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Conditional Characteristic Functions of Molchan-Golosov Fractional Lévy Processes with Application to Credit Risk

Part of: Stochastic processes Stochastic analysis Mathematical finance

Published online by Cambridge University Press:  30 January 2018

Holger Fink
Holger Fink*
Affiliation:
Technische Universität München
*
∗ Postal address: Center for Mathematical Sciences, Technische Universität München, Parkring 13, D-85748 Garching, Germany. Email address: [email protected]
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Abstract

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Molchan-Golosov fractional Lévy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on an n-dimensional driving Lévy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Lévy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

Keywords

Conditional characteristic functionmacroeconomic variables processlong-range dependencefractional Brownian motionfractional Lévy processprediction

MSC classification

Primary: 60G10: Stationary processes 60G22: Fractional processes, including fractional Brownian motion 60G51: Processes with independent increments; L'evy processes 60H10: Stochastic ordinary differential equations 60H20: Stochastic integral equations 91G40: Credit risk
Secondary: 60G15: Gaussian processes 91G30: Interest rates (stochastic models) 91G60: Numerical methods (including Monte Carlo methods)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 983 - 1005
Copyright
© Applied Probability Trust 

References

Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press.Google Scholar
Benassi, A., Cohen, S. and Istas, J. (2002). Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8, 97–115.Google Scholar
Bender, C. and Elliott, R. J. (2003). {On the Clark–Ocone theorem for fractional Brownian motions with Hurst parameter bigger than a half}. Stoch. Stoch. Rep. 75, 391–405.Google Scholar
Bender, C. and Marquardt, T. (2008). {Stochastic calculus for convoluted Lévy processes}. Bernoulli 14, 499–518.Google Scholar
Bender, C. and Marquardt, T. (2009). {Integrating volatility clustering into exponential Lévy models}. J. Appl. Prob. 46, 609–628.Google Scholar
Biagini, F., Fink, H. and Klüppelberg, C. (2013). A fractional credit model with long range dependent default rate. Stoch. Process. Appl. 123, 1319–1347.CrossRefGoogle Scholar
Biagini, F., Fuschini, S. and Klüppelberg, C. (2011). Credit contagion in a long range dependent macroeconomic factor model. In Advanced Mathematical Methods in Finance, eds Di Nunno, G. and Øksendal, B., Springer, Heidelberg, pp. 105–132.CrossRefGoogle Scholar
Bielecki, T. R. and Rutkowski, M. (2002). Credit Risk: Modelling, Valuation and Hedging. Springer, Berlin.Google Scholar
Blumenthal, R. M. and Getoor, R. K. (1961). Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10, 493–516.Google Scholar
Buchmann, B. and Klüppelberg, C. (2006). Fractional integral equations and state space transforms. Bernoulli 12, 431–456.Google Scholar
Duffie, D. (2004). Credit risk modeling with affine processes. 2002 Cattedra Galileana Lecture, Stanford University and Scuola Normale Superiore, Pisa.Google Scholar
Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.Google Scholar
Duncan, T. E. (2006). Prediction for some processes related to a fractional Brownian motion. Statist. Prob. Lett. 76, 128–134.Google Scholar
Elliott, R. J. and van der Hoek, J. (2003). A general fractional white noise theory and applications to finance. Math. Finance 13, 301–330.Google Scholar
Filipovic, D. (2003). Term-Structure Models. Springer, New York.Google Scholar
Fink, H. and Klüppelberg, C. (2011). Fractional Lévy-driven Ornstein–Uhlenbeck processes and stochastic differential equations. Bernoulli 17, 484–506.Google Scholar
Fink, H. and Scherr, C. (2012). Credit risk with long memory. Submitted.Google Scholar
Fink, H., Klüppelberg, C. and Zähle, M. (2013). Conditional distributions of processes related to fractional Brownian motion. J. Appl. Prob. 50, 166–183.Google Scholar
Frey, R. and Backhaus, J. (2008). Pricing and hedging of portfolio credit derivatives with interacting default intensities. Internat. J. Theoret. Appl. Finance 11, 611–634.Google Scholar
Henry, M. and Zaffaroni, P. (2003). The long-range dependence paradigm for macroeconomics and finance. In Theory and Applications of Long-Range Dependence, eds Doukhan, P., Oppenheim, G., and Taqqu, M., Birkhäuser, Boston, MA, pp. 417–438.Google Scholar
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam.Google Scholar
Jost, C. (2006). {Transformation formulas for fractional Brownian motion}. Stoch. Process. Appl. 116, 1341–1357.Google Scholar
Klüppelberg, C. and Matsui, M. (2010). Generalized fractional Lévy processes with fractional Brownian motion limit. Submitted. Available at www-m4.ma.tum.de/Papers.Google Scholar
Marcus, M. B. and Rosiński, J. (2005). Continuity and boundedness of infinitely divisible processes: a Poisson point process approach. J. Theoret. Prob. 4, 109–160.Google Scholar
Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12, 1099–1126.Google Scholar
Marquardt, T. (2007). {Multivariate fractionally integrated CARMA processes}. J. Multivariate Analysis 98, 1705–1725.CrossRefGoogle Scholar
Monroe, I. (1972). On the γ-variation of processes with stationary independent increments. Ann. Math. Statist. 43, 1213–1220.Google Scholar
Ohashi, A. (2009). Fractional term structure models: no-arbitrage and consistency. Ann. Appl. Prob. 19, 1553–1580.Google Scholar
Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Prob. Theory Relat. Fields 118, 251–291.Google Scholar
Pipiras, V. and Taqqu, M. S. (2001). Are classes of deterministic integrands for fractional Brownian motion on an interval complete? Bernoulli 7, 873–897.Google Scholar
Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–487.Google Scholar
Russo, F. and Vallois, P. (2007). {Elements of stochastic calculus via regularisation}. In Séminaire de Probabilités XL (Lecture Notes Math. 1899), Springer, Berlin, pp. 147–185.CrossRefGoogle Scholar
Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Sato, K.-I. (2006). {Additive processes and stochastic integrals}. Illinois J. Math. 50