Super-replication under proportional transaction costs: From discrete to continuous-time models

In this paper, we study the problem of finding the minimal initial capital (i.e. super-replication value) needed in order to hedge (without risk) European contingent claims in a Markov setting under proportional transaction costs. The main result is that the cheapest (trivial) buy-and-hold strategy is optimal. Such a negative result has been derived previously in different contexts. First, we focus on discrete-time binomial models. We prove that the continuous-time limit of the super-replication value is the cost of the cheapest buy-and-hold strategy. Then, the result is proved in a multivariate continuous-time model with Brownian filtration. As a direct consequence, we obtain an explicit characterization of the hedging set, i.e. the set of all initial positions in the market assets from which the contingent claim can be hedged through some admissible portfolio strategy.     

Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks

We formulate discrete-time analogues of integrodifferential equations modelling bidirectional neural networks studied by Gopalsamy and He. The discrete-time analogues are considered to be numerical discretizations of the continuous-time networks and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the equilibria of the continuous-time networks. By constructing a Lyapunov-type sequence, we obtain easily verifiable sufficient conditions under which every solution of the discrete-time analogue converges exponentially to the unique equilibrium. The sufficient conditions are identical to those obtained by Gopalsamy and He for the uniqueness and global asymptotic stability of the equilibrium of the continuous-time network. By constructing discrete-time versions of Halanay-type inequalities, we obtain another set of easily verifiable sufficient conditions for the global exponential stability of the unique equilibrium of the discrete-time analogue. The latter sufficient conditions have not been obtained in the literature of continuous-time bidirectional neural networks. Several computer simulations are provided to illustrate the advantages of our discrete-time analogue in numerically simulating the continuous-time network with distributed delays over finite intervals.     

Optimal Investment and Consumption with Proportional Transaction Costs in Regime-Switching Model

This paper is concerned with an infinite-horizon problem of optimal investment and consumption with proportional transaction costs in continuous-time regime-switching models. An investor distributes his/her wealth between a stock and a bond and consumes at a non-negative rate from the bond account. The market parameters (the interest rate, the appreciation rate, and the volatility rate of the stock) are assumed to depend on a continuous-time Markov chain with a finite number of states (also known as regimes). The objective of the optimization problem is to maximize the expected discounted total utility of consumption. We first show that for a class of hyperbolic absolute risk aversion utility functions, the value function is a viscosity solution of the Hamilton–Jacobi–Bellman equation associated with the optimization problem. We then treat a power utility function and generalize the existing results to the regime-switching case.     

Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups

We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation arguments, valid for both geometric and subgeometric cases with essentially the same ideas. In particular, we present very simple new proofs of the geometric case.     

Discrete–time delta hedging and the Black–Scholes model with transaction costs

The paper deals with the problem of discrete–time delta hedging and discrete-time option valuation by the Black–Scholes model. Since in the Black–Scholes model the hedging is continuous, hedging errors appear when applied to discrete trading. The hedging error is considered and a discrete-time adjusted Black–Scholes–Merton equation is derived. By anticipating the time sensitivity of delta in many cases the discrete-time delta hedging can be improved and more accurate delta values dependent on the length of the rebalancing intervals can be obtained. As an application the discrete-time trading with transaction costs is considered. Explicit solution of the option valuation problem is given and a closed form delta value for a European call option with transaction costs is obtained.     

Power utility maximization in exponential Lévy models: convergence of discrete-time to continuous-time maximizers

We consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Lévy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strategies to the continuous-time counterpart. In addition, we provide and compare qualitative properties of the discrete-time and continuous-time optimizers.     

A Mathematical Analysis of Technical Analysis

In this paper, we investigate trading strategies based on exponential moving averages (ExpMAs) of an underlying risky asset. We study both logarithmic utility maximization and long-term growth rate maximization problems and find closed-form solutions when the drift of the underlying is modelled by either an Ornstein-Uhlenbeck process or a two-state continuous-time Markov chain. For the case of an Ornstein-Uhlenbeck drift, we carry out several Monte Carlo experiments in order to investigate how the performance of optimal ExpMA strategies is affected by variations in model parameters and by transaction costs.     

