Esquível, Manuel L.; Gaspar, Raquel M

Sousa, J. Beleza; Esquível, Manuel L.; Gaspar, Raquel M. £Pulled-to-par returns for zero-coupon bonds historical simulation value at risk. (English) ¢Zbl 1444.91214 ¡ J. Stat. Theory Pract. 14, No. 2, Paper No. 30, 17 p. (2020).

Summary: Due to bond prices pull-to-par, zero-coupon bond historical returns are not stationary, as they tend to zero as time to maturity approaches. Given that the historical simulation method for computing value at risk (VaR) requires a stationary sequence of historical returns, zero-coupon bonds’ historical returns cannot be used to compute VaR by historical simulation. Their use would systematically overes- timate VaR, resulting in invalid VaR sequences. In this paper, we propose an adjustment of zero-coupon bonds’ historical returns. We call the adjusted returns “pulled-to-par” returns. We prove that when the zero-coupon bonds’ continuously compounded yields-to-maturity are stationary, the adjusted pulled-to- par returns allow VaR computation by historical simulation. We ﬁrstly illustrate the VaR computation in a simulation scenario, and then, we apply it to real data on eurozone STRIPS.

MSC: 91G20 Derivative securities (option pricing, hedging, etc.) 91G70 Statistical methods; risk measures

Keywords: historical simulation; value at risk; zero-coupon bond

Software: Dowd; QRM

Full Text: DOI

References: [1] International convergence of capital measurement and capital standards a revised framework comprehensive version (2006), Basel: Bank for International Settlements Bank for International Settlements, Basel [2] STANDARDS Minimum capital requirements for market risk (2016), Basel: Bank for International Settlements, Basel [3] Mehta, A.; Neukirchen, M.; Pfetsch, S.; Poppensieker, T., Managing market risk: today and tomorrow. McKinsey working papers on risk, 32 (2012), London: McKinsey \& Company, London [4] Abad, P.; Benito, S.; López, C., A comprehensive review of value at risk methodologies, Span Rev Financ Econ, 12, 1, 15-32 (2014) [5] Fabozzi, FJ; Choudhry, M., The handbook of European fixed income securities (2004), Berlin: Wiley, Berlin [6] Björk, T., Arbitrage theory in continuous time (2004), Oxford: Oxford University Press, Oxford · Zbl 1140.91038 [7] Sousa JB, Esquıvel ML, Gaspar RM, Real PC (2014) Historical VaR for bonds—a new approach. In: Coelho L, Peixinho R (eds) Proceedings of the 8th finance conference of the Portuguese finance network, pp 1951-1970 [8] Dowd, K., Measuring market risk (2007), Berlin: Wiley, Berlin [9] McNeil, AJ; Frey, R.; Embrechts, P., Quantitative risk management: concepts, techniques, and tools (2005), Princeton: Princeton University Press, Princeton · Zbl 1089.91037 [10] Papoulis, A., Probability, random variables, and stochastic processes (1984), London: McGraw-Hill, London · Zbl 0191.46704 [11] Christoffersen, PF, Evaluating interval forecasts, Int Econ Rev, 39, 841-862 (1998) [12] Daníelsson, J., Financial risk forecasting: the theory and practice of forecasting market risk, with implementation in R and matlab (2015), Berlin: Wiley, Berlin [13] Afonso, A.; Rault, C., Short-and long-run behaviour of long-term sovereign bond yields, Appl Econ, 47, 37, 3971-3993 (2015) [14] Alexander, C., Market risk analysis, value at risk models (2009), Berlin: Wiley, Berlin This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original

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Esquível, M. L.; Guerreiro, G. R.; Fernandes,£ J. M. Open Markov chain scheme models. (English) ¢Zbl 1370.60116 ¡ REVSTAT 15, No. 2, 277-297 (2017).

Summary: We introduce a schematic formalism for the time evolution of a random open population divided into classes. With a Markov chain model, allowing for population entrances, we consider the ﬂow of incoming members modeled by a time series – either ARIMA for the number of new incomings or SARMA for the residuals of a deterministic sigmoid type trend – and we detail the time series structure of the elements in each class. A practical application to real data from a credit portfolio is presented.

MSC: 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P05 Applications of statistics to actuarial sciences and ﬁnancial mathematics

Keywords: Markov chains; open Markov chain models; second order processes; ARIMA; SARMA; credit risk

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Mota, Pedro P.; Esquível, Manuel L. £ Model selection for stock prices data. (English) ¢Zbl 07281661 ¡ J. Appl. Stat. 43, No. 16, 2977-2987 (2016).

Summary: The geometric Brownian motion (GBM) is very popular in modeling the dynamics of stock prices. However, the constant volatility assumption is questionable and many models with nonconstant volatility have been developed. In the papers [“On some auto-induced regime switching double-threshold glued diffusions”, J. Stat. Theory Pract. 8, No. 4, 760–771 (2014; doi:10.1080/15598608.2013.854184); Quant. Finance 14, No. 8, 1479–1488 (2014; Zbl 1402.91806)] the authors introduce a regime switching process where in each regime the process is driven by GBM and the change in regime is defined by the crossing of a threshold. In this paper we used Akaike’s and Bayesian information criteria to show that the GBM with regimes provides a better fit than the GBM. We also perform a forecasting comparison of the models for two selected companies.

MSC: 60J60 Diﬀusion processes 62B10 Statistical aspects of information-theoretic topics 62F10 Point estimation 62P05 Applications of statistics to actuarial sciences and ﬁnancial mathematics 91B25 Asset pricing models (MSC2010)

Keywords: geometric Brownian motion; maximum likelihood estimator; regimes

Full Text: DOI

References: [1] H. Akaike, \textit{Information theory and an extension of the maximum likelihood principle}, in \textit{Proceedings of the 2nd International Symposium on Information Theory}, B.N. Petrov and F. Csaki, eds., Akademiai Kiado, Budapest, 1973, pp. 267-281. [2] H. Akaike, \textit{A new look at the statistical model identiﬁcation}, IEEE Trans. Autom. Control 19 (1974), pp. 716-723. doi: 10.1109/TAC.1974.1100705 · Zbl 0314.62039 [3] T. Bjork, \textit{Arbitrage Theory in Continuous Time}, Oxford University Press, New York, 1998.

