Continuous-time Setting Research Articles

Optimal consumption and investment with Lévy processes

We study the intertemporal consumption and investment problem in a continuous time setting when the security prices follow a Geometric Lévy process. Using stochastic calculus for semimartingales we obtain conditions for the existence of optimal consumption policies. Also, we give a charaterization of the equivalent martingale measures.

Coalescence times for the branching process

We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a branching process founded t units of time ago, in both the discrete and continuous (time and state-space) settings. We obtain limiting distributions as t→∞ in the subcritical case. In the continuous setting, these distributions are specified for quadratic branching mechanisms (corresponding to Brownian motion and Brownian motion with positive drift), and we also extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.

Coalescence times for the branching process

We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a branching process foundedtunits of time ago, in both the discrete and continuous (time and state-space) settings. We obtain limiting distributions ast→∞ in the subcritical case. In the continuous setting, these distributions are specified for quadratic branching mechanisms (corresponding to Brownian motion and Brownian motion with positive drift), and we also extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.

Chapter 11, Private Workouts and Corporate Debt Pricing under Asymmetric Information

This paper studies the valuation of risky corporate debt, the firm's financing decisions and firm strategy during financial distress under asymmetric information. Using Chapter 11 as a costly state verification mechanism in a simple continuous-time setting, we characterize which firms will choose to file for the Chapter 11 Bankruptcy procedure and which will choose private workouts (in the form of strategic debt service) when the firm is in financial distress. The model predicts that firms with more intangible assets, higher book value and less information asymmetry are more likely to restructure their debt privately, and their stockholders are systematically better off. We also provide a closed-form solution to the debt pricing problem. The formula reduces to the well-known results of Merton, Black and Cox, Leland and others if the private information can be costlessly revealed.

A tool for model-checking Markov chains

Markov chains are widely used in the context of the performance and reliability modeling of various systems. Model checking of such chains with respect to a given (branching) temporal logic formula has been proposed for both discrete [34, 10] and continuous time settings [7, 12]. In this paper, we describe a prototype model checker for discrete and continuous-time Markov chains, the Erlangen-Twente Markov Chain Checker EIMC2, where properties are expressed in appropriate extensions of CTL. We illustrate the general benefits of this approach and discuss the structure of the tool. Furthermore, we report on successful applications of the tool to some examples, highlighting lessons learned during the development and application of EIMC2.

Debt Policy, Corporate Taxes, and Discount Rates

This paper studies the valuation of assets with debt tax shields when debt policy is a general time-dependent function of the asset's unlevered cash flows, value, and history. In a continuous-time setting, it shows that the value of a project's debt tax shield satisfies a partial differential equation, which simplifies to an easily solved ordinary differential equation for most plausible debt policies. A large class of cases exhibits closed-form solutions for the value of a levered asset, the value of its tax shield, and the appropriate tax-adjusted cost of capital for discounting unlevered cash flows.

Segmented risk sharing in a continuous-time setting

The economy we study is comprised of a continuum of individuals. Each has a stochastic endowment that evolves continuously and independently of all other individuals' endowment processes. Individuals are risk averse and would therefore like to insure their endowment processes. The mutual independence of their endowment processes makes it feasible for them to obtain this insurance by pooling their endowments. We investigate whether such a scheme would survive as an equilibrium in a noncooperative setting.

Ambiguity, Risk, and Asset Returns in Continuous Time

Models of utility in stochastic continuous–time settings typically assume that beliefs are represented by a probability measure, hence ruling out a priori any concern with ambiguity. This paper formulates a continuous–time intertemporal version of multiple–priors utility, where aversion to ambiguity is admissible. In a representative agent asset market setting, the model delivers restrictions on excess returns that admit interpretations reflecting a premium for risk and a separate premium for ambiguity.

On the Coincidence of the Feedback Nash and Stackelberg Equilibria in Economic Applications of Differential Games

In this paper the scope of the applicability of the Stackelberg equilibrium concept in differential games is investigated. Firstly, it is showed that for a class of differential games with state-interdependence the stationary feedback Nash equilibrium coincides with the stationary feedback Stackelberg equilibrium independently of the player being the leader of the game. Secondly, sufficient conditions for obtaining the coincidence between the two equilibria are defined. A review of different economic models shows that this coincidence is going to occur for a good number of economic applications of differential games. This result appears because of the continuous-time setting in which differential games are defined. In this setting the first movement advantage of the leader may disappears and then both equilibria coincide.

