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Probability Measure (probability + measure)
Selected AbstractsASSET PRICING WITH NO EXOGENOUS PROBABILITY MEASUREMATHEMATICAL FINANCE, Issue 1 2008Gianluca Cassese In this paper, we propose a model of financial markets in which agents have limited ability to trade and no probability is given from the outset. In the absence of arbitrage opportunities, assets are priced according to a probability measure that lacks countable additivity. Despite finite additivity, we obtain an explicit representation of the expected value with respect to the pricing measure, based on some new results on finitely additive measures. From this representation we derive an exact decomposition of the risk premium as the sum of the correlation of returns with the market price of risk and an additional term, the purely finitely additive premium, related to the jumps of the return process. We also discuss the implications of the absence of free lunches. [source] A General Formula for Valuing Defaultable SecuritiesECONOMETRICA, Issue 5 2004P. Collin-Dufresne Previous research has shown that under a suitable no-jump condition, the price of a defaultable security is equal to its risk-neutral expected discounted cash flows if a modified discount rate is introduced to account for the possibility of default. Below, we generalize this result by demonstrating that one can always value defaultable claims using expected risk-adjusted discounting provided that the expectation is taken under a slightly modified probability measure. This new probability measure puts zero probability on paths where default occurs prior to the maturity, and is thus only absolutely continuous with respect to the risk-neutral probability measure. After establishing the general result and discussing its relation with the existing literature, we investigate several examples for which the no-jump condition fails. Each example illustrates the power of our general formula by providing simple analytic solutions for the prices of defaultable securities. [source] Cautious hierarchical switching control of stochastic linear systemsINTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 4 2004M.C. Campi Abstract Standard switching control methods are based on the certainty equivalence philosophy in that, at each switching time, the supervisor selects the candidate controller that is better tuned to the currently estimated process model. In this paper, we propose a new supervisory switching logic that takes into account the uncertainty on the process description when performing the controller selection. Specifically, a probability measure describing the likelihood of the different models is computed on-line based on the collected data and, at each switching time, the supervisor selects the candidate controller that, according to this probability measure, performs the best on the average. If the candidate controller class is hierarchically structured so that for each model one has available several controllers with distinct levels of robustness, the supervisor automatically selects the controller that suitably compromises robustness versus performance, given the current level of model uncertainty. The use of randomized algorithms makes the supervisor implementation computationally tractable. Copyright © 2004 John Wiley & Sons, Ltd. [source] Semiparametric Bayesian classification with longitudinal markersJOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES C (APPLIED STATISTICS), Issue 2 2007Rolando De la Cruz-Mesía Summary., We analyse data from a study involving 173 pregnant women. The data are observed values of the , human chorionic gonadotropin hormone measured during the first 80 days of gestational age, including from one up to six longitudinal responses for each woman. The main objective in this study is to predict normal versus abnormal pregnancy outcomes from data that are available at the early stages of pregnancy. We achieve the desired classification with a semiparametric hierarchical model. Specifically, we consider a Dirichlet process mixture prior for the distribution of the random effects in each group. The unknown random-effects distributions are allowed to vary across groups but are made dependent by using a design vector to select different features of a single underlying random probability measure. The resulting model is an extension of the dependent Dirichlet process model, with an additional probability model for group classification. The model is shown to perform better than an alternative model which is based on independent Dirichlet processes for the groups. Relevant posterior distributions are summarized by using Markov chain Monte Carlo methods. [source] ASSET PRICING WITH NO EXOGENOUS PROBABILITY MEASUREMATHEMATICAL FINANCE, Issue 1 2008Gianluca Cassese In this paper, we propose a model of financial markets in which agents have limited ability to trade and no probability is given from the outset. In the absence of arbitrage opportunities, assets are priced according to a probability measure that lacks countable additivity. Despite finite additivity, we obtain an explicit representation of the expected value with respect to the pricing measure, based on some new results on finitely additive measures. From this representation we derive an exact decomposition of the risk premium as the sum of the correlation of returns with the market price of risk and an additional term, the purely finitely additive premium, related to the jumps of the return process. We also discuss the implications of the absence of free lunches. [source] Stability of quantization dimension and quantization for homogeneous Cantor measuresMATHEMATISCHE NACHRICHTEN, Issue 8 2007Marc Kesseböhmer Abstract We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Limits of zeros of orthogonal polynomials on the circleMATHEMATISCHE NACHRICHTEN, Issue 12-13 2005Barry Simon Abstract We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a result of Alfaro and Vigil (which answered a question of P. Turán): namely, for n < N , one can freely prescribe the n -th polynomial and N , n zeros of the N -th one. We shall also describe all possible limit sets of zeros within the unit disk. