This article provides a Markov model for the term structure of credit risk spreads. The model is based on Jarrow and Turnbull (1995), with the bankruptcy process following a discrete state space Markov chain in credit ratings. The parameters of this process are easily estimated using observable data. This model is useful for pricing and hedging corporate debt with imbedded options, for pricing and hedging OTC derivatives with counterparty risk, for pricing and hedging (foreign) government bonds subject to default risk (e.g., municipal bonds), for pricing and hedging credit derivatives, and for risk management. This article presents a simple model for valuing risky debt that explicitly incorporates a firm's credit rating as an indicator of the likelihood of default. As such, this article presents an arbitrage-free model for the term structure of credit risk spreads and their evolution through time. This model will prove useful for the pricing and hedging of corporate debt with We would like to thank John Tierney of Lehman Brothers for providing the bond index price data, and Tal Schwartz for computational assistance. We would also like to acknowledge helpful comments received from an anonymous referee. Send all correspondence to Robert A. Jarrow, Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853. The Review of Financial Studies Summer 1997 Vol. 10, No. 2, pp. 481--523 1997 The Review of Financial Studies 0893-9454/97/$1.50 imbedded options, for the pricing and hedging of OTC derivatives with counterparty risk, for the pricing and hedging of (foreign) government bonds subject to default risk (e.g., municipal bonds), and for the pricing and hedging of credit derivatives (e.g. credit sensitive notes and spread adjusted notes). This model can also...

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. The deviation matrix of an ergodic, continuous-time Markov chain with transition probability matrix P (.) and ergodic matrix # is the matrix D # R # 0 (P (t) - #)dt. We give conditions for D to exist and discuss properties and a representation of D. The deviation matrix of a birth-death process is investigated in detail. We also describe a new application of deviation matrices by showing that a measure for the convergence to stationarity of a stochastically increasing Markov chain can be expressed in terms of the elements of the deviation matrix of the chain. Keywords and phrases: birth-death process, convergence to stationarity, deviation matrix, ergodic Markov chain, ergodic potential, first entrance time, first return time, fundamental matrix, group inverse. 2000 Mathematics Subject Classification: Primary 60J27, Secondary 60J10 60J35 60J80 1 Introduction In what follows X # {X(t), t # 0} is a time-homogeneous, continuous-time Markov chain with a discrete state space...

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SUMMARY. Consider the situation in which subjects arrive sequentially in a clinical trial. There are K possibly unrelated treatments. Suppose balls in an urn are labeled with treatments. When a subject arrives, a ball is drawn randomly from the urn, the subject receives the treatment indicated on the ball and the ball is returned to the urn. If the treatment is successful, one ball of the same type is added to the urn. Otherwise, one ball of the same type is taken out from the urn. Under certain assumptions, the urn process can be embedded in a continuous time birth and death process, and associated distributional results can thereby be obtained. Since certain treatments can “die out, ” we introduce a Poisson immigration process to replenish the urn. We derive the maximum likelihood estimators of the success probabilities, prove their consistency and obtain their limiting distributions using martingale theory. We then derive inference procedures for the comparison of the K treatments. 1.

Abstract. In distributed and mobile systems with volatile bandwidth and fragile connectivity, non-functional aspects like performance and reliability become more and more important. To formalise, measure, and predict these properties, stochastic methods are required. At the same time such systems are characterised by a high degree of architectural reconfiguration. Viewing the architecture of a distributed system as a graph, this is naturally modelled by graph transformations. To address these two concerns, stochastic graph transformation systems have been introduced associating with each rule its application rate—the rate of the exponential distribution governing the delay of its application. Deriving continuous-time Markov chains, Continuous Stochastic Logic is used to specify reliability properties and verify them through model checking. In particular, we study a protocol for the reconfiguration of P2P networks intended to improve their reliability by adding redundant connections. The modelling of this protocol as a (stochastic) graph transformation system takes advantage of negative application and conditions path expressions. This ensuing high-level style of specification helps to reduce the number of states and increases the capabilities for automated analysis. 1

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Most of SDEwMSs do not have explicit solutions so it is important to have numerical solutions. It is surprising that there are not any numerical methods established for SDEwMSs yet, although the numerical methods for stochastic differential equations (SDEs) have been well studied. The main aim of this paper is to develop a numerical scheme for SDEwMSs and estimate the error between the numerical and exact solutions. This is the first paper in this direction and the emphasis lies on the error analysis. © 2003 IMACS. Published by Elsevier B.V. All rights reserved.

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> jjX 0 = i) = p (t) ij , where P (t) = PP : : : P is the t-fold matrix product. Write P i (\Delta) and E i (\Delta) for probabilities and expectations for the chain started at state i and time 0. More generally, write P ae (\Delta) and E ae (\Delta) for probabilities and expectations for the chain started at time 0 with distribution ae. Write T i = minft 0 : X t = ig 1 for the first hitting time on state i, and write T + i =