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 2 [4] F. Black and M. Scholes, \textit{The pricing of options and corporate liabilities}, J. Political Econ. 81 (1973), pp. 637-654. doi: 10.1086/260062 · Zbl 1092.91524 [5] K.P. Burnham and D.R. Anderson, \textit{Model Selection and Inference: A Practical Information - Theoretical Approach}, 2nd ed., Springer, New York, 2002. [6] J.E. Cavanaugh and A.A. Neath, \textit{Generalizing the derivation of the Schwarz information criterion}, Commun. Stat - Theory Methods 28 (1999), pp. 49-66. doi: 10.1080/03610929908832282 · Zbl 1083.62506 [7] M.L. Esquível and P.P. Mota, \textit{On some auto-induced regime switching double-threshold glued diffusions}, J. Stat. Theory Pract. 8 (2014), pp. 760-771. doi: 10.1080/15598608.2013.854184. [8] J. Hull and A. White, \textit{The pricing of options on assets with stochastic volatilities}, J. Financ. 42 (1987), pp. 281-300. doi: 10.1111/j.1540-6261.1987.tb02568.x [9] R.E. Kass and A.E. Raftery, \textit{Bayes factors}, J. Am. Stat. Assoc. 90 (1995), pp. 773-795. doi: 10.1080/01621459.1995.10476572 [10] S.G. Kou, \textit{A jump-diffusion model for option pricing}, Manage. Sci. 48 (2002), pp. 1086-1101. doi: 10.1287/mnsc.48.8.1086.166 · Zbl 1216.91039 [11] R.C. Merton, \textit{Option pricing when underlying stock returns are discontinuous}, J. Financ. Econom. 3 (1976), pp. 125-144. doi: 10.1016/0304-405X(76)90022-2 · Zbl 1131.91344 [12] P. P. Mota and M.L. Esquível, \textit{On a continuous time stock price model with regime switching, delay, and threshold}, Quant. Financ. 14 (2014), pp. 1479-1488. doi: 10.1080/14697688.2013.879990. · Zbl 1402.91806 [13] A.A. Neath and J.E. Cavanaugh, \textit{Regression and time series model selection using variants of the Schwarz information criterion}, Commun. Stat. - Theory Methods 26 (1997), pp. 559-580. doi: 10.1080/03610929708831934 · Zbl 1030.62532 [14] A.A. Neath and J.E. Cavanaugh, \textit{The Bayesian information criterion: Background, derivation, and applications}, WIREs Comput. Stat. 4 (2012), pp. 199-203. doi: 10.1002/wics.199 [15] B. Oksendal, \textit{Stochastic Differential Equations}, 5th ed., Springer, Berlin, 1998. [16] S.M. Ross, \textit{An Elementary Introduction to Mathematical Finance}, 3rd ed., Cambridge University Press, New York, 2011. · Zbl 1221.91001 [17] A. Sayyareh, R. Obeidi, and A. Bar-Hen, \textit{Empirical comparison between some model selection criteria}, Commun. Stat. - Simul. Comput. 40 (2010), pp. 72-86. doi: 10.1080/03610918.2010.530367 · Zbl 1209.62109 [18] G. Schwarz, \textit{Estimating the dimension of a model}, Ann. Stat. 6 (1978), pp. 461-464. doi: 10.1214/aos/1176344136 · Zbl 0379.62005 [19] P. Zhang, \textit{On the convergence rate of model selection criteria}, Commun. Stat. - Theory Methods 22 (1993), pp. 2765-2775. doi: 10.1080/03610929308831184 · Zbl 0785.62073 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

Esquível, Manuel L.; Mota, Pedro P.; Mexia, João Tiago £ On some statistical models with a random number of observations. (English) ¢Zbl 1426.62103 ¡ J. Stat. Theory Pract. 10, No. 4, 805-823 (2016).

Summary: We extend some classical statistical inference to the case of a random number of observations with a stabilized distribution: namely, in the normal model, inference for the mean with known and unknown variance and inference for the variance. We describe some useful models for the number of observations obtained by truncation or translation of usual models given by integer-valued random variables: Poisson, binomial, geometric, and negative binomial. We present an eﬃcient random search algorithm for the computation of the quantiles of the relevant statistics, we describe an interval estimation procedure for the extended model, and we propose a parametric bootstrap simulation study to validate the proposed procedure.

MSC: 62G05 Nonparametric estimation 62G09 Nonparametric statistical resampling methods 62G08 Nonparametric regression and quantile regression

Keywords: statistical inference; random number of observations; interval estimation; hypothesis tests power function

Software: fastR

References: [1] Barra, J.-R. 1981. \textit{Mathematical basis of statistics}, transl. and ed. L. Herbach. New York: Academic Press. · Zbl 0523.62001 [2] Barsotti, F.; Philippe, A.; Rochet, P., Hypothesis testing for markovian models with random time observations, Journal of Statistical Planning and Inference, 173, 87-98, (2016) · Zbl 1332.62276 [3] Bunge, J. A.; Nagaraja, H. N., The distributions of certain record statistics from a random number of observations, Stochastic Processes and Their Applications, 38, 167-83, (1991) · Zbl 0734.62018 [4] Capistrano, G.; Nunes, C.; Ferreira, D.; Ferreira, S. S.; Mexia, J. T., One-way random effects ANOVA with random sample sizes: An application to a brazilian database on cancer registries, AIP Conference Proceedings, 1648, 110009, (2015) [5] Casella, G., and R. L. Berger. 2002. \textit{Statistical inference}, 2nd ed. Pacific Grove: Brooks/Cole Cengage Learning. · Zbl 0699.62001 [6] Esquível, M. L., Probability generating functions for discrete real-valued random variables, Theory of Probability and Its Applications, 52, 40-57, (2008) · Zbl 1147.60010 [7] Ivchenko, G.I., and Y. I. Medvedev. 1992. \textit{Mathematical statistics}, 2nd ed. Moscow, USSR: Vysshaya Shkola. · Zbl 0875.62001 [8] Korolev, V. Y.; Zeifman, A. I., On normal variance-mean mixtures as limit laws for statistics with random sample sizes, Journal of Statistical Planning and Inference, 169, 34-42, (2016) · Zbl 1329.60035 [9] Lin’kov, Y. N. 2005. \textit{Lectures in mathematical statistics. Parts 1 and 2}, trans. O. Klesov and V. Zayats. Providence, RI: American Mathematical Society. [10] Mexia, J. T.; Moreira, E. E., Randomized sample size f tests for the oneway layout, AIP Conference Proceedings, 1281, 1248-1251, (2010) [11] Mexia, J. T.; Nunes, C.; Ferreira, D.; Ferreira, S. S.; Moreira, E., Orthogonal fixed effects ANOVA with random sample sizes, 84-90, (2011), Stevens Point, WI [12] Moreira, E. E.; Mexia, J. T.; Minder, C. E., F tests with random sample sizes. Theory and applications, Statistics \& Probability Letters, 83, 1520-1526, (2013) · Zbl 1278.62173 [13] Nguyen Bac Van, On the statistical analysis of a random number of observations, Acta Mathematica Vietnamica, 13, 55-61, (1988) · Zbl 0683.62054 [14] Nunes, C.; Ferreira, D.; Ferreira, S. S.; Mexia, J. T., F tests with random sample sizes, AIP Conference Proceedings, 1281, 1241-1244, (2010) [15] Nunes, C.; Ferreira, D.; Ferreira, S. S.; Mexia, J. T., Generalized \(f\) distributions with random non-centrality parameters: convolution of gamma and beta variables, Far East Journal of Mathematical Sciences, 62, 1-14, (2012) · Zbl 1254.62013 [16] Nunes, C.; Capistrano, G.; Ferreira, D.; Ferreira, S. S., ANOVA with random sample sizes: An application to a Brazilian database on cancer registries, AIP Conference Proceedings, 1558, 825-828, (2013) [17] Nunes, C.; Ferreira, D.; Ferreira, S. S.; Mexia, J. T., Fixed effects ANOVA: An extension to samples with random size, Journal of Statistical Computation and Simulation, 84, 2316-2328, (2014) [18] Nunes, C.; Capistrano, G.; Ferreira, D.; Ferreira, S. S.; Mexia, J. T., One-way fixed effects ANOVA with missing observations, AIP Conference Proceedings, 1648, 110008, (2015) [19] Nunes, C.; Ferreira, D.; Ferreira, S. S.; Mexia, J. T., F-tests with a rare pathology, Journal of Applied Statistics, 39, 551-561, (2012) [20] Nunes, C.; Ferreira, D.; Ferreira, S. S.; Mexia, J. T., Control of the truncation errors for generalized F distributions, Journal of Statistical Computation and Simulation, 82, 165-171, (2012) · Zbl 1431.62226 [21] Nunes, C., D. Ferreira, S. S. Ferreira, M. M. Oliveira, and J. T. Mexia. 2012d. One-way random effects ANOVA: An extension to samples with random size. In \textit{Numerical analysis and applied mathematics (ICNAAM 2012) Vols A and B}, vol. 1479 of \textit{AIP Conference Proceedings}, ed T.E. Simos, G. Psihoyios, C. Tsitouras, and Z. Anastassi, 1678-81. American Institute of Physics. [22] Pestman, W. R. 1998. \textit{Mathematical statistics. An introduction}. Berlin, Germany: de Gruyter. · Zbl 0911.62001 [23] Pruim, R. 2011. \textit{Foundations and applications of statistics. An introduction using R}. Providence, RI: American Mathematical Society. · Zbl 1304.62013 [24] Shiryaev, A. N. 1973. \textit{Statistical sequential analysis: Optimal stopping rules}. Providence, RI: American Mathematical Society. · Zbl 0267.62039 [25] Singh, M.; Gupta, V. K., On the estimation of mean parameter with random number of observations on each unit, Biometrical Journal, 22, 41-45, (1980) · Zbl 0457.62012 [26] Weerahandi, S., Generalized confidence intervals, Journal of the American Statistical Association, 88, 899-905, (1993) · Zbl 0785.62029 [27] Weerahandi, S. 1996. \textit{Exact statistical methods for data analysis}, 2nd printing. Berlin, Germany: Springer-Verlag. · Zbl 0912.62002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 4 paper as accurately as possible without claiming the completeness or perfect precision of the matching.

