Source: http://www.wias-berlin.de/research/ats/volatility/?lang=1
Timestamp: 2019-04-18 15:09:56+00:00

Document:
The valuation and risk assessment of financial products requires appropriate analysis of the characteristics involved in the modeling. Particular examples are volatility and characteristics of extreme events. Usually the models under consideration are multi-dimensional and have time dependent parameters.
The developed approaches for estimating time dependent volatility end extreme events have to be extended to the multivariate case. In this context suitable dimension reduction methods will play an important role. The results will be needed for modeling complex financial structures.
Statistical procedures for inference of non-stationary time series are developed in the univariate case. In particular, new procedures for estimating time dependent volatilities based on local adaptive smoothing techniques are proposed.
G. Dos Reis, A. Réveillac, J. Zhang, FBSDEs with time delayed generators: $L^p$-solutions, differentiability, representation formulas and path regularity, Stochastic Processes and their Applications, 121 (2011), pp. 2114--2150.
A simple and commonly used method to approximate the total claim distribution of a (possible weakly dependent) insurance collective is the normal approximation. In this article, we investigate the error made when the normal approximation is plugged in a fairly general distribution-invariant risk measure. We focus on the rate of the convergence of the error relative to the number of clients, we specify the relative error's asymptotic distribution, and we illustrate our results by means of a numerical example. Regarding the risk measure, we take into account distortion risk measures as well as distribution-invariant coherent risk measures.
D. Spivakovskaya, A.W. Heemink, J.G.M. Schoenmakers, Two-particle models for the estimation of the mean and standard deviation of concentrations in coastal waters, Stochastic Environmental Research and Risk Assessment (SERRA), 21 (2007), pp. 235--251.
E. VAN DEN Berg, A.W. Heemink, H.X. Lin, J.G.M. Schoenmakers, Probability density estimation in stochastic environmental models using reverse representations, Stochastic Environmental Research & Risk Assessment, 20 (2006), pp. 126--139.
The estimation of probability densities of variables described by stochastic differential equations has long been done using forward time estimators, which rely on the generation of forward in time realizations of the model. Recently, an estimator based on the combination of forward and reverse time estimators has been developed. This estimator has a higher order of convergence than the classical one. In this article, we explore the new estimator and compare the forward and forward? reverse estimators by applying them to a biochemical oxygen demand model. Finally, we show that the computational efficiency of the forward?reverse estimator is superior to the classical one, and discuss the algorithmic aspects of the estimator.
D. Spivakovskaya, A.W. Heemink, G.N. Milstein, J.G.M. Schoenmakers, Simulation of the transport of particles in coastal waters using forward and reverse time diffusion, Advances in Water Resources, 28 (2005), pp. 927--938.
Particle models are often used to simulate the spreading of a pollutant in coastal waters in case of a calamity at sea. Here many different particle tracks starting at the point of release are generated to determine the particle concentration at some critical locations after the release. This Monte Carlo method however consumes a large CPU time. Recently, Milstein, Schoenmakers and Spokoiny (2003) introduced the concept of reverse-time diffusion. They derived a reverse system from the original forward simulation model and showed that the Monte Carlo estimator can also be based on realizations of this reverse system. In this paper we apply this concept to estimate particle concentrations in coastal waters. The results for the experiments considered show that the CPU time compared with the classical method is reduced orders of magnitude.
H. Haaf, O. Reiss, J.G.M. Schoenmakers, Numerically stable computation of CreditRisk+, Phys. Rev. E (3), 6 (2004), pp. 1--10.
The CreditRisk+ model launched by Credit Suisse First Boston in 1997 is widely used by practitioners in the banking sector as a simple means for the quantification of credit risk, primarily of the loan book. We present an alternative numerical recursion scheme for CreditRisk+, equivalent to an algorithm recently proposed by Giese, that is based on well-known expansions of the logarithm and the exponential of a power series. We show that it is advantageous for the Panjer recursion advocated in the original CreditRisk+ document, in that it is numerically stable. The crucial stability arguments are explained in detail. We explain how to apply the suggested recursion scheme to incorporate stochastic exposures into the CreditRisk+ model as introduced by Tasche (2004). Finally, the computational complexity of the resulting algorithm is stated and compared with other methods for computing the CreditRisk+ loss distribution.
To determine the probability of exceedence Monte Carlo simulation of stochastic models is often used. Mathematically this requires the evaluation of an expectation of some function of a solution of a stochastic model. This can be reformulated as a Kolmogorov final value problem. It can thus be calculated numerically by either solving a deterministic partial differential equation (Kolmogorov's Backwards equations) or by simulating a large number of trajectories of the stochastic differential equation. Here we discuss a composite method of variance reduced Monte Carlo simulation. The variance reduction is obtained by the Girsanov transformation to modify the stochastic model by a correction term that is obtained from an approximate solution of the partial differential equation computed by a classical numerical method. The composite method is more efficient than either the standard Monte Carlo or the classical numerical method. The approach is applied to estimate the probability of exceedence in a model for biochemical-oxygen demand.
