use concave utility functions and convex risk measure to characterize their...

The aim of this paper is to provide several examples of convex risk measures necessary for the application of the general framework for portfolio theory of Maier-Paape and Zhu (2018), presented in Part I of this series. As an alternative to classical portfolio risk measures such as the standard deviation, we, in particular, construct risk measures...

Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two-dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of Markowitz (1959) and the...

Many investment firms and portfolio managers rely on backtests (ie, simulations of performance based on historical market data) to select investment strategies and allocate capital. Standard statistical techniques designed to prevent regression overfitting, such as hold-out, tend to be unreliable and inaccurate in the context of investment backtest...

We discuss Lagrange multiplier rules from a variational perspective. This allows us to highlight many of the issues involved and also to illustrate how broadly an abstract version can be applied.

In mathematical finance, backtest overfitting relates to the usage of historical market data (a backtest) to develop an investment strategy, where the strategy profits from random patterns rather than variables’ signals. Backtest overfitting is now thought to be a primary reason why quantitative investment models and strategies that look good on pa...

Growth Optimal Portfolio (GOP) theory determines the path of bet sizes that maximize long-term wealth. How it is also known in practice GOP is too risky. We explain in this talk that the reason is in practice the investment horizon is finite and practitioners account for risk more explicitly. We develop risk adjusted growth portfolio and discuss ho...

We carry out several test cases to illustrate how the Probability of Backtest Overfitting (PBO) performs under different scenarios. We also assess the accuracy of PBO using two alternative approaches (Monte Carlo Methods and Extreme Value Theory). The paper "The Probability of Backtest Overfitting" to which these Appendices apply is available at th...

The purpose of this paper is to survey and to provide a unified framework to connect a diverse group of results, currently scattered in the literature, that can be usefully viewed as consequences of applying variational methods to problems involving symmetry. Here, variational methods refer to mathematical treatment by way of constructing an approp...

Inflection points in the the function for determining estimated geometric gain with respect to capital committed emerge when planning an investment for a finite time horizon. These are points where marginal increases in reward with respect to risk peaks. We analyze the properties of inflection points for a mix of one risky asset and one risk-free a...

Most firms and portfolio managers rely on backtests (or historical simulations of performance) to select investment strategies and allocate them capital. Standard statistical techniques designed to prevent regression over-fitting, such as hold-out, tend to be unreliable and inaccurate in the context of investment backtests. We propose a framework t...

Growth Optimal Portfolio (GOP) theory determines the path of bet sizes that maximize long-term wealth. This multi-horizon goal makes it more appealing among practitioners than myopic approaches, like Markowitz's mean-variance or risk parity. The GOP literature typically considers risk-neutral investors with an infinite investment horizon. In this p...

We develop the theory of Kelly and Thorp in analyzing the optimal bet sizes for blackjack by incorporating the practical considerations of players wherein only a finite number of plays shall occur as well as pursuing maximizing risk-adjusted returns. We show that the ratio of return to bet size is approximately proportional to the return / drawdown...

We discuss here a dynamic implementation of the leverage space portfolio theory in the form of DJ-LSP position sizing index implemented the Dow Jones indexes. Our emphasis in this case study is on the practical issues related to the implementation.

Using the language of convex analysis, we describe key results in several important areas of finance: portfolio theory, financial derivative trading and pricing and consumption based asset pricing theory. We hope to emphasize the importance of convex analysis in financial mathematics and also draw the attention of researchers in convex analysis to...

We develop an optimal trend following trading rule in a bull-bear switching market, where the drift of the stock price switches between two parameters corresponding to an uptrend (bull market) and a downtrend (bear market) according to an unobservable Markov chain. We consider a finite horizon investment problem and aim to maximize the expected ret...

We study the term structure of interest rates in the presence of consumption commitments using an equilibrium model. Under rea-sonable assumptions we prove the existence and uniqueness of the equilibrium and develop computation methods. Examples are ana-lyzed to illustrate the effect of consumption commitments on the term structure and its manifest...