Esquível, Manuel L.; Lita Da Silva, João; Tiago Mexia, João; Ramos, Luís The rate of convergence of some asymptotic£ expansions for distribution approximations via an Esseen type estimate. (English) ¢Zbl 06599049 ¡ Commun. Stat., Theory Methods 43, No. 2, 266-290 (2014).

Summary: Some asymptotic expansions not necessarily related to the central limit theorem are studied. We first observe that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation. We then present several instances of this observation. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to (g(X+µn))n∈N, where g is some smooth function, X is a random variable and (µn)n∈N is a sequence going to infinity; a multivariate version is also stated and proved. We finally present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas; namely, a generic Laplace’s type integral, randomized by the sequence (µnX)n∈N, X being a Gamma distributed random variable.

MSC: 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 60B10 Convergence of probability measures 60F05 Central limit and other weak theorems

Keywords: high precision measurements; Kolmogorov distance; rate of convergence; uniform asymptotic weak approximations

Full Text: DOI

References: [1] DOI: 10.1016/j.stamet.2007.09.001 · Zbl 1248.62233 · doi:10.1016/j.stamet.2007.09.001 [2] DOI: 10.1017/S1446788700008570 · Zbl 1071.62016 · doi:10.1017/S1446788700008570 [3] Bhattacharya R., Normal Approximation and Asymptotic Expansions (1976) · Zbl 0331.41023 [4] Bingham N. H., Regular Variation 27 (1989) · Zbl 0667.26003 [5] DOI: 10.1051/aas:1998221 · doi:10.1051/aas:1998221 [6] Cramer H., Mathematical Methods of Statisitics (1999) [7] DOI: 10.1002/mana.19911530124 · Zbl 0795.60022 · doi:10.1002/mana.19911530124 [8] De Bruijn N. G., Asymptotic Methods in Analysis (1981) · Zbl 0556.41021 [9] Edgeworth F. Y., Trans. Cambridge Philos. Soc. 20 (35) pp 113– (1905) [10] Esquível M. L., Proc. 6th St. Petersburg Workshop Simul. pp 444– (2009) [11] DOI: 10.1007/BF02392223 · Zbl 0060.28705 · doi:10.1007/BF02392223 [12] Evans L. C., Measure Theory and Fine Properties of Functions (1992) · Zbl 0804.28001 [13] Federer H., Geometric Measure Theory (1969) · Zbl 0176.00801 [14] DOI: 10.1063/1.3497920 · doi:10.1063/1.3497920 [15] Feller W., An Introduction to Probability Theory and its Applications 2 (1971) · Zbl 0219.60003 [16] DOI: 10.1214/aos/1176347259 · Zbl 0681.62033 · doi:10.1214/aos/1176347259 [17] DOI: 10.1016/0047-259X(89)90040-7 · Zbl 0676.62020 · doi:10.1016/0047-259X(89)90040-7 [18] Fujikoshi Y., Sugaku Expositions 3 (1) pp 75– (1990) [19] DOI: 10.1016/j.jmva.2004.09.002 · Zbl 1074.62010 · doi:10.1016/j.jmva.2004.09.002 [20] DOI: 10.1090/S0002-9939-1973-0323963-5 · doi:10.1090/S0002-9939-1973-0323963-5 [21] DOI: 10.1007/s10959-005-7525-3 · Zbl 1089.60017 · doi:10.1007/s10959-005-7525-3 [22] Hoﬀmann-Jørgensen , J. ( 1992 ).Asymptotic Likelihood Theory.Functional Analysis III, Proc. Dubrovnik 1989, Matematisk Institut, Aarhus Universitet, Var. Publ. Ser. no 40, pp. 5–192 .

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 5 [23] DOI: 10.1007/978-1-4899-3019-4 · doi:10.1007/978-1-4899-3019-4 [24] Inzhevitov P. G., Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 2 pp 88– (1983) [25] Itô K., Encyclopedic Dictionary of Mathematics., 2. ed. (1993) · Zbl 0674.00025 [26] Kristof G., Litovsk. Mat. Sb. 22 (2) pp 137– (1982) [27] DOI: 10.1016/j.spl.2004.06.018 · Zbl 1069.60073 · doi:10.1016/j.spl.2004.06.018 [28] DOI: 10.1007/978-1-4612-4946-7 · doi:10.1007/978-1-4612-4946-7 [29] DOI: 10.1017/CBO9780511623813 · doi:10.1017/CBO9780511623813 [30] DOI: 10.1016/j.jspi.2009.09.014 · Zbl 1177.62019 · doi:10.1016/j.jspi.2009.09.014 [31] Oehlert G. W., Amer. Statistician 46 (1) pp 27– (1992) [32] Pestana , D. ( 2007 ). Personal communication . [33] Ramos , L. P. C. ( 2007 ). Quase-normalidade e Inferência para Séries de Estudos Emparelhadas. Ph.D.’s dissertation in Mathematics, speciality Statistics Faculty of Science and Technology, New University of Lisbon . [34] Ramos L. P. C., Far East J. Math. Sci. 68 (2) pp 287– (2012) [35] DOI: 10.1007/978-1-4757-2539-1 · doi:10.1007/978-1-4757-2539-1 [36] DOI: 10.1214/aoms/1177706528 · Zbl 0086.34004 · doi:10.1214/aoms/1177706528 [37] Zorich V. A., Mathematical Analysis II (2004) · Zbl 1071.00003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identiﬁers and may contain data conversion errors. It attempts to reﬂect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

Mota, Pedro P.; Esquível, Manuel L. £On a continuous time stock price model with regime switching, delay, and threshold. (English) ¢Zbl 1402.91806 ¡ Quant. Finance 14, No. 8, 1479-1488 (2014).