On the relationship between uniqueness and stability in sum-aggregative,symmetric and general differentiable games

This article explores the relationship between uniqueness and stability in differentiable regular games, with a major focus on the important classes of sum-aggregative, two-player and symmetric games. We consider three types of popular dynamics, continuous-time gradient dynamics as well as continuous- and discrete-time best-reply dynamics, and include aggregate-taking behavior as a non-strategic behavioral variant. We show that while in general games stability conditions are only sufficient for uniqueness, they are likely to be necessary as well in models with sum-aggregative or symmetric payoff functions. In particular, a unique equilibrium always verifies the stability conditions of all dynamics if strategies are equilibrium complements, and this also holds for both continuous-time dynamics if strategies are equilibrium substitutes with bounded slopes. These findings extend to the case of aggregate-taking equilibria. We further analyze the stability relations between the various dynamics, and demonstrate that the restrictive nature of the discrete dynamics originates from simultaneity of adjustments. Asynchronous decisions or heterogeneous forward thinking may stabilize the adjustment process.     

Optimal impulse control of a portfolio with a fixed transaction cost

The aim of this work is to investigate a portfolio optimization problem in presence of fixed transaction costs. We consider an economy with two assets: one risky, modeled by a geometric Brownian motion, and one risk-free which grows at a certain fixed rate. The agent is fully described by his/her utility function and the objective is to maximize the expected utility from the liquidation of wealth at a terminal date. We deal with different forms of utility functions (power, logarithmic and exponential utility), describing in each case how the fixed transaction costs influence the agent’s behavior. We show when it is optimal to recalibrate his/her portfolio and which are the best adjusted portfolios. We also analyze how the optimal strategy is influenced by the risk-aversion, as well as other model parameters.     

Fitting continuous-time and discrete-time models using discrete-time data and their applications

An exponential function scheme, which is an extension of the time-domain prony method, and a mixed-matching method are developed for fitting the coefficients of both continuous-time and discrete-time transfer functions, using the discrete-time data of either continuous-time or discrete-time systems. When the discrete-time data are obtained from a continuous-time (discrete-time) system and the discrete-time (continuous-time) models are desirable, the proposed method can be applied to perform the model conversions. If the discrete-time data are obtained from a high-degree system, the proposed method can be applied to determine the reduced-degree models.     

Digital Stabilizer Design for a Switched Linear Control Delay System

The problem of designing a digital controller stabilizing a continuous-time switched linear control delay system is studied. The approach to stabilization successively includes the construction of a continuous-time–discrete-time closed-loop system with a digital controller, the transition to its discrete-time model, and the construction of a discrete-time controller by simultaneous stabilization methods.     

Unbounded knapsack problem with controllable rates: the case of a random demand for items

This paper focuses on a dynamic, continuous-time control generalization of the unbounded knapsack problem. This generalization implies that putting items in a knapsack takes time and has a due date. Specifically, the problem is characterized by a limited production horizon and a number of item types. Given an unbounded number of copies of each type of item, the items can be put into a knapsack at a controllable production rate subject to the available capacity. The demand for items is not known until the end of the production horizon. The objective is to collect items of each type in order to minimize shortage and surplus costs with respect to the demand. We prove that this continuous-time problem can be reduced to a number of discrete-time problems. As a result, solvable cases are found and a polynomial-time algorithm is suggested to approximate the optimal solution with any desired precision.     

Model conversions of uncertain linear system using the improved block-pulse function approach

This paper presents an improved block-pulse function approach to convert a continuous-time (respectively, discrete-time) structured uncertain linear system into an equivalent discrete-time (respectively, continuous-time) structured uncertain linear model. The concept of the principle of equivalent areas is utilized for the uncertain model conversions. This allows the use of well-established theorems and algorithms in the discrete-time (respectively, continuous-time) domain to indirectly solve the continuous-time (respectively, discrete-time) domain problems. A numerical example is given to demonstrate the effectiveness of the proposed method.     

