Showing papers on "Semimartingale published in 1988"
TL;DR: In this paper, the authors consider a class of reflecting Brownian motions on the non-negative orthant in R K, and show that the boundary processes appearing in this decomposition do not charge the set of times that the process is at the intersection of two or more faces.
Abstract: We consider a class of reflecting Brownian motions on the non-negative orthant inR K . In the interior of the orthant, such a process behaves like Brownian motion with a constant covariance matrix and drift vector. At each of the (K-1)-dimensional faces that form the boundary of the orthant, the process reflects instantaneously in a direction that is constant over the face. We give a necessary condition for the process to have a certain semimartingale decomposition, and then show that the boundary processes appearing in this decomposition do not charge the set of times that the process is at the intersection of two or more faces. This boundary property plays an essential role in the derivation (performed in a separate work) of an analytical characterization of the stationary distributions of such semimartingale reflecting Brownian motions.
140 citations
TL;DR: Schneider et al. as mentioned in this paper used the Black-Scholes formula in terms of martingale theory and developed a general regression scheme in discrete time, first in an elementary two-period model, and then in a multiperiod model which involves martelingale considerations and sets the stage for extensions to continuous time.
Abstract: It is widely acknowledge that there has been a major breakthrough in the mathematical theory of option trading. This breakthrough, which is usually summarized by the Black–Scholes formula, has generated a lot of excitement and a certain mystique. On the mathematical side, it involves advanced probabilistic techniques from martingale theory and stochastic calculus which are accessible only to a small group of experts with a high degree of mathematical sophistication; hence the mystique. In its practical implications it offers exciting prospects. Its promise is that, by a suitable choice of a trading strategy, the risk involved in handling an option can be eliminated completely.Since October 1987, the mood has become more sober. But there are also mathematical reasons which suggest that expectations should be lowered. This will be the main point of the present expository account. We argue that, typically, the risk involved in handling an option has an irreducible intrinsic part. This intrinsic risk may be much smaller than the a priori risk, but in general one should not expect it to vanish completely. In this more sober perspective, the mathematical technique behind the Black–Scholes formula does not lose any of its importance. In fact, it should be seen as a sequential regression scheme whose purpose is to reduce the a priori risk to its intrinsic core.We begin with a short introduction to the Black–Scholes formula in terms of currency options. Then we develop a general regression scheme in discrete time, first in an elementary two-period model, and then in a multiperiod model which involves martingale considerations and sets the stage for extensions to continuous time. Our method is based on the interpretation and extension of the Black–Scholes formula in terms of martingale theory. This was initiated by Kreps and Harrison; see, e.g. the excellent survey of Harrison and Pliska (1981,1983). The idea of embedding the Black-Scholes approach into a sequential regression scheme goes back to joint work of the first author with D. Sondermann. In continuous time and under martingale assumptions, this was worked out in Schweizer (1984) and Follmer and Sondermann (1986). Schweizer (1988) deals with these problems in a general semimartingale model.
114 citations
01 Jan 1988
Abstract: In this thesis, we investigate the mathematical theory of option pricing and option hedging which is a generaUzation of the Black-Scholes approach. We consider a very general stochastic model where the price process X is a semimartingale. In contrast to the usual Uterature, we do not assume completeness, i.e., we work without the restriction that X generates the space of options in the sense of stochastic integration theory. Our aim is a sequential reduction of risk for both European and American options; in the latter case, this also involves a problem of optimal stopping. In discrete time, this leads to a sequential conditional regression problem which is expücitly solved by using backwards induction. In continuous time, we develop new results on the differentiation of semimartingales which allow us to derive a nonlinear stochastic optimaUty equation of a new kind. In the case of a European option, we solve this optimality equation by a suitable Cameron-MartinMaruyama-Girsanov transformation.
97 citations
TL;DR: In this paper, the authors considered the time reversal of semimartingales defined on a Levy process framework and established an expansion-of-filtrations result for Levy processes.
Abstract: Time reversal of semimartingales defined on a Levy process framework is considered. Usually semimartingales cannot be time-reversed such that the reversed process is still a semimartingale. An expansion-of-filtrations result for Levy processes is established and then it is used to give sufficient conditions such that a semimartingale defined on a Levy process can be time-reversed and still remain a semimartingale.
72 citations
TL;DR: Godambe's (1985) approach is employed to obtain estimates for random signals for a continuous semimartingale model, which leads to estimation for nonconjugate prior situations which is computationally attractive as well as optimal in a restricted sense as discussed by the authors.
12 citations
TL;DR: In this paper, the Grossman-Shiller generalisation of Breeden's consumption-based asset pricing model is further generalised to the class of semimartingale stochastic processes for the arguments of marginal utility and share prices.
11 citations
TL;DR: In this paper, the Levy-Prokhorov distance between the law of an arbitrary semimartingaie and the process with independent increments is estimated. But the distance is not defined.
Abstract: We give an estimate oi the Levy-Prokhorov distance between the law of an arbitrary semimartingaie and the law of a process with independent increments. This estimate allows to recover the "most general" criterion of weak convergence of semimartingales to processes with independent increments without fixed times of discontinuity, in terms of their characteristics. It also provides criteria for uniform convergence for a sequence of semimartingales depending on a parameter
8 citations
TL;DR: In this paper, a convolution-smoothed estimate of the cumulative hazard function in the censored case was proposed, an open problem proposed by Mack (Bulletin of Informatics and Cybernetics 21 (1984) 29-35).
6 citations
TL;DR: In this article, a recursive estimation algorithm for a semimartingale model based on the theory of optimal estimating functions is presented, which may be used to handle the problems of missing observations and censored observations.
Abstract: A recursive estimation algorithm is presented for a semimartingale model based on the theory of optimal estimating functions. Strong consistency and asymptotic normality of the recursive estimate generated by the algorithm are established. This recursive algorithm may be used to handle the problems of missing observations and censored observations.
4 citations
Book•
01 Jan 1988
TL;DR: In this article, Jacobsen et al. presented partial likelihood methods for counting processes and their applications to statistical inference. But they did not consider the problem of estimating the branching process with immigration.
Abstract: Partially specified semimartingale experiments by P. E. Greenwood Censoring, truncation, and filtering in statistical models based on counting processes by P. K. Andersen, O. Borgan, R. D. Gill, and N. Keiding Right censoring and the Kaplan-Meier and Nelson-Aalen estimators. Summary of results by M. Jacobsen Partial likelihood: applications, ramifications, generalizations by D. Oakes Multiple regression with integrated time series by P. C. B. Phillips Analysis of grouped duration data by N. M. Kiefer Asymptotic theory for weighted least squares estimators in Aalen's additive risk model by I. W. McKeague Some applications in statistics of semimartingale weak convergence theorems by M. J. Phelan Censoring, martingales and the Cox model by Y. Ritov and J. A. Wellner Composite likelihood methods by B. G. Lindsay Fixed sample and asymptotic optimality for classes of estimating functions by C. C. Heyde Statistical inference from sampled data for stochastic processes by B. L. S. P. Rao Optimal properties of SPRT for some stochastic processes by B. R. Bhat Estimation theory for the branching process with immigration by J. Winnicki A sequential approach for reducing curved exponential families of stochastic processes to noncurved exponential ones by V. T. Stefanov Palm distributions of point processes and their applications to statistical inference by A. F. Karr The mathematical structure of error correction models by S. Johansen.
2 citations
TL;DR: In this paper, a ratio-limit comparison between ξt, the solution of an SDE driven by a semimartingale, and Ht, an associated ODE, is proved on the set where limt→∞ξt=∞.