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Slow mixing of Glauber dynamics for the hard-core model on regular bipartite graphsRANDOM STRUCTURES AND ALGORITHMS, Issue 4 2006David Galvin Abstract Let , = (V,E) be a finite, d -regular bipartite graph. For any , > 0 let ,, be the probability measure on the independent sets of , in which the set I is chosen with probability proportional to ,|I| (,, is the hard-core measure with activity , on ,). We study the Glauber dynamics, or single-site update Markov chain, whose stationary distribution is ,,. We show that when , is large enough (as a function of d and the expansion of subsets of single-parity of V) then the convergence to stationarity is exponentially slow in |V(,)|. In particular, if , is the d -dimensional hypercube {0,1}d we show that for values of , tending to 0 as d grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006 [source] Reversible coagulation,fragmentation processes and random combinatorial structures: Asymptotics for the number of groupsRANDOM STRUCTURES AND ALGORITHMS, Issue 2 2004Michael M. Erlihson Abstract The equilibrium distribution of a reversible coagulation-fragmentation process (CFP) and the joint distribution of components of a random combinatorial structure (RCS) are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (= components) in the case a(k) = qkp,1, k , 1, q, p > 0, where a(k), k , 1, is the parameter function that induces the invariant measure. The result obtained is compared with the ones for logarithmic RCS's and for RCS's, corresponding to the case p < 0. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004 [source] Bayesian Nonparametric Estimation of Continuous Monotone Functions with Applications to Dose,Response AnalysisBIOMETRICS, Issue 1 2009Björn Bornkamp Summary In this article, we consider monotone nonparametric regression in a Bayesian framework. The monotone function is modeled as a mixture of shifted and scaled parametric probability distribution functions, and a general random probability measure is assumed as the prior for the mixing distribution. We investigate the choice of the underlying parametric distribution function and find that the two-sided power distribution function is well suited both from a computational and mathematical point of view. The model is motivated by traditional nonlinear models for dose,response analysis, and provides possibilities to elicitate informative prior distributions on different aspects of the curve. The method is compared with other recent approaches to monotone nonparametric regression in a simulation study and is illustrated on a data set from dose,response analysis. [source] Hybrid Dirichlet mixture models for functional dataJOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 4 2009Sonia Petrone Summary., In functional data analysis, curves or surfaces are observed, up to measurement error, at a finite set of locations, for, say, a sample of n individuals. Often, the curves are homogeneous, except perhaps for individual-specific regions that provide heterogeneous behaviour (e.g. ,damaged' areas of irregular shape on an otherwise smooth surface). Motivated by applications with functional data of this nature, we propose a Bayesian mixture model, with the aim of dimension reduction, by representing the sample of n curves through a smaller set of canonical curves. We propose a novel prior on the space of probability measures for a random curve which extends the popular Dirichlet priors by allowing local clustering: non-homogeneous portions of a curve can be allocated to different clusters and the n individual curves can be represented as recombinations (hybrids) of a few canonical curves. More precisely, the prior proposed envisions a conceptual hidden factor with k -levels that acts locally on each curve. We discuss several models incorporating this prior and illustrate its performance with simulated and real data sets. We examine theoretical properties of the proposed finite hybrid Dirichlet mixtures, specifically, their behaviour as the number of the mixture components goes to , and their connection with Dirichlet process mixtures. [source] CASH SUBADDITIVE RISK MEASURES AND INTEREST RATE AMBIGUITYMATHEMATICAL FINANCE, Issue 4 2009Nicole El Karoui A new class of risk measures called cash subadditive risk measures is introduced to assess the risk of future financial, nonfinancial, and insurance positions. The debated cash additive axiom is relaxed into the cash subadditive axiom to preserve the original difference between the numéraire of the current reserve amounts and future positions. Consequently, cash subadditive risk measures can model stochastic and/or ambiguous interest rates or defaultable contingent claims. Practical examples are presented, and in such contexts cash additive risk measures cannot be used. Several representations of the cash subadditive risk measures are provided. The new risk measures are characterized by penalty functions defined on a set of sublinear probability measures and can be represented using penalty functions associated with cash additive risk measures defined on some extended spaces. The issue of the optimal risk transfer is studied in the new framework using inf-convolution techniques. Examples of dynamic cash subadditive risk measures are provided via BSDEs where the generator can locally depend on the level of the cash subadditive risk measure. [source] RISK MEASURES ON ORLICZ HEARTSMATHEMATICAL FINANCE, Issue 2 2009Patrick Cheridito Coherent, convex, and monetary risk measures were introduced in a setup where uncertain outcomes are modeled by bounded random variables. In this paper, we study such risk measures on Orlicz hearts. This includes coherent, convex, and monetary risk measures on Lp -spaces for 1 ,p < , and covers a wide range of interesting examples. Moreover, it allows for an elegant duality theory. We prove that every coherent or convex monetary risk measure on an Orlicz heart which is real-valued on a set with non-empty algebraic interior is real-valued on the whole space and admits a robust representation as maximal penalized expectation with respect to different probability measures. We also show that penalty functions of such risk measures have to satisfy a certain growth condition and that our risk measures are Luxemburg-norm Lipschitz-continuous in the coherent case and locally Luxemburg-norm