Indifference pricing of pure endowments and life annuities under stochastic hazard and interest rates

We study indifference pricing of mortality contingent claims in a fully stochastic model. We assume both stochastic interest rates and stochastic hazard rates governing the population mortality. In this setting we compute the indifference price charged by an insurer that uses exponential utility and sells k contingent claims to k independent but homogeneous individuals. Throughout we focus on the examples of pure endowments and temporary life annuities. We begin with a continuous-time model where we derive the linear pdes satisfied by the indifference prices and carry out extensive comparative statics. In particular, we show that the price-per-risk grows as more contracts are sold. We then also provide a more flexible discrete-time analog that permits general hazard rate dynamics. In the latter case we construct a simulation-based algorithm for pricing general mortality-contingent claims and illustrate with a numerical example.     

Digital modeling and hybrid control of sampled-data uncertain system with input time delay using the law of mean

This paper utilizes an interval Pade approximate method together with interval arithmetic operation to convert a continuous-time uncertain system with input time-delay to an equivalent discrete-time interval model and transforms the robust control law of a continuous-time uncertain system with input time delay into an equivalent one of a sampled-data uncertain system with input time delay. The developed discrete-time interval model tightly encloses the exact discrete-time uncertain system with input time delay. Based on the law of mean and inclusion theory, a perturbed digital control law of input time-delay sampled-data uncertain system is newly presented, so that the states of the digitally controlled sample-data uncertain system closely match those of the originally well-designed continuous-time uncertain system.     

Radner equilibrium in incomplete Lévy models

We construct continuous-time equilibrium models based on a finite number of exponential utility investors. The investors’ income rates as well as the stock’s dividend rate are governed by discontinuous Lévy processes. Our main result provides the equilibrium (i.e., bond and stock price dynamics) in closed-form. As an application, we show that the equilibrium Sharpe ratio can be increased and the equilibrium interest rate can be decreased (simultaneously) when the investors’ income streams cannot be traded.     

A Retrial BMAP/PH/N System

A multi-server retrial queueing model with Batch Markovian Arrival Process and phase-type service time distribution is analyzed. The continuous-time multi-dimensional Markov chain describing the behavior of the system is investigated by means of reducing it to the corresponding discrete-time multi-dimensional Markov chain. The latter belongs to the class of multi-dimensional quasi-Toeplitz Markov chains in the case of a constant retrial rate and to the class of multi-dimensional asymptotically quasi-Toeplitz Markov chains in the case of an infinitely increasing retrial rate. It allows to obtain the existence conditions for the stationary distribution and to elaborate the algorithms for calculating the stationary state probabilities.     

On the limit of sampled signal extrapolation with a wavelet signal model ∗

The extrapolation of sampled signals from a given interval using a wavelet model with various sampling rates is examined in this research. We present sufficient conditions on signals and wavelet bases so that the discrete-time extrapolated signal converges to its continuous-time counterpart when the sampling rate goes to infinity. Thus, this work provides a practical procedure to implement continuous-time signal extrapolation, which is important in wideband radar and sonar signal processing, with a discrete one via carefully choosing the sampling rate and the wavelet basis. A numerical example is given to illustrate our theoretical result.     

有交易成本的利率期限结构的分析

本文利用无套利均衡方法对存在着交易费，赋税，以及买卖差价等交易成本的债券市场进行分析．用数学方法严格地证明了一个基本结论：在有交易成本的债券市场上，弱无套利性与相容期限结构的存在性是等价的．     

Trust,assurance, and inequality: A rational choice model of mutual trust 1

The rational choice approach to trust has three problems: it has not explicitly explained findings verified in social psychological study of trust; it stands on a limited assumption of asymmetric relationship between a truster and a trustee; and it has not dealt with situations in which a rich person encounters a poor person. We build a game theoretic model of mutual trust to solve these problems. Then we analyze an equilibrium of the model and derive some implications from it. That is, the ratio between the transaction costs and the opportunity costs affects actor' trustfulness; a more trustworthy actors finds it easier to leave his/her group in search of higher returns; and a rich actor is more willing to trust his/her counterpart than a poor actor.