关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

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Methods Of Modern Mathematical Physics

https://www.lef.wiwi.uni-due.de/fileadmin/fileupload/BWL-LEF/Seminarreihe/RWeron14_Essen.pdf

Electricity price forecasting: A review of the state-of-the-art with a look into the future

A variety of methods and ideas have been tried for electricity price forecasting (EPF) over the last 15 years, with varying degrees of success. This review article aims at explaining the complexity of available solutions, their strengths and weaknesses, and the opportunities and treats that the forecasting tools offer or that may be encountered. The paper also looks ahead and speculates on the directions EPF will or should take in the next decade or so. In particular, it postulates the need for objective comparative EPF studies involving (i) the same datasets, (ii) the same robust error evaluation procedures and (iii) statistical testing of the significance of the outperformance of one model by another.

This paper develops a novel class of hybrid credit-equity models with state-dependent jumps, local-stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time-changed Markov diffusion process with state-dependent local volatility and killing rate (default intensity). When the time change is a Levy subordinator, the stock price process exhibits jumps with state-dependent Levy measure. When the time change is a time integral of an activity rate process, the stock price process has local-stochastic volatility and default intensity. When the time change process is a Levy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state-dependent jumps, local-stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time-changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump-to-default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local-stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.

https://papers.ssrn.com/sol3/Delivery.cfm/mafi_411.pdf?abstractid=1681711&mirid=1

Time-Changed Markov Processes in Unified Credit-Equity Modeling

We show that by modeling the time series of mortality rate changes rather than mortality rate levels we can better model human mortality. Leveraging on this, we propose a model that expresses log mortality rate changes as an age group dependent linear transformation of a mortality index. The mortality index is modeled as a Normal Inverse Gaussian. We demonstrate, with an exhaustive set of experiments and data sets spanning 11 countries over 100 years, that the proposed model significantly out performs existing models. We further investigate the ability of multiple principal components, rather than just the first component, to capture differentiating features of different age groups and find that a two component NIG model for log mortality change best fits existing mortality rate data.

Modeling and Forecasting Mortality Rates

We show that by modeling the time series of mortality rate changes rather than mortality rate levels we can better model human mortality. Leveraging on this, we propose a model that expresses log mortality rate changes as an age group dependent linear transformation of a mortality index. The mortality index is modeled as a Normal Inverse Gaussian. We demonstrate, with an exhaustive set of experiments and data sets spanning 11 countries over 100 years, that the proposed model significantly outperforms existing models. We further investigate the ability of multiple principal components, rather than just the first component, to capture differentiating features of different age groups and find that a two component NIG model for log mortality change best fits existing mortality rate data.

Modeling and forecasting mortality rates

This paper develops a novel class of hybrid credit-equity models with state-dependent jumps, local-stochastic volatility and default intensity based on time changes of Markov pro- cesses with killing. We model the defaultable stock price process as a time changed Markov difiusion process with state-dependent local volatility and killing rate (default intensity). When the time change is a Levy subordinator, the stock price process exhibits jumps with state-dependent Levy measure. When the time change is a time integral of an activity rate process, the stock price process has local-stochastic volatility and default intensity. When the time change process is a Levy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state-dependent jumps, local-stochastic volatility and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace trans- form of the time change are both available in closed form, the expectation operator of the time changed process is expressed in closed form as a single integral in the complex plane. If the payofi is square-integrable, the complex integral is further reduced to a spectral ex- pansion. To illustrate our general framework, we time change the jump-to-default extended CEV model (JDCEV) of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local-stochastic volatility and default intensity. These models can be used to jointly price equity and credit derivatives.

http://users.iems.northwestern.edu/~rmendoza/Rafael%20Mendoza-Paper-1-TC.pdf

Time-changed markov processes in unified credit-equity modeling

The present paper introduces a jump-diffusion extension of the classical diffusion default intensity model by means of subordination in the sense of Bochner. We start from the bi-variate process of the diffusion state variable and default indicator process (X,D) in the diffusion intensity model and time change it with a Levy subordinator T. We then give a detailed characterization of the resulting time changed process (Xφ,Dφ)=(X(T),D(T)) as a Markovian Ito semimartingale and, in particular, show from the Doob-Meyer decomposition of Dφ that the default time in the time-changed model has a jump-diffusion or a pure jump intensity. In particular, when X is a CIR diffusion, the default intensity of the subordinate model (SubCIR) turns out to be a jump-diffusion or a pure jump process with two-sided mean-reverting jumps that stays non-negative. The SubCIR default intensity model is fully analytically tractable by means of the explicitly computed eigenfunction expansion of the relevant semigroups. This allows explicit closed-form pricing of credit-sensitive securities.

Rafael Mendoza-Arriaga

Papers

Time-Changed Markov Processes in Unified Credit-Equity Modeling

Modeling and Forecasting Mortality Rates

Modeling and forecasting mortality rates

Time-changed markov processes in unified credit-equity modeling

Time-Changed CIR Default Intensities with Two-Sided Mean-Reverting Jumps