Lipschitz-continuous in the convex monetary case. In the second part of the paper we investigate cash-additive hulls of transformed Luxemburg-norms and expected transformed losses. They provide two general classes of coherent and convex monetary risk measures that include many of the currently known examples as special cases. Explicit formulas for their robust representations and the maximizing probability measures are given. [source] COHERENT ACCEPTABILITY MEASURES IN MULTIPERIOD MODELSMATHEMATICAL FINANCE, Issue 4 2005Berend Roorda The framework of coherent risk measures has been introduced by Artzner et al. (1999; Math. Finance 9, 203,228) in a single-period setting. Here, we investigate a similar framework in a multiperiod context. We add an axiom of dynamic consistency to the standard coherence axioms, and obtain a representation theorem in terms of collections of multiperiod probability measures that satisfy a certain product property. This theorem is similar to results obtained by Epstein and Schneider (2003; J. Econ. Theor. 113, 1,31) and Wang (2003; J. Econ. Theor. 108, 286,321) in a different axiomatic framework. We then apply our representation result to the pricing of derivatives in incomplete markets, extending results by Carr, Geman, and Madan (2001; J. Financial Econ. 32, 131,167) to the multiperiod case. We present recursive formulas for the computation of price bounds and corresponding optimal hedges. When no shortselling constraints are present, we obtain a recursive formula for price bounds in terms of martingale measures. [source] Weighted isoperimetric inequalities on ,n and applications to rearrangementsMATHEMATISCHE NACHRICHTEN, Issue 4 2008M. Francesca Betta Abstract We study isoperimetric inequalities for a certain class of probability measures on ,n together with applications to integral inequalities for weighted rearrangements. Furthermore, we compare the solution to a linear elliptic problem with the solution to some "rearranged" problem defined in the domain {x: x1 < , (x2, ,, xn)} with a suitable function , (x2, ,, xn). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] estim 1.0: a computer program to infer population parameters from one- and two-locus gene identity probabilitiesMOLECULAR ECOLOGY RESOURCES, Issue 4 2001R. Vitalis Abstract Estimating effective population size is an important issue in population and conservation genetics. Recently, we proposed a new method to infer effective size and migration rate from one- and two-locus identity probability measures. We now announce the release of a user-friendly Microsoft® Windows program that uses this method to provide joint estimates of local effective population size and immigration rate for each subpopulation in a population genetics data set. [source] Negative correlation and log-concavityRANDOM STRUCTURES AND ALGORITHMS, Issue 3 2010J. Kahn Abstract We give counterexamples and a few positive results related to several conjectures of R. Pemantle (Pemantle, J Math Phys 41 (2000), 1371,1390) and D. Wagner (Wagner, Ann Combin 12 (2008), 211,239) concerning negative correlation and log-concavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. (Borcea et al., J Am Math Soc 22 (2009), 521,567). We also give short proofs of a pair of results from (Pemantle, J Math Phys 41 (2000), 1371,1390) and (Borcea et al., J Am Math Soc 22 (2009), 521,567); prove that "almost exchangeable" measures satisfy the "Feder-Mihail" property, thus providing a "non-obvious" example of a class of measures for which this important property can be shown to hold; and mention some further questions. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010 [source] Negative association in uniform forests and connected graphsRANDOM STRUCTURES AND ALGORITHMS, Issue 4 2004G. R. Grimmett Abstract We consider three probability measures on subsets of edges of a given finite graph G, namely, those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a uniform forest, and a related conjecture is posed for a uniform connected subgraph. The former conjecture is verified numerically for all graphs G having eight or fewer vertices, or having nine vertices and no more than 18 edges, using a certain computer algorithm which is summarized in this paper. Negative association is known already to be valid for a uniform spanning tree. The three cases of uniform forest, uniform spanning tree, and uniform connected subgraph are special cases of a more general conjecture arising from the random-cluster model of statistical mechanics. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004 [source] Pricing VIX futures: Evidence from integrated physical and risk-neutral probability measuresTHE JOURNAL OF FUTURES MARKETS, Issue 12 2007Yueh-Neng Lin This study derives closed-form solutions to the fair value of VIX (volatility index) futures under alternate stochastic variance models with simultaneous jumps both in the asset price and variance processes. Model parameters are estimated using an integrated analysis of integrated volatility and VIX time series from April 21, 2004 to April 18, 2006. The stochastic volatility model with price jumps outperforms for the short-dated futures, whereas additionally including a state-dependent volatility jump can further reduce out-of-sample pricing errors for other futures maturities. Finally, adding volatility jumps enhances hedging performance except for the short-dated futures on a daily-rebalanced basis. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:1175,1217, 2007 [source] Heat flow on Finsler manifoldsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2009Shin-ichi Ohta This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM , ,+ on each tangent space. Mostly, we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: as gradient flow on L2(M, m) for the energy as gradient flow on the reverse L2 -Wasserstein space ,,2(M) of probability measures on M for the relative entropy Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove ,,1, , regularity for solutions to the (nonlinear) heat equation on the Finsler space (M, F, m). Typically solutions to the heat equation will not be ,,2. Moreover, we derive pointwise comparison results à la Cheeger-Yau and integrated upper Gaussian estimates à la Davies. © 2008 Wiley Periodicals, Inc. [source] | ||||||||||||||||