A minimum principle for chaotic dynamical systems

Discrete time dynamical systems generated by the iteration of nonlinear maps, such as the logistic map or the tent map, provide interesting examples of chaotic systems. But what is the physical principle behind the emergence of these maps? In the continuous time settings, differential equations of mechanics arise from the minimization of the energy function (Hamiltonian). However, there is no general physical principle for the discrete time analogue of differential equations, namely, maps. In this note, we present an approach to this problem. Using a natural definition of energy for chaotic systems, we minimize energy subject to the constraint that the observed dynamical system has a known entropy. We consider the case where the natural invariant measure is Lebesgue. Invoking the Euler–Lagrange equation, we derive a nonlinear second order differential equation whose solution is the chaotic map that minimizes energy.

Predictive pole-placement control with linear models

The predictive pole-placement control method introduced in this paper embeds the classical pole-placement state feedback design into a quadratic optimisation-based model-predictive formulation. This provides an alternative to model-predictive controllers which are based on linear–quadratic control. The theoretical properties of the controller in a linear continuous-time setting are presented and a number of illustrative examples are given. These results provide the foundation for novel linear and nonlinear constrained predictive control methods based on continuous-time models.

Resilience: An Evolutionary Approach to Spatial Economic Systems

The concept of resilience has received a great deal of attention in the past decades. Starting from the first fundamental definitions offered by Holling, Pimms and Perrings in an economic-ecological modeling context, the present paper explores the ‘evolution’ of the resilience concept—as well as related different measures—in both a continuous and discrete time setting.

Portfolio choice and equity characteristics: characterizing the hedging demands induced by return predictability

This paper examines portfolio allocation across equity portfolios formed on the basis of characteristics like size and book-to-market. In particular, the paper assesses the impact of return predictability on portfolio choice for a multi-period investor with a coefficient of relative risk aversion of 4. Compared to the investor's allocation in her last period, return predictability with dividend yield causes the investor early in life to tilt her risky-asset portfolio away from high book-to-market stocks and away from small stocks. These results are explained using Merton's (Econometrica 41 (1973) 867) characterization of portfolio allocation by a multi-period investor in a continuous time setting. Abnormal returns relative to the investor's optimal early life portfolio are also calculated. These abnormal returns are found to exhibit the same cross-sectional patterns as abnormal returns calculated relative to the market portfolio: higher for small rather than large firms, and higher for high rather than low book-to-market firms. Thus, hedging demand may be a partial explanation for the high expected returns documented for small firms and high book-to-market firms. However, even with this hedging demand, the investor wants to sell short the low book-to market portfolio to hold the high book-to-market portfolio. The utility costs of using a value-weighted equity index or of ignoring predictability are also calculated. An investor using a value-weighted equity index would give up a much larger fraction of her wealth to have access to book-to-market portfolio than size portfolios. Finally, an investor would give up a much larger fraction of her wealth to have access to dividend yield information than term spread information.

A discrete-time algorithm for pricing double barrier options

A double knock-out European option is an option that expires if the underlying asset price reaches a lower or an upper barrier before maturity. Otherwise, the option payoff at maturity is the same as that of a standard European option. The problem of pricing these options was tackled by Kunimoto and Ikeda (1992) and Geman andYor (1996) who proposed accurate algorithms developed in a continuous-time setting under the usual assumptions of the Black–Scholes analysis. Both these models are based on the assumption of a continuous monitoring of the option contract. This means that, at any time in the life of the option, the contract is checked to verify if the underlying asset price has touched or crossed the barriers. However, in many cases this assumption is not realistic and only a discrete monitoring of the contract is possible. In these situations a discrete-time algorithm is needed to give the correct answer to the problem of pricing barrier options. In this paper we present a discrete-time method for pricing a double knock-out European option. In this framework, the underlying asset process

DYNAMIC COOPERATIVE GAMES

We consider dynamic cooperative games in characteristic function form in the sense that the characteristic function evolves over time in accordance with a difference or differential equation that is influenced not only by the current ("instantaneous") characteristic function but also by the solution concept used to allocate the benefits of cooperation among the players. The latter solution concept can be any one of a number of now standard solution concepts of cooperative game theory but, for demonstration purposes, we focus on the core and the Shapley value. In the process, we introduce some new mechanisms by which players may regard the evolution of cooperative game over time and analyse them with respect to the goal of attaining time consistency either in discrete or in continuous time setting. In discrete time, we illustrate the phenomena that can arise when an allocation according to a given solution concept is used to adapt the values of coalitions at successive time points. In continuous time, we introduce the notion of an "instantaneous" game and its integration over time.