Summary: Motivated by the need to describe bear-bull market regime switching in stock prices, we introduce and study a stochastic process in continuous time with two regimes, threshold and delay, given by a stochastic differential equation. When the difference between the regimes is simply given by a different set of real valued parameters for the drift and diffusion coefficients, with changes between regimes depending only on these parameters, we show that if the delay is known there are consistent estimators for the threshold as long we know how to classify a given observation of the process as belonging to one of the two regimes. When the drift and diffusion coefficients are of geometric Brownian motion type we obtain a model with parameters that can be estimated in a satisfactory way, a model that allows differentiating regimes in some of the NYSE 21 stocks analyzed and also, that gives very satisfactory results when compared to the usual Black-Scholes model for pricing call options.

MSC: 91G20 Derivative securities (option pricing, hedging, etc.) Cited in 1 Review 60H10 Stochastic ordinary diﬀerential equations (aspects of stochastic analy- Cited in 6 Documents sis) 93E24 Least squares and related methods for stochastic control systems

Keywords: stock price model; option pricing; consistent estimator; continuous processes; delay; regime switching; threshold

Full Text: DOI

References: [1] DOI: 10.1214/aos/1176349040 · Zbl 0786.62089 · doi:10.1214/aos/1176349040 [2] DOI: 10.1093/biomet/85.2.413 · Zbl 0938.62089 · doi:10.1093/biomet/85.2.413 [3] DOI: 10.1080/15598608.2013.854184 · doi:10.1080/15598608.2013.854184 [4] Freidlin M., Finance Stoch. 36 pp 337– (1998) [5] DOI: 10.1007/PL00013523 · Zbl 0963.60037 · doi:10.1007/PL00013523

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 6 [6] DOI: 10.1007/978-1-4612-0949-2 · doi:10.1007/978-1-4612-0949-2 [7] Lamberton D., Introduction to Stochastic Calculus Applied to Finance, 2. ed. (2008) · Zbl 1167.60001 [8] Mota, P.P., Brownian motion with drift threshold model. PhD Dissertation, FCT/UNL, 2008. [9] Øksendal B., Stochastic Differential Equations (2007) [10] DOI: 10.1111/j.1467-9892.1986.tb00494.x · Zbl 0601.62110 · doi:10.1111/j.1467-9892.1986.tb00494.x [11] Tong H., Non-linear Time Series – A Dynamical System Approach (1990) · Zbl 0716.62085 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

Esquível, Manuel L.; Fernandes, José M.; Guerreiro, Gracinda R. On the evolution and asymptotic£ analysis of open Markov populations: application to consumption credit. (English) ¢Zbl 1320.60128 ¡ Stoch. Models 30, No. 3, 365-389 (2014).

Summary: In this paper, we study, by means of randomized sampling, the long-run stability of some open Markov population fed with time-dependent Poisson inputs. We show that the state probabilities within transient states converge – even when the overall expected population dimension increases without bound – under general conditions on the transition matrix and input intensities. Following the convergence results, we obtain ML estimators for a particular sequence of input intensities, where the sequence of new arrivals is modeled by a sigmoidal function. These estimators allow the forecast, by conﬁdence intervals, of the evolution of the relative population structure in the transient states. Applying these results to the study of a consumption credit portfolio, we estimate the implicit default rate.

MSC: 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 62F86 Parametric inference and fuzziness 91B70 Stochastic models in economics 91G40 Credit risk

Keywords: Markov chains; open populations; parameter inference; consumption credit

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References: [1] Bartholomew D. J., Stochastic Models for Social Processes, 3rd Ed. (1982) · Zbl 0578.92026 [2] DOI: 10.1016/S0167-6687(00)00082-2 · Zbl 1055.91021 · doi:10.1016/S0167-6687(00)00082-2 [3] Cramér H., Mathematical Methods of Statistics (1999) · Zbl 0985.62001 [4] Da Cunha D., Recueil de Problèmes de Calcul des Probabilités, 2nd Ed. (1970) [5] Da Cunha D., Probabilités et Statistiques. Tome 2 (1983) [6] Feller W., An Introduction to Probability Theory and Its Applications, 2nd Ed. (1960) · Zbl 0039.13201 [7] DOI: 10.2307/2982224 · doi:10.2307/2982224 [8] Guerreiro G.R., Discussiones Mathematicae, Probability and Statistics 24 (2) pp 197– (2004) [9] Guerreiro G.R., Discussiones Mathematicae, Probability and Statistics 28 (2) pp 209– (2008) [10] DOI: 10.1080/15598608.2010.10411988 · Zbl 1216.62135 · doi:10.1080/15598608.2010.10411988 [11] DOI: 10.1080/15598608.2012.719741 · doi:10.1080/15598608.2012.719741 [12] Guerreiro, G.R.; Mexia, J.T.; Miguens, M.F. Preliminary Results on Conﬁdence Intervals for Open Bonus Systems, Springer: Berlin, 199–206, 2014. [13] DOI: 10.1017/asb.2013.26 · Zbl 1284.62639 · doi:10.1017/asb.2013.26

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 7 [14] DOI: 10.2307/3212838 · Zbl 0336.60073 · doi:10.2307/3212838 [15] DOI: 10.2307/3213233 · Zbl 0374.60101 · doi:10.2307/3213233 [16] DOI: 10.2307/3212978 · Zbl 0435.60092 · doi:10.2307/3212978 [17] DOI: 10.2307/3213465 · Zbl 0372.60147 · doi:10.2307/3213465 [18] Mood A., Introduction to the Theory of Statistics, 3rd edition (1974) · Zbl 0277.62002 [19] DOI: 10.1093/biomet/53.3-4.397 · Zbl 0144.43901 · doi:10.1093/biomet/53.3-4.397 [20] DOI: 10.2307/3212315 · doi:10.2307/3212315 [21] DOI: 10.2307/2343757 · doi:10.2307/2343757 [22] DOI: 10.1111/j.1467-842X.1979.tb01149.x · Zbl 0425.62086 · doi:10.1111/j.1467-842X.1979.tb01149.x [23] DOI: 10.1080/03461238.1980.10408642 · Zbl 0431.62067 · doi:10.1080/03461238.1980.10408642 [24] Schott J.R., Matrix analysis for statistics, 2nd Ed. (2005) · Zbl 1076.15002 [25] DOI: 10.1239/jap/1032374251 · Zbl 0951.92019 · doi:10.1239/jap/1032374251 [26] DOI: 10.2307/3212231 · Zbl 0236.60069 · doi:10.2307/3212231 [27] DOI: 10.1002/(SICI)1099-0747(199806)14:2<165::AID-ASM344>3.0.CO;2-# · Zbl 0927.60072 · doi:10.1002/(SICI)1099-0747(199806)14:2<165::AID- ASM344>3.0.CO;2-# [28] de Oliveira J., Publications de l’Institut de Statistique de l’Univiversité de Paris 27 pp 49– (1982) [29] DOI: 10.2307/3213497 · Zbl 0484.60058 · doi:10.2307/3213497 [30] DOI: 10.2307/3213839 · Zbl 0498.60075 · doi:10.2307/3213839 [31] DOI: 10.2307/3214820 · Zbl 0718.60082 · doi:10.2307/3214820 [32] DOI: 10.1016/0024-3795(94)90470-7 · Zbl 0813.60084 · doi:10.1016/0024-3795(94)90470-7 [33] DOI: 10.1002/(SICI)1099-0747(199709/12)13:3/4<159::AID-ASM309>3.0.CO;2-Q · Zbl 0918.60062 · doi:10.1002/(SICI)1099- 0747(199709/12)13:3/4<159::AID-ASM309>3.0.CO;2-Q [34] Yakasai B.M., J. Nigerian Assoc. Math. Physics 9 pp 395– (2005) [35] Zorich V.A., Mathematical Analysis I (2009) · Zbl 1151.00002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identiﬁers and may contain data conversion errors. It attempts to reﬂect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

Esquível, Manuel L.; Dimas, Luís; Mexia, João Tiago; Didier, Philippe£ Small perturbations with large eﬀects on value-at-risk. (English) ¢Zbl 1315.60021 ¡ Discuss. Math., Probab. Stat. 33, No. 1-2, 151-169 (2013).