This paper is concerned with the optimality of a trend following trading rule. The idea is to catch a bull market at its early stage, ride the trend, and liquidate the position at the first evidence of the subsequent bear market. We characterize the bull and bear phases of the markets mathematically using the conditional probabilities of the bull m...

We show that vector majorization and its related preference sets can be used to establish useful option pricing bounds for a robust option replacement investment strategy. This robust trading strategy can help to overcome some of the difficulties in implementing arbitrage option trading strategies when there exists model inaccuracy.

Consider an investment system with a nonnegative expected return in a one period economy. We show that, for an option with a given strike price, there exists a pricing interval [p^c,p^w] such that replacing the original investment with the option will benefit judging by the Kelly criterion only when the price of the option lies...

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function. A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization proble...

Consider an investment system with a nonnegative expected return in a one period economy. We show that, for an option with a given strike price, there exists a pricing interval $[p^C, p^W]$ such that replacing the original investment with the option will benefit judging by the Kelly criterion only when the price of the option lies outside of the in...

We deduce control equations governed by ordinary differential equations that determine the moments of the inventory levels in a flow control model of a flexible manufacturing system. Then we discuss how to use these control equations to study the expected inventory level set and the optimal inventory level control problem with quadratic cost functi...

Consider an investment system with a nonnegative expected re-turn in a one period economy. We show that, for an option with a given strike price, there exists a pricing interval [p C , p W ] such that replacing the original investment with the option will benefit judging by the Kelly criterion only when the price of the option lies outside of the i...

We use variational methods to provide a concise development of a number of basic results in convex and functional analysis. This illuminates the parallels between convex analysis and smooth subdifferential theory.

Investment systems are studied using a framework that emphasize their profiles (the cumulative probability distribution on all the possible percentage gains of trades) and their log return functions (the expected average return per trade in logarithmic scale as a function of the investment size in terms of the percentage of the available capital)....

We give a generalization of the classical Helly's theorem on inter-section of convex sets in R N for the case of manifolds of nonpositive curvature. In particular, we show that if any N + 1 sets from a family of closed convex sets on N-dimensional Cartan-Hadamard manifold contain a common point, then all sets from this family contain a com-mon poin...

Multidirectional mean value inequalities provide estimates of the difference of the extremal value of a function on a given bounded set and its value at a given point in terms of its (sub)-gradient at some intermediate point. A generalization of such multidirectional mean value inequalities is derived by using new infinitesimal conditions for a wea...

This note provides a short variational proof of the Birkhoff's theorem asserting that the extreme points of the convex set of doubly stochastic matrices are the permutation matrices.

Nonsmooth analysis, differential analysis for functions without differentiability, has witnessed a rapid growth in the past several decades stimulated by intrinsic nonsmooth phenomena in control theory, optimization, mathematical economics and many other fields. In the past several y ears many problems in control theory, matrix analysis and geometr...

Nonsmooth analysis, differential analysis for functions without differentiability, has witnessed a rapid growth in the past several decades stimulated by intrinsic nonsmooth phenomena in control theory, optimization, mathematical economics and many other fields. In the past several years many problems in control theory, matrix analysis and geometry...

We derive a nonconvex separation theorem for multifunctions that generalizes an early result of Borwein and Jofré and show that this result is equivalent to several other subdifferential calculus results in smooth Banach spaces. Then we apply this nonconvex separation theorem to improve a second welfare theorem in economics and a necessary optimali...

 We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker type necessary optimality conditions under additional constraint qualifications including the c...

We show that lower semicontinuous Lyapunov functions can be used to determine both stable and attractive sets of differential equations with a short proof similar to that of the original Lyapunov indirect method. Several examples illustrate the flexibility of using such lower semicontinuous Lyapunov functions.

We develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization-related problems. The main difference bet...

We derive necessary optimality conditions for constrained minimization problems with lower semicontinuous inequality constraints, continuous equality constraints, and a set constraint in smooth Banach spaces. Then we apply these necessary optimality conditions to derive subdifferential characterizations of singular normal vectors to the epigraph an...