H. Haaf, O. Reiss, J.G.M. Schoenmakers, Numerically stable computation of Credit Risk+, in: CreditRisk+ in the Banking Industry, M. Gundlach, F. Lehrbass, eds., XII, Springer, Berlin Heidelberg, 2004, pp. 67-76.
D. Spivakovskaya, A.W. Heemink, J.G.M. Schoenmakers, G.N. Milstein, Stochastic modeling of transport in coastal waters using forward and reverse time diffusion, in: Computational Methods in Water Resources, Proceedings of the 15th International Conference on Computational Methods in Water Resources (CMWR XV), June 13--17 2004, Chapel Hill, North Carolina, C.T. Miller, M.W. Farthing, W.G. Gray, G.F. Pinder, eds., 55 of Developments in Water Science, Elsevier Science, Amsterdam, 2004, pp. 1813--1824.
V. Panov, Affine stochastic volatility models: asymptotic behavior of the characteristic function and estimation of the Blumenthal - Getoor index, Tagung des SFB 649 "Ökonomisches Risiko" in Motzen, June 30 - July 2, 2011, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, July 1, 2011.
J. Zhang, Lp-solutions of BSDEs with time delayed generators, Mathematikkolloquium, Universität Innsbruck, Fachbereich Mathematik, Austria, April 28, 2011.
J. Zhang, Solvability and numerical simulation of BSDEs related to BSPDEs with applications to utility maximization, Universität Innsbruck, Fachbereich Mathematik, Austria, April 26, 2011.
V. Krätschmer, A uniform central limit theorem for distortions of empirical distributions with applications to nonparametric estimation of distribution-invariant risk measures, Leipziger Stochastik Tage, March 1 - 5, 2010, Universität Leipzig, Fakultät für Mathematik und Informatik, March 2, 2010.
V. Krätschmer, Error caused by normal approximation of the total claim distribution when plugged in risk measures, 4th International Conference Mathematical and Statistical Methods for Actuarial Sciences and Finance, April 7 - 9, 2010, Università degli Studi di Salerno, Dipartimento di Scienze Economiche e Statistiche, Ravello, Italy, April 8, 2010.
V. Krätschmer, Error caused by normal approximation of the total claim distribution when plugged in risk measures, DAGStat2010, March 23 - 26, 2010, Technische Universität Dortmund, Fakultät Statistik, March 23, 2010.
V. Krätschmer, Nichtparametrische Schätzung verteilungsinvarianter Risikomaße, Universität Bayreuth, Fachbereich Mathematik, July 27, 2010.
V. Krätschmer, Nichtparametrische Schätzung verteilungsinvarianter Risikomaße, Universität Mannheim, Institut für Mathematik und Informatik, May 4, 2010.
V. Krätschmer, Nichtparametrische Schätzung verteilungsinvarianter Risikomaße, Heinrich Heine Universität Düsseldorf, Institut für Mathematik, April 23, 2010.
V. Krätschmer, Nichtparametrische Schätzung von verteilungsinvarianten Risikomaßen, Mathematisches Kolloquium, Universität Duisburg-Essen, Fakultät für Mathematik, October 19, 2010.
V. Krätschmer, Nonparametric estimation of low invariant risk measures for time series, Statistische Woche Nürnberg 2010, September 14 - 17, 2010, Deutschen Statistischen Gesellschaft (DStatG), September 16, 2010.
P. Friz, Stochastic analysis and quantitative finance, Microsoft Research Cambridge, UK, January 5, 2010.
J.G.M. Schoenmakers, Numerically stable computation of CreditRisk+, Karlsruher Stochastik-Tage 2004, March 23 - 26, 2004, Universität Karlsruhe, March 25, 2004.
J.G.M. Schoenmakers, Transition density estimation for stochastic differential equations via forward-reverse representations, Tandem-Workshop Stochastik-Numerik, June 11 - March 26, 2004, Humboldt-Universität zu Berlin, June 11, 2004.
J.G.M. Schoenmakers, Kreditrisiko Portfolio-Modelle, Kreditanstalt für Wiederaufbau, Frankfurt, July 18, 2003.
In the last few years there has been renewed interest in the classical control problem of de Finetti for the case that underlying source of randomness is a spectrally negative Levy process. In particular a significant step forward is made in an article of Loeffen where it is shown that a natural and very general condition on the underlying Levy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Levy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem but now with the restriction that control strategies are absolutely continuous with respect to Lebesgue measure. This problem has been considered by Asmussen and Taksar, Jeanblanc and Shiryaev and Boguslavskaya in the diffusive case and Gerber and Shiu for the case of a Cramer-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Levy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.

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