Option pricing when asset returns jump interruptedly

This paper proposes an extension of Merton's jump‐diffusion model to reflect the time inhomogeneity caused by changes of market states. The benefit is that it simultaneously captures two salient features in asset returns: heavy tailness and volatility clustering. On the basis of an empirical analysis where jumps are found to happen much more frequently in risky periods than in normal periods, we assume that the Poisson process for driving jumps is governed by a two‐state on‐off Markov chain. This makes jumps happen interruptedly and helps to generate different dynamics under these two states. We provide a full analysis for the proposed model and derive the recursive formulas for the conditional state probabilities of the underlying Markov chain. These analytical results lead to an algorithm that can be implemented to determine the prices of European options under normal and risky states. Numerical examples are given to demonstrate how time inhomogeneity influences return distributions, option prices, and volatility smiles. The contrasting patterns seen in different states indicate the insufficiency of using time‐homogeneous models and justify the use of the proposed model. Copyright © 2012 John Wiley & Sons, Ltd.     

Nested Benders decomposition and dynamic programming for reservoir optimisation

This paper presents a computational comparison of nested Benders decomposition and dynamic programming (DP) for stochastic optimisation problems arising from the optimisation of hydro-electric generation from hydraulically linked reservoirs. The examples considered have between 3 and 17 reservoirs, two weather states, three runoff patterns and five periods. The examples are solved exactly by the simplex method and nested Benders decomposition and solved approximately by discrete dynamic programming (DP). A full version of DP is used for examples with 3 and 4 reservoirs, and a decomposition method is used for all examples. The full DP results are within 1% of optimal and the DP decomposition results are within 3.2% of optimal. Timings are given for serial and parallel versions of the algorithms. An analysis is given of how the different methods scale with the number of periods, reservoirs, weather states and runoff patterns, and also how applicable they are to more general problems.     

A positional game for an overlapping generation economy

We develop a model with intra-generational consumption externalities, based on the overlapping generation model by Diamond (1965). More specifically, we consider a two-period lived overlapping generation economy, assuming that the utility of each consumer depends also on the average consumption level by the consumers in the same generation. We suppose that such level is not taken as a parameter by agents, who behave strategically. We characterize the consumption and saving choices for the two periods in the Nash equilibrium path and we determine a dynamic equation for capital accumulation. For the associated dynamical system, we find a unique positive steady state for capital and we investigate how its position, as well as that of the steady states for consumption in both periods, change with respect to variations in the degree of interaction in the two periods. We finally compare the steady states for capital with and without social interaction.     

Using a Markov reward model to estimate spend-down costs for a geriatric department

A Markov model is used to describe movements of geriatric patients within a hospital system where the states of the Markov chain are acute/rehabilitative, long-stay care, discharge or death. By assigning costs to the states of this model, we can estimate the spend-down costs of running down services given that there are no more admissions and different costs are assigned to acute/rehabilitative and long-stay care. The model is used to estimate the spend-down costs using data previously validated for three Departments of Geriatric Medicine in the South West Thames Region of England. Our approach allows hospital planners to identify cost-effective strategies which take into account the fact that some geriatric patients remain in long-stay care for very long periods of time.     

Punctuality and idleness in just-in-time scheduling

In scheduling problems with irregular cost functions, such as deviation functions in earliness–tardiness scheduling problems, optimal solutions usually contain some idleness periods during which no activity is processed. Then, minimizing the penalties for not delivering on time and minimizing the idleness cost are two complementary criteria for a schedule. A Dynamic Programming procedure, in which the states are represented by continuous two-dimensional piecewise linear functions, is proposed to compute the cost of the Pareto optimal schedules.     

Optimal consumption and investment strategies with liquidity risk and lifetime uncertainty for Markov regime-switching jump diffusion models

In this paper, we consider the optimal consumption and investment strategies for households throughout their lifetime. Risks such as the illiquidity of assets, abrupt changes of market states, and lifetime uncertainty are considered. Taking the effects of heritage into account, investors are willing to limit their current consumption in exchange for greater wealth at their death, because they can take advantage of the higher expected returns of illiquid assets. Further, we model the liquidity risks in an illiquid market state by introducing frozen periods with uncertain lengths, during which investors cannot continuously rebalance their portfolios between different types of assets. In liquid market, investors can continuously remix their investment portfolios. In addition, a Markov regime-switching process is introduced to describe the changes in the market’s states. Jumps, classified as either moderate or severe, are jointly investigated with liquidity risks. Explicit forms of the optimal consumption and investment strategies are developed using the dynamic programming principle. Markov chain approximation methods are adopted to obtain the value function. Numerical examples demonstrate that the liquidity of assets and market states have significant effects on optimal consumption and investment strategies in various scenarios.     