. This survey is an account of the current status of subdifferential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied analysis areas who might profitably use subdifferentials as tools. Key words. Viscosity subdifferential, proximal subdifferential, subdifferential calcul...

This survey is an account of the current status of subdifferential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied analysis areas who might profitably use subdifferentials as tools.

Warga's concept of an abundant set and the related controllability-extremality alternative theorem provides a powerful framework for deriving necessary optimality conditions for various optimal control problems. This method enables us to focus on the essential issue of proving the set of original controls is abundant in an appropriately constructed...

Evolutionary programs are capable of finding good solutions to difficult optimization problems. Previous analysis of their convergence properties has normally assumed the strategy parameters are kept constant, although in practice these parameters are dynamically altered. In this paper, we propose a modified version of the 1/5-success rule for self...

It was shown in Part I of this work that the Gateaux differentiability of a convex unitarily invariant function is characterized by that of a similar induced rearrangement invariant function on the corresponding spectral space. A natural question is then whether this is also the case for Fréchet differentibility. In this paper we show the answer is...

This paper concerns with generalized differentiation of set-valued and nonsmooth mappings between Banach spaces. We study the so-called viscosity coderivatives of multifunctions and their limiting behavior under certain geometric assumptions on spaces in question related to the existence of smooth bump functions of any kind. The main results includ...

For a flexible manufacturing system, determining the feasible production set at a given time is essential for production management. In this paper, we propose methods for estimating the production set for a general multi–part type failure prone flexible manufacturing system. We also discuss how to get an open loop control that can ensure the system...

Nonsmooth analysis had its origins in the early 1970s when control theorists and nonlinear programmers attempted to deal with necessary optimality conditions for problems with nonsmooth data or with nonsmooth functions that arise even in many problems with smooth data. Two simple examples that illustrate how such intrinsic nonsmoothness arises in p...

We consider functions on the space of compact self-adjoint Hilbert space operators. Specifically, we study those extended-real functions which depend only on the operators' spectral sequences. Examples include the norms of the Schatten p-spaces, the Calder'on norms, the k'th largest eigenvalue, and some infinite-dimensional self-concordant barriers...

. This paper discusses Hamiltonian necessary conditions for a nonsmooth multiobjective optimal control problem with endpoint constraints related to a general preference. The transversality condition in our necessary conditions is stated in terms of a normal cone to the level sets of the preference. Examples for a number of useful preferences are di...

These lectures center on the structure of real—valued Lipschitz functions, and their generalized derivatives on Banach spaces. We pay some attention to the role of measure and category and will try to illustrate a number of different techniques. These published notes are much more detailed and comprehensive than the lectures as given. Much of this...

We show that assuming all the summand functions to be lower semicontinuous is not sucient to ensure a (strong) fuzzy sum rule for subdif- ferentials in any innite dimensional Banach space. From this we deduce that additional assumptions are also needed on functions for chain rules, multiplier rules for constrained minimization problems and Clarke-L...

. We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit func...

This paper discusses recent improvements on subdifferential fuzzy sum rules in nonsmooth analysis and their applications to optimal control problems

. Several different basic properties are used for developing a system of calculus for subdifferentials. They are a nonlocal fuzzy sum rule in [5, 25], a multidirectional mean value theorem in [7, 8], local fuzzy sum rules in [14, 15] and an extremal principle in [19, 21]. We show that all these basic results are equivalent and discuss some interest...

In previous papers Warga and Zhu have proven the conjecture of Rosenblueth and Vinter that a particular abstract model characterizes the setR̄of relaxed controls that are limits of ordinary controls with (possibly time-dependent) delays. Rosenblueth and Vinter have also provided a simple and concrete characterization ofR̄for the case of constant de...

The generalized bilevel programming problem (GBLP) is a bilevel mathematical program where the lower level is a variational inequality. In this paper we prove that if the objective function of a GBLP is uniformly Lipschitz continuous in the lower level decision variable with respect to the upper level decision variable, then using certain uniform p...