Summary: We show that in the delta-normal model there exist perturbations of the Gaussian multivariate distribution of the returns of a portfolio such that the initial marginal distributions of the returns are statistically undistinguishable from the perturbed ones and such that the perturbed value-at-risk (VR) is close to the worst possible VR which, under some reasonable assumptions, is the sum of the Vs of each of the portfolio assets.

MSC: 60E05 Probability distributions: general theory 91B30 Risk theory, insurance (MSC2010) 91G40 Credit risk

Keywords: Gaussian perturbation; value-at-risk; delta-normal model

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Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 8 Lita da Silva, João (ed.); Caeiro, Frederico (ed.); Natário, Isabel (ed.); Braumann, Carlos A. (ed.); Esquível, Manuel L. (ed.); Mexia, João Tiago (ed.) Advances in regression, survival analysis, extreme values, Markov processes and other statistical applications. Selected papers based on the presentations of the 17th congress of the £Portuguese Statistical Society, Sesimbra, Portugal, September 30–October 3, 2009. (English) ¢Zbl 1270.62010 ¡ Studies in Theoretical and Applied Statistics. Selected Papers of the Statistical Societies. Berlin: Springer (ISBN 978-3-642-34903-4/hbk; 978-3-642-34904-1/ebook). xvii, 471 p. (2013).

The articles of mathematical interest will be reviewed individually. Indexed articles: Mejza, Stanisław Franciszek; Kuriki, Shinji, Youden square with split units, 3-10 [Zbl 1402.62180] Vilares, Manuel J.; Coelho, Pedro S., Likelihood and PLS estimators for structural equation modeling: an assessment of sample size, skewness and model misspeciﬁcation eﬀects, 11-33 [Zbl 1402.62342]

MSC: 62-06 Proceedings, conferences, collections, etc. pertaining to statistics 60-06 Proceedings, conferences, collections, etc. pertaining to probability theory

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Sousa, J. Beleza; Esquível, M. L.; Gaspar, R. M. £Machine learning Vasicek model calibration with Gaussian processes. (English) ¢Zbl 1262.91162 ¡ Commun. Stat., Simulation Comput. 41, No. 6, 776-786 (2012).

Summary: We calibrate the Vasicek interest rate model under the risk neutral measure by learning the model parameters using Gaussian processes for machine learning regression. The calibration is done by maximizing the likelihood of zero coupon bond log prices, using mean and covariance functions computed analytically, as well as likelihood derivatives with respect to the parameters. The maximization method used is the conjugate gradients. The only prices needed for calibration are zero coupon bond prices and the parameters are directly obtained in the arbitrage free risk neutral measure.

MSC: 91G80 Financial applications of other theories Cited in 1 Document 91G30 Interest rates, asset pricing, etc. (stochastic models) 60G15 Gaussian processes 68T05 Learning and adaptive systems in artiﬁcial intelligence

Software: Mathematica

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References: [1] DOI: 10.1093/0199271267.001.0001 · Zbl 1140.91038 · doi:10.1093/0199271267.001.0001 [2] Brigo D., Interest Rate Models – Theory and Practice: With Smile, Inflation and Credit., 2. ed. (2006) · Zbl 1109.91023 [3] Rasmussen C. E., Advanced Lectures on Machine Learning: ML Summer Schools 2003, Revised Lectures (2004) [4] Rasmussen C. E., Gaussian Processes for Machine Learning (2005) [5] DOI: 10.1016/0304-405X(77)90016-2 · Zbl 1372.91113 · doi:10.1016/0304-405X(77)90016-2 [6] Wolfram Research , I. ( 2009 ).Mathematica Edition: Version7.01.0 . This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 9 Veiga, Carlos; Wystup, Uwe; Esquível, Manuel L.£ Unifying exotic option closed formulas. (English) ¢Zbl 1256.91056 ¡ Rev. Deriv. Res. 15, No. 2, 99-128 (2012).

Summary: This paper aims to unify exotic option closed formulas by generalizing a large class of existing formulas and by setting a framework that allows for further generalizations. The formula presented covers options from the plain vanilla to most, if not all, mountain range exotic options and is developed in a multi-asset, multi-currency Black–Scholes model with time dependent parameters. It particular, it focuses on payoffs that depend on the distributions of the underlyings prices at multiple but set time horizons. The general formula not only covers existing cases but also enables the combination of diverse features from different types of exotic options. It also creates implicitly a language to describe payoffs that can be used in industrial applications to decouple the functions of payoff definition from pricing functions. Examples of several exotic options are presented, benchmarking the closed formulas’ performance against Monte Carlo simulations. Results show a consistent over performance of the closed formula reducing calculation time by double digit factors.

MSC: 91G20 Derivative securities (option pricing, hedging, etc.) 91G60 Numerical methods (including Monte Carlo methods) 65C05 Monte Carlo methods

Keywords: exotic options; mountain range; discrete lookback; closedformula; payoﬀ language; multi-asset multi- currency model; C02

Software: QSIMVN

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References: [1] Björk T. (1998) Arbitrage theory in continuous time. Oxford University Press, New York · Zbl 1140.91038 [2] Björk, T., \& Landén, C. (2002). On the term structure of futures and forward prices. In Mathematical Finance Bachelier Congress 2000, pp. 111–150. · Zbl 1012.91016 [3] Carr P. (1995) Two extensions to barrier options valuation. Applied Mathematical Finance 2(3): 173–209 · doi:10.1080/13504869500000010 [4] Carr P. (2001) Deriving derivatives of derivative securities.. Journal of Computational Finance, 4(2): 5–29 [5] Conze A., Conze A. (1991) Path dependent options: The case of lookback options. Journal of Finance, 46(5): 1893–1907 · doi:10.1111/j.1540-6261.1991.tb04648.x [6] Dudley R. (2002) Real analysis and probability. Cambridge University Press, UK · Zbl 1023.60001 [7] Geman H., El Karoui N., Rochet J. C. (1995) Changes of Numéraire, changes of probability measure and option pricing. Journal of Applied Probability 32: 443–458 · Zbl 0829.90007 · doi:10.2307/3215299 [8] Genz A. (1992) Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statis- tics 1(2): 141–149 [9] Goldman M., Soson H., Gatto M. (1979) Path dependent options: Buy at the low, sell at the high. Journal of Finance 34(5): 1111–1127 [10] Harrison J., Kreps D. (1979) Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20: 381–408 · Zbl 0431.90019 · doi:10.1016/0022-0531(79)90043-7 [11] Harrison J., Pliska S. (1981) Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11: 215–260 · Zbl 0482.60097 · doi:10.1016/0304-4149(81)90026-0 [12] Hakala J., Wystup U. (2002) Foreign exchange risk. Risk Books, London [13] Haug E. (1998) The complete guide to option pricing formulas. McGraw-Hill Professional, New York [14] Heynen R., Kat H. (1994) Partial barrier options. Journal of Financial Engineering 3(3): 253–274 · Zbl 1153.91507 · doi:10.1007/BF02425804 [15] Hunt, P., \& Kennedy, J. (2004). Financial derivatives in theory and practice (revised ed.). England: Wiley · Zbl 1140.91045 [16] Johnson H. (1987) Options on the minimum and maximum of several assets. Journal of Financial and Quantitative Analysis 22: 277–283 · doi:10.2307/2330963