A Stochastic Model of Sugar Cane Crop Rotation

An identifying model of sugar cane crop rotation has been developed to optimize return to a single sugar cane farm by scheduling harvesting times and implicitly defining the crop growth periods. A Markov decision process is used to provide a simple structure which retains the problem's dynamic nature. Stochastic influence on the system is incorporated within this structure by defining states representing crops at discrete quality levels with probabilistic transitions between states. Non-discounted and discounted versions of the Markov process are solved using a linear programming formulation. Problem formulation and interpretation of the solution is complicated by an extra constraint on the Markov process due to sugar industry restriction. This constraint is treated by two different means and a rule is presented to aid its interpretation.     

An entropy-based measurement of working time flexibility

Working time flexibility is of major concern to companies, and must be established by reaching arrangements with workers. Comparing flexibility arrangements is far from easy. Measures of flexibility that assume a probability distribution of future demand are not really measuring flexibility, but rather risk. We define a space of states made up of possible working hours over several periods, and apply three measures of flexibility to it: the proportion of feasible states, the average cost and a new entropy-based measure of flexibility (EMF). The EMF is based on Shannon’s entropy. We propose the use of three measures simultaneously for comparing the flexibility provided by a working time arrangement. Two examples are given: one that assesses the flexibility generated by using time accounts and overtime in a working time accounts (WTAs) modality, and one that compares the WTAs and hiring and firing (H&F) modalities.     

Sampled-data distributed protocol for coordinated aggregation of multi-agent systems subject to communication delays

This paper studies the coordinated aggregation problem of a multi-agent system. Particularly, all the agents reach a consensus within a pre-specified target region. However, only a subset of agents have access to this target region, and each agent merely interacts with its neighbors by communication. Moreover, there exist unknown heterogeneous delays in communication channels. The underlying communication topology is characterized by a digraph. To accommodate the practical digital disposal, a sampled-data distributed protocol is proposed, where the sampling is asynchronous in the sense that the sampling periods of distinct agents are heterogeneous. The resulting closed-loop system from the proposed sampled-data distributed protocol is in a hybrid fashion that the continuous system is fed-back by using discrete states at sampling instants. The convergence performance of this hybrid closed-loop system is analyzed based on the contraction theory. More specifically, it is first shown that all the states are coordinated to aggregate within the target region, i.e., coordinated aggregation. With this result, it is next shown that all the states are coordinated towards a consensus, i.e., state agreement. These together guarantee the fulfillment of the concerned coordinated aggregation objective. Finally, a simulation example is given to validate the theoretical results.     

Analysis of job transfer policies in systems with unreliable servers

We consider a system where incoming jobs may be executed at different servers, each of which goes through alternating periods of being available and unavailable. Neither the states of the servers nor the relevant queue sizes are known at moments of arrival. Hence, a load balancing mechanism that relies on random time-out intervals and job transfers from one queue to another is adopted. The object is to minimize a cost function which may include holding costs and transfer costs. A model of a single queue with an unreliable server and timeouts is analyzed first. The results are then used to obtain an approximate solution for arbitrary number of queues. Several transfer policies are evaluated and compared.     

状态转移下的利率预期与误差学习假说

采用线性回归和状态转移模型讨论利率预期的"误差学习假说"和不同期限利率预测误差之间的关联性,并考虑带有风险溢价时预测误差的作用.分析发现,1年期限的"误差学习假说"在银行间国债市场中是成立的,不同期限的利率预期对预测误差的反应呈现一定的独立性,不同的状态中预测误差和风险溢价对利率预期的影响是不同的,考虑风险溢价有助于提升对利率预期变动的解释.     

Mean Buffer Contents in Discrete-Time Single-Server Queues with Heterogeneous Sources

We consider discrete-time single-server queues fed by independent, heterogeneous sources with geometrically distributed idle periods. While being active, each source generates some cells depending on the state of the underlying Markov chain. We first derive a general and explicit formula for the mean buffer contents in steady state when the underlying Markov chain of each source has finite states. Next we show the applicability of the general formula to queues fed by independent sources with infinite-state underlying Markov chains and discrete phase-type active periods. We then provide explicit formulas for the mean buffer contents in queues with Markovian autoregressive sources and greedy sources. Further we study two limiting cases in general settings, one is that the lengths of active periods of each source are governed by an infinite-state absorbing Markov chain, and the other is the model obtained by the limit such that the number of sources goes to infinity under an appropriate normalizing condition. As you will see, the latter limit leads to a queue with (generalized) M/G/∞ input sources. We provide sufficient conditions under which the general formula is applicable to these limiting cases.AMS subject classification: 60K25, 60K37, 60J10This revised version was published online in June 2005 with corrected coverdate     

Detecting shifts in the mean of a simulation output process

The application of the correct simulation output analysis technique requires a knowledge of the model's behaviour. Traditionally, only two types of model behaviour are discussed for discrete event simulations: transient and steady state. In this paper, a third type of behaviour, denoted shifting steady state, is considered in which the model passes through successive periods of different steady states. A heuristic technique for identifying the shifts in the mean of a time series is applied to the output data from simulation models with known shifting steady-state behaviour in order to test its effectiveness in detecting the shifts. The heuristic performs well, indicating that it may be a valuable additional technique for output analysis.     