In this paper, we derive an estimate for the G-subderivative of the value function associated with a perturbed optimization problem with differential inclusion constraints. We then apply this result to obtain a necessary condition for a solution to a bilevel dynamic problem.

. We refine and extend the Clarke-Ledyaev mean value inequality to smooth Banach spaces. Keywords: Mean value inequalities, fuzzy sum rule and smooth spaces. Short Title: Mean Value Inequalities in Smooth Spaces. AMS (1991) subject classification: Primary 26B05. Acknowledgement: I thank A. ' Swiech and P. Wolenski for a stimulating conversation. 1...

In Gâteaux or bornologically differentiable spaces there are two natural generalizations of the concept of a Fréchet subderivative. In this paper we study the viscosity subderivative (which is the more robust of the two) and establish refined fuzzy sum rules for it in a smooth Banach space. These rules are applied to obtain comparison results for v...

In this paper, we define expected reachability and reachable set for the class of piecewise linear deterministic systems. We develop formulas for calculating the reachable set and analyze its variance and discuss how to implement these formulas numerically. We also discuss a sufficient condition for the system to be completely reachable. Our defini...

. We consider nonsmooth constrained optimization problems with semicontinuous and continuous data in Banach space and derive necessary conditions without constraint qualification in terms of smooth subderivatives and normal cones. These results, in different versions, are set in reflexive and smooth Banach spaces. Key Words. Constrained optimizatio...

The purpose of this paper is to develop an age dependent hedging point policy and to show that it gives a better performance than the classical one for an optimal machine age to start stocking parts

. In Gateaux or bornologically differentiable spaces there are two natural generalizations of the concept of a Fr'echet subderivative: In this paper we study the viscosity subderivative (which is the more robust of the two) and establish refined fuzzy sum rules for it in a smooth Banach space. These rules are applied to obtain comparison results fo...

The value function plays an important role in optimization; it measures the sensitivity of the problem to perturbations of the objective function and the various constraints. Particularly interesting is the derivative of the value function, a measure of so called “differential stability”. When the value function is differentiable, it plays the role...

Because relaxed controls form a convex and compact subset of a normed vector space, it is generally much easier to derive controllability results and corresponding optimization results for such problems. For standard and nonstandard unrelaxed control problems, these properties of relaxed controls are used to divide the essential part of the derivat...

We consider the following integral-inclusion u(t) ∈ F(t, ξ(x)(t)) a.e. inT in Banach space which, in particular, includes a control system defined by partial differential equations with delayed or shifted controls and an ordinary differential inclusion as special cases. We prove a relaxation theorem for this integral-inclusion and discuss some prop...

Control problems defined by ordinary differential equations with right-hand sides that are unbounded functions of the control variables are considered. These problems can be reformulated in terms of bounded (relaxed or unrelaxed) differential inclusions by introducing a new independent variable (which is a function of the old state and control func...

. We provide a refined sensitivity analysis for finite and infinite horizon control problems where in both cases the perturbation space is L 1 . Our underlying technique relies on a recent sequential description of both the generalized gradient of Clarke and of the approximate G--subdifferential of functions defined on a smooth Banach space. We als...

In this paper, we consider a maintenance and production model of a flexible manufacturing system. The maintenance activity involves lubrication, routine adjustments, etc., which reduce the machine failure rates and therefore reduce the aging of the machines. The objective of the problem is to choose the rate of maintenance and the rate of productio...

. In this paper, we derive necessary optimality conditions for optimization problems defined by non-convex differential inclusions with endpoint constraints. We do this in terms of parametrizations of the convexified form of the differential inclusion and, under additional assumptions, in terms of the inclusion itself. 1 Introduction Consider the o...

In this paper we establish a theoretical basis for a class of flexible mannfacturing system (FMS) production and maintenance planning problem. Results include local Lipschitzian property of the optimal cost function for the FMS production and maintenance planning problem, existence of the optimal control policy, and necessary and sufficient conditi...