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 10 [17] Margrabe W. (1978) The value of an option to exchange one asset for another. Journal of Finance 33: 177–186 · doi:10.1111/j.1540- 6261.1978.tb03397.x [18] Nelken I. (1995) The handbook of exotic options: Instruments, analysis, and applications. McGraw-Hill Professional, New York [19] Reiß O., Wystup U.(2001). Efficient computation of option price sensitivities using homogeneity and other tricks. The Journal of Derivatives, 9(2), 41–53 (Winter). · doi:10.3905/jod.2001.319174 [20] Rubinstein M., Reiner E. (1991) Breaking down barriers. Risk Magazine 4(9): 28–35 [21] Shreve S. (2004) Stochastic calculus for finance II, continuous–Time models. Springer, New York · Zbl 1068.91041 [22] Stulz R. (1982) Options on the minimum and maximum of two risky assets. Journal of Financial Economics 10: 161–185 · doi:10.1016/0304-405X(82)90011-3 [23] Večeř J. (2001) A new PDE approach for pricing arithmetic average Asian options. Journal of Computational Finance 4(4): 105–113 [24] Veiga, C. (2004). Expanding further the universe of exotic options closed pricing formulas. Proceedings of the 2004 international conference on stochastic finance. [25] Veiga C., Wystup U. (2009) Closed formula for options with discrete dividends and its derivatives. Applied Mathematical Finance 16(6): 517–531 · Zbl 1186.91220 · doi:10.1080/13504860903075498 [26] Vorst T. (1992) Prices and hedge ratios of average exchange rate options. International Review of Financial Analysis 1: 179–193 · doi:10.1016/1057-5219(92)90003-M [27] Wystup U. (2003) The market price of one-touch options in foreign exchange markets. Derivatives Week 12(13): 8–9 [28] Zhang P. (1997) Exotic options. World Scientific, Singapore · Zbl 0934.91030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

Esquível, Manuel L. Some applications£ of probability generating function based methods to statistical estimation. (English) ¢Zbl 1208.62028 ¡ Discuss. Math., Probab. Stat. 29, No. 2, 131-153 (2009).

Summary: After recalling previous work on probability generating functions for real valued random variables we extend to these random variables uniform laws of large numbers and functional limit theorem for the empirical probability generating function. We present an application to the study of continuous laws, namely, estimation of parameters of Gaussian, gamma and uniform laws by means of a minimum contrast estimator that uses the empirical probability generating function of the sample. We test the procedure by simulation and we prove the consistency of the estimator.

MSC: 62F10 Point estimation 60E10 Characteristic functions; other transforms 62G05 Nonparametric estimation 60F15 Strong limit theorems 60F17 Functional limit theorems; invariance principles 65C60 Computational problems in statistics (MSC2010)

Keywords: empirical laws; estimation of parameters of continuous laws

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Esquível, M. L. £Probability generating functions for discrete real-valued random variables. (English) ¢Zbl 1147.60010 ¡ Theory Probab. Appl. 52, No. 1, 40-57 (2008) and Teor. Veroyatn. Primen. 52, No. 1, 129-149 (2007).

Summary: The probability generating function is a powerful technique for studying the law of ﬁnite sums of independent discrete random variables taking integer positive values. For real-valued discrete random

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 11 variables, the well-known elementary theory of Dirichlet series and the symbolic computation packages available nowadays, such as Mathematica 5, allow us to extend this technique to general discrete random variables. Being so, the purpose of this work is twofold. First, we show that discrete random variables taking real values, nonnecessarily integer or rational, may be studied with probability generating functions. Second, we intend to draw attention to some practical ways of performing the necessary calculations.

MSC: 60E10 Characteristic functions; other transforms Cited in 1 Document 05A15 Exact enumeration problems, generating functions 60E05 Probability distributions: general theory

Keywords: probability generating functions; ﬁnite sums of independent real-valued discrete random variables; Dirich- let series

Software: Mathematica

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Esquível, Manuel L. £A conditional Gaussian martingale algorithm for global optimization. (English) ¢Zbl 1172.90454 ¡ Gavrilova, Marina (ed.) et al., Computational science and its applications – ICCSA 2006. International conference, Glasgow, UK, May 8–11, 2006. Proceedings, Part III. Berlin: Springer (ISBN 3-540-34075- 0/pbk). Lecture Notes in Computer Science 3982, 841-851 (2006).

Summary: A new stochastic algorithm for determination of a global minimum of a real-valued continuous function deﬁned on K, a compact set of Rn, having an unique global minimizer in K is introduced and studied, a context discussion is presented and implementations are used to compare the performance of the algorithm with other algorithms. The algorithm may be thought to belong to the random search class but although we use Gaussian distributions, the mean is changed at each step to be the intermediate minimum found at the preceding step and the standard deviations, on the diagonal of the covariance matrix, are halved from one step to the next. The convergence proof is simple relying on the fact that the sequence of intermediate random minima is an uniformly integrable conditional Gaussian martingale. For the entire collection see [Zbl 1107.68010].

MSC: 90C15 Stochastic programming Cited in 1 Document 90C59 Approximation methods and heuristics in mathematical programming

Software: Global Optimization Toolbox For Maple; PGSL

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Grosshinho, Maria do Rosário (ed.); Shiryaev, Albert N. (ed.); Esquível, Manuel L. (ed.); Oliveira, Paulo E. (ed.) Stochastic ﬁnance. Selected papers based on the presentations at the international confer- £ence on stochastic ﬁnance 2004, Lisbon, Portugal, September 26–30, 2004. (English) ¢Zbl 1134.60005 ¡ New York, NY: Springer (ISBN 0-387-28262-9/hbk; 978-1-4419-3932-6/pbk; 0-387-28359-5/ebook). xiv, 364 p. (2006).

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 12 The articles of this volume will be reviewed individually. Indexed articles: Aït-Sahalia, Yacine; Mykland, Per A.; Zhang, Lan, How often to sample a continuous-time process in the presence of market microstructure noise, 3-72 [Zbl 1151.62365] Barndorff-Nielsen, Ole E.; Shephard, Neil, Multipower variation and stochastic volatility, 73-82 [Zbl 1144.60314] Bielecki, Tomasz R.; Jeanblanc, Monique; Rutkowski, Marek, Completeness of a general semimartingale market under constrained trading, 83-106 [Zbl 1143.91330] Fasen, Vicky; Klüppelberg, Claudia; Lindner, Alexander, Extremal behavior of stochastic volatility models, 107-155 [Zbl 1159.62068] Platen, Eckhard, Capital asset pricing for markets with intensity based jumps, 157-182 [Zbl 1156.60051] Pliska, Stanley R., Mortgage valuation and optimal refinancing, 183-196 [Zbl 1143.91351] Runggaldier, Wolfgang J.; di Emidio, Sara, Computing efficient hedging strategies in discontinuous market models, 197-212 [Zbl 1143.60043] Yu, Lian; Zhang, Shuzhong; Zhou, Xun Yu, A downside risk analysis based on financial index tracking models, 213-236 [Zbl 1143.91348] Borovka, Svetlana; Permana, Ferry Jaya, Modelling electricity prices by the potential jump-diffusion, 239-263 [Zbl 1154.60055] Gaspar, Raquel M., Finite dimensional Markovian realizations for forward price term structure models, 265-320 [Zbl 1153.60041] Irle, Albrecht; Sass, Jörn, Good portfolio strategies under transaction costs: a renewal theoretic approach, 321-341 [Zbl 1143.91017] Woerner, Jeannette H. C., Power and multipower variation: inference for high frequency data, 343-364 [Zbl 1142.62095]

MSC: 60-06 Proceedings, conferences, collections, etc. pertaining to probability theory 91-06 Proceedings, conferences, collections, etc. pertaining to game theory, economics, and ﬁnance 00B25 Proceedings of conferences of miscellaneous speciﬁc interest

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Esquível, Manuel L. On the asymptotic£ behavior of the second moment of the Fourier transform of a random measure. (English) ¢Zbl 1082.60041 ¡ Int. J. Math. Math. Sci. 2004, No. 61-64, 3423-3434 (2004).