Behavioral modeling in weight loss interventions

Designing systems with human agents is difficult because it often requires models that characterize agents’ responses to changes in the system’s states and inputs. An example of this scenario occurs when designing treatments for obesity. While weight loss interventions through increasing physical activity and modifying diet have found success in reducing individuals’ weight, such programs are difficult to maintain over long periods of time due to lack of patient adherence. A promising approach to increase adherence is through the personalization of treatments to each patient. In this paper, we make a contribution toward treatment personalization by developing a framework for predictive modeling using utility functions that depend upon both time-varying system states and motivational states evolving according to some modeled process corresponding to qualitative social science models of behavior change. Computing the predictive model requires solving a bilevel program, which we reformulate as a mixed-integer linear program (MILP). This reformulation provides the first (to our knowledge) formulation for Bayesian inference that uses empirical histograms as prior distributions. We study the predictive ability of our framework using a data set from a weight loss intervention, and our predictive model is validated by comparison to standard machine learning approaches. We conclude by describing how our predictive model could be used for optimization, unlike standard machine learning approaches that cannot.     

A structural model of the higher‐order Markov process incorporating reversion effects

The paper presents a formal approach which may increase the realism and parsimony of higher‐order Markov models applied to certain human behaviors. Often in behavioral applications, any improvements in fit available from increasing the order of a Markov model would be more than offset by interpretive problems caused by the very rapid increase in the number of independent parameters. The model proposed here for the higher‐order process greatly reduces the number of independent parameters, replacing them with sociologically relevant effects of persistence in and reversion to previous conditions.

The general model is called the “reversion model.” In it, individuals are allowed to carry along some information about their pasts, for a number of periods corresponding to the order of the model. The parameters describing residence histories are constructed to give each individual an underlying set of first‐order transition probabilities, which are modified by experience of the various states of the system. When an individual occupies a particular state, his relative probability of future residence there (vis‐a‐vis the other states as a group) is permitted to change. But occupation of a particular state is not permitted to affect the relative chances of residence among the other states. With suitable constraints, the number of parameters of this higher‐order process no longer increases geometrically with the order, but only arithmetically.

Maximum likelihood estimation formulas are derived for the reversion model, which is then applied to longitudinal data on the work activities of U.S. Ph.D. physicists and chemists in 1960–1966, and is found to fit well using likelihood ratio tests.     

Period integrals,quantum numbers,and confinement in Susy QCD

We compute the period integrals on degenerate Seiberg—Witten curves for supersymmetric QCD explicitly and also show how these periods determine the changes in the quantum numbers of the states when passing from the weak- to strong-coupling domains in the mass moduli space of the theory. We discuss the confinement of monopoles at a strong coupling and demonstrate that the ambiguities in choosing the path in the moduli space do not affect the physical conclusions on confinement of monopoles in the phase with condensed light dyons.     

Simulation-based parametric optimization for long-term asset allocation using behavioral utilities

This article presents a simulation-based methodology for finding optimal investment strategy for long-term financial planning. The problem becomes intractable due to its size or the properties of the utility function of the investors. One approach is to make simplifying assumptions regarding the states of the world and/or utility functions in order to obtain a solution. These simplifications lead to the true solution of an approximate problem. Our approach is to find a good approximate solution to the true problem. We approximate the optimal decision in each period with a low dimensional parameterization, thus reformulating the problem as a non-linear, simulation-based optimization in the parameter space. The dimension of the reformulated optimization problem becomes linear in the number of periods. The approach is extendable to other problems where similar solution characteristics are known.     

General existence principles for Stieltjes differential equations with applications to mathematical biology

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper.     

Stability and Bifurcation Analysis of a Nonlinear DDE Model for Drilling

We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of cells and in lattices (and periodic arrays) of cells that cannot readily be explained by conventional symmetry considerations. We also show that different types of dynamics can coexist robustly in single solutions of systems of coupled identical cells. The examples include a three-cell system exhibiting equilibria, periodic, and quasiperiodic states in different cells; periodic $2n\times 2n$ arrays of cells that generate $2^n$ different patterns of synchrony from one symmetry-generated solution; and systems exhibiting multirhythms (periodic solutions with rationally related periods in different cells). Our theoretical results include the observation that reduced equations on a center manifold of a skew product system inherit a skew product form.     

Some Curious Phenomena in Coupled Cell Networks

We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of cells and in lattices (and periodic arrays) of cells that cannot readily be explained by conventional symmetry considerations. We also show that different types of dynamics can coexist robustly in single solutions of systems of coupled identical cells. The examples include a three-cell system exhibiting equilibria, periodic, and quasiperiodic states in different cells; periodic 2n × 2n arrays of cells that generate 2ⁿ different patterns of synchrony from one symmetry-generated solution; and systems exhibiting multirhythms (periodic solutions with rationally related periods in different cells). Our theoretical results include the observation that reduced equations on a center manifold of a skew product system inherit a skew product form.