Segmented Risk Sharing in a Continuous-Time Setting

In an economy with a continuum of individuals, each individual has a stochastic, continuously evolving endowment process. Individuals are risk averse and would therefore like to insure their endowment processes. It is feasible to obtain insurance by pooling endowments across individuals because the processes are mutually independent. We characterize the payoff from an insurance contracting scheme of this type, and we investigate whether such a scheme would survive as an equilibrium in a noncooperative setting.

DESIGN AND VALUATION OF CORPORATE SECURITIES WITH STRATEGIC DEBT SERVICE AND ASYMMETRIC INFORMATION

This paper studies the effects of strategic debt service, asymmetric information and their interaction on the valuation of corporate securities and on corporate financing decisions. By introducing information asymmetry into a continuous-time setting, our model is able to integrate these two factors in a unified framework. Such a model allows for obtaining valuation results in a separating equilibrium. The basic results of this paper imply that the risk premium of debt could be partly contributed by information effect. This part of risk premium could be very significant for those good firms with a project which will produce much higher cash flows than what the market expects. We also find that a firm's financing decision depends on its primitives: firms are more apt to rely on equity if they have: (1) high growth potential, (2) riskier projects, (3) higher ratio of intangible assets to total assets and (4) lesser information asymmetry; firms would prefer debt, otherwise.

Portfolio Choice and Equity Characteristics: Characterizing the Hedging Demands Induced by Return Predictability

This paper examines portfolio allocation across equity portfolios formed on the basis of characteristics like size and book-to- market. In particular, the paper assesses the impact of return predictability on portfolio choice for a multi-period investor with a coefficient of relative risk aversion of 4. Compared to the investor s allocation in her last period, return predictability with dividend yield causes the investor early in life to tilt her risky-asset portfolio away from high book-to-market stocks and away from small stocks. These results are explained using Merton s (1973) characterization of portfolio allocation by a multiperiod investor in a continuous time setting. Abnormal returns relative to the investor s optimal early-life portfolio are also calculated. These abnormal returns are found to exhibit the same cross-sectional patterns as abnormal returns calculated relative to the market portfolio: higher for small than large firms, and higher for high than low book-to-market firms. Thus, hedging demand may be a partial explanation for the high expected returns documented empirically for small firms and high book-to-market firms. However, even with this hedging demand, the investor wants to short-sell the low book-to market portfolio to hold the high book-to-market portfolio. The utility costs of using a value-weighted equity index or of ignoring predictability are also calculated. An investor using a value-weighted equity index would give up a much larger fraction of her wealth to have access to book-to-market portfolios than size portfolios. Finally, while an investor would give up a much larger fraction of her wealth to have access to dividend yield information than term spread information, term spread does have incremental benefits over and above just using dividend yield alone.

統計モデルを用いた合意形成過程の定量的分析

The purpose of this study is to build a statistical model, which describes consensus formation process, and to analyze its process quantitatively. The model built in this study is using Markovian process with continuous time and discrete set of events The state of the process will be the number of people in the society convinced about plan. There are three parameters in this model namely preference, adaptation trend and flexibility. These parameters are considered as factors of individual decision-making process. Applying this model to the real society, we can estimate these three parameters. With the experimental consideration, it should be clear that this model could describe the real world.

A local time curiosity in random environment

In random environments, the most elementary processes are Sinai’s simple random walk and Brox’s diffusion process, respectively in discrete and continuous time settings. The two processes are often considered as a kind of companions, somewhat in the same way as the usual random walk and Brownian motion are. In this paper, we study the maximum local times for the Sinai and Brox processes. A somewhat peculiar asymptotic behaviour is observed.