The author obtains an estimate for the asymptotic behavior of the second moment of the Fourier transform of the limit random measure in the theory of multiplicative chaos. After looking at the behavior at inﬁnity of the Fourier transform of some remarkable functions and measures, the author proves a formula essentially due to Frostman, involving the Riesz kernels. Reviewer: Ferenc Weisz (Budapest)

MSC: 60G57 Random measures 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

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Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 13 Esquível, Manuel L. Some risk£ processes associated to the dept function of a loan with variable interest rates. (English) ¢Zbl 0925.90125 ¡ Z. Angew. Math. Mech. 76, Suppl. 3, 419-420 (1996).

MSC: 91B30 Risk theory, insurance (MSC2010)

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References: [1] ; : Handbook of mathematical functions. Dover Publ. Inc., New York 1972. [2] ; : Topologie. Springer-Verlag, Berlin 1935; reprinted: Chelsea, New York 1965. [3] Hoenders, Computing 30 pp 137– (1983) [4] : Classical electrodynamics. John Wiley and Sons, New York 1975. · Zbl 0997.78500 [5] : Applications of Lie groups to differential equations. Springer-Verlag, New York 1993. · doi:10.1007/978-1-4612-4350-2 [6] Picard, J. de Math. Pure et Appl., 4e série 8 pp 5– (1892) [7] : Traité d’analyse. 3rd ed., gauthier-Villars, Paris 1922. [8] Vrahatis, ACM Trans. Math. Software 14 pp 312– (1988) [9] Vrahatis, ACM Trans. Math. Software 14 pp 330– (1988) [10] Vrahatis, Proc. Amer. Math. Soc. 107 pp 701– (1989) [11] ; ; ; : On the localization and computation of zeros of Based functions. Submitted. [12] : A treatise on the theory of Bassel functions. Cambridge University Press, Cambridge 1966. [13] : Linear and nonlinear waves. John Wiley and Sons, New York 1974. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

Esquível, Manuel Leote Applications£ of Fourier methods to the analysis of some stochastic processes. (English, Por- tuguese) ¢Zbl 0898.60047 ¡ Lisboa: Univ. Nova de Lisboa, Faculdade de Ciências e Tecnologia, xv, 101 p. (1996).

Summary: In the first chapter, a class of random periodic Schwartz distributions is introduced, some examples, elementary properties and a characterization result are studied and three applications are presented. A random Schwartz periodic distribution is, for us, just a function defined in a complete probability space and taking values in the space of Schwartz distributions over the line, that are left invariant by an integer translation, endowed with the natural algebraic and topological structures. The second chapter deals, primarily, with an extension of the methods of Kahane, as applied to the Brownian sheet, in what concerns analogs of the rapid points. After presenting the Brownian sheet process, by way of Gaussian white noise, some results, on the local behavior of this process and for some other processes associated with the sheet, are derived using the Schauder series representation. In the third chapter, we prove a formula essentially due to Frostman, we look at the behavior at infinity of the Fourier transform of some remarkable functions and measures and, finally, we study the asymptotic behavior of the second moment of the Fourier transform of a random measure that appears in the theory of multiplicative chaos. In the last chapter, a class of random tempered distributions on the line is introduced by considering random series, in the usual Hermite functions, having as coefficients random variables which satisfy certain growth conditions. This class is shown to be exactly the class of random Schwartz distributions having a mean. We present also a study on a possible converse of a result on Brownian distributions, that leads to a moment problem.

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 14 MSC: 60G20 Generalized stochastic processes 60G17 Sample path properties 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 46F25 Distributions on inﬁnite-dimensional spaces

Keywords: random Schwartz periodic distribution; Brownian sheet; Fourier transform of a random measure; moment problem

Esquível, Manuel L. £ On the local behavior of the Brownian sheet. (English) ¢Zbl 0862.60030 ¡ Fouque, Jean-Pierre (ed.) et al., Stochastic analysis: random ﬁelds and measure-valued processes. Papers of the binational France-Israel symposium on the Brownian sheet, September 1993, and the conference on measure-valued branching and superprocesses, May 1995, Ramat Gan, Israel. Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 10, 81-89 (1996).

Summary: In his seminal book “Some random series of functions” (1985; Zbl 0571.60002), J.-P. Kahane has shown, in a systematic way, how to take advantage of Paul Lévy’s construction of the Brownian process, using the Haar functions, in order to study the local behavior of this process. To reach this goal Kahane looks at the Haar’s interpolation of the Brownian process done by Lévy, as a series expansion in the Schauder system, having Gaussian random variables as coefficients, and exploits this series representation with sharp estimates of the distribution function of the maximum of a finite subfamily of a normal sequence. With this method Kahane gets easily the results corresponding to the existence of rapid points and slow points [which were first discovered by S. Orey and S. J. Taylor, Proc. Lond. Math. Soc., III. Ser. 28, 174-192 (1974; Zbl 0292.60128) and J.-P. Kahane, in: Conf. Harmonic Analysis in honor of A. Zygmund 1, 67-83 (1983; Zbl 0532.42001), respectively]. The present work deals with an extension of the methods of Kahane as applied to the Brownian sheet, in what concerns an analog of the rapid points. For the entire collection see [Zbl 0851.00070].

MSC: 60G17 Sample path properties 42C15 General harmonic expansions, frames

Keywords: Brownian process; Haar functions; Brownian sheet

Esquível, Manuel L. Points of£ rapid oscillation for the Brownian sheet via Fourier-Schauder series representation. (English) ¢Zbl 0842.60052 ¡ Kalton, Nigel (ed.) et al., Interaction between functional analysis, harmonic analysis, and probability. Proceedings of a conference held at the University of Missouri, Columbia, MO, USA, May 29-June 3, 1994. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 175, 153-162 (1996).

Summary: The representation of the Brownian sheet as a sum of a series, which converges uniformly almost surely, of Schauder functions having as coefficients normal random variables, is a simple conse- quence of the definition of the Brownian sheet using Gaussian white noise. Some results on the local behavior of the Brownian sheet and for some other processes associated with the sheet, can be derived by using this representation. Namely, a uniform modulus of continuity, nondifferentiability results and at some points, faster oscillation than the one prescribed by the laws of iterated logarithm. In previous work [the author, “On the local behavior of the Brownian sheet”, in: Isr. Math. Conf. Proc., AMS 1994] rapid points and almost sure everywhere nondifferentiability for the location homogeneous part of the Fourier-Schauder series representation were presented. Here we show the existence of rapid points for the independent increments of the Brownian sheet, using the same method. This method, first used by J.-P.

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 15 Kahane [“Some random series of functions” (1968; Zbl 0192.53801)] to deal with similar properties of the Brownian unidimensional time process, consists on exploiting the Fourier-Schauder representation with sharp estimates of the distribution function of the maximum of a ﬁnite subfamily of a normal sequence. Some results on the usual increments behavior are also presented. For the entire collection see [Zbl 0827.00042].

MSC: 60G60 Random ﬁelds 60B05 Probability measures on topological spaces 60J65 Brownian motion

Keywords: Brownian sheet; modulus of continuity; series representation; Fourier-Schauder representation; increments behavior

Esquível, Manuel L. £ An introduction to limited polynomial expansions. (Portuguese) ¢Zbl 0846.41007 ¡ Bol. Soc. Port. Mat. 30, 1-26 (1994).

This is a basically expository article. Its main objective is to show the advantages of the well-known representation of the real functions f : R → R by means of their limited polynomial expansions (Taylor theorem). With illustrative character several elementary examples, comments and applications are given so that the didactic content of the paper is a feature to be remarked. Reviewer: N.Hayek (La Laguna)

MSC: 41A10 Approximation by polynomials 26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions

Keywords: Taylor theorem; real functions; limit polynomial expansions

Esquível, Manuel L. On a class of periodic random distributions.£ (Sur une classe de distributions aléatoires périodiques.) (French. Extended English abstract) ¢Zbl 0809.46033 ¡ Ann. Sci. Math. Qué. 17, No. 2, 169-186 (1993).

Summary: The theoretical foundations for the study and application of periodic random distributions have been established since at least the sixties. In the context of the fractal geometry of Benoit Mandelbrot, mathematical models of irregular surfaces that can be obtained by computer have led us to consider stochastic processes arising from Fourier series with random coefficients. In this article, we introduce a class of such periodic random distributions. Three examples of this case are: random mass on the unit circle, Brownian motion (classical and fractional) and classical Schwartz distributions that are randomized by translations. We begin with a result giving conditions which characterize the distributions of this class. These conditions are easy to verify and this is done for the three previous examples. Two important questions in harmonic analysis are considered: uniqueness of the representation by Fourier series and differentiability. Also, we examine the statistical problem of the existence of a generalized first moment for these random distributions. As an application, a classical result in Fourier analysis, useful for constructing particular solutions of ordinary differential equations with constant coefficients, is generalized to this class of periodic random distributions. This last result is applied to obtain the Fourier-Wiener- Schwartz series of a particular solution of a generalized Langevin equation. We conclude with a comment on the regularity of the solution.

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 16 MSC: 46F10 Operations with distributions and generalized functions 60E99 Distribution theory 28A80 Fractals

Keywords: periodic random distributions; fractal geometry of Benoit Mandelbrot; Fourier series with random co- eﬃcients; random mass on the unit circle; Brownian motion; classical Schwartz distributions that are randomized by translations; existence of a generalized ﬁrst moment; Fourier-Wiener-Schwartz series; Langevin equation; regularity

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Esquível, Manuel L.; de O. Martins, Maria Ana F. £ On a basic theorem on the geometry of convex sets. (Portuguese) ¢Zbl 0828.52003 ¡ Bol. Soc. Port. Mat. 23, 22-32 (1992).

This is an expository article. First some simple properties of convex sets in n-space are given and then some theorems on the separation of convex sets from aﬃne subspaces and other convex sets are proved. Reviewer: Bernd Wegner (Berlin)

MSC: 52A20 Convex sets in n dimensions (including convex hypersurfaces)

Keywords: convex sets; separation; aﬃne subspaces

Esquível, Manuel L. On some applications£ of harmonic analysis of a class of random distributions. (Portuguese. English summary) ¢Zbl 0743.46039 ¡ Analysis, Proc. 15th Port.-Span. Meet. Math., Évora/Port. 1990, Vol. II, 285-290 (1991).

Summary: [For the entire collection see Zbl 0741.00014.] Fractional Brownian movement is shown to be an example of a class of random Schwarz periodic distributions introduced and studied by the author in [Sur une application de l’analyse harmonique d’une class de distributions aleatoires, Relatório Técnico 890927, Mat. Esq. 1- F.C.T.-U.N.L.]. Using results there reported, a particular solution for a generalized Langevin equation is represented as a Fourier-Wiener- Schwartz series. A comment on the regularity of the solution is given.

MSC: 46F25 Distributions on inﬁnite-dimensional spaces 60G20 Generalized stochastic processes

Keywords: Fractional Brownian movement; random Schwarz periodic distributions; particular solution for a generalized Langevin equation; Fourier-Wiener- Schwartz series; regularity of the solution

Esquível, Manuel L. Sur la méthode des séries de Fourier dans les équations différentielles à coefficients con- stants. (On£ the method of Fourier series for differential equations with constant coefficients). (French) ¢Zbl 0676.35011 ¡ Trab. Invest. 2, 35 p. (1987).

Edited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities © 2020 FIZ Karlsruhe GmbH Total: 27 Documents, Page 17 The author extends the method of Fourier series to obtain solutions of the differential equation P (D)u = f (P(D) = differential polynomial with constant coefficients) to the case where f does not satisfy the so called compatibility conditions P (n) = 0 ⇒ fˆ(n) = 0 (f(n)̂ = coefficient of the Fourier-Schwartz transform). Reviewer: R.Salvi

MSC: 35E20 General theory of PDEs and systems of PDEs with constant coeﬃcients 35C10 Series solutions to PDEs

Keywords: Fourier series; diﬀerential polynomial; constant coeﬃcients; Fourier- Schwartz transform

Esquível, Manuel L. £ 1 ⊂ 1 1 ⊂ 1 Note sur les inclusions Lµ Lλ. (A note on the inclusions Lµ Lλ). (French) ¢Zbl 0606.28001 ¡ Trab. Invest. 1, 5 p. (1985).

In this note we explicitly enunciate and prove an easy and perhaps known condition on the Radon- Nikodým derivative of one measure relative to another, in order to get set inclusion of their respective 1 L spaces. Let λ and µ be two σ-ﬁnite measures over a measure space and let dµ = hdλ + dµ1 be the Lebesgue Radon-Nikodým decomposition of µ with respect to λ. A necessary and suﬃcient condition for 1 ⊆ 1 ∃ { } Lµ Lλ is that: K > 0λ( h < K ) = 0. No priority research about this subject has been done by the author.

MSC: 28A15 Abstract diﬀerentiation theory, diﬀerentiation of set functions 28A25 Integration with respect to measures and other set functions 46E30 Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords: L1 inclusions; Radon-Nikodým derivative; Lebesgue Radon- Nikodým decomposition

Gamas, Carlos D.; Esquível, Manuel L. £ A property of periodic functions. (Portuguese) ¢Zbl 0582.10024 ¡ Bol. Soc. Port. Mat. 5, 56-59 (1982).

Die Autoren ”zeigen”, daß für irrationales α die Folge nα direkt modulo 1 ist (Satz 1), für rationale Zahlen aber nicht (Satz 2). Zum Beweis zitieren sie noch das entsprechende Resultat über die Gleichverteilung der Folge (nα) !! Ein Aprilscherz ? Die zitierten Resultate und Namen sind selten richtig geschrieben. Reviewer: H.Rindler

MSC: 11J71 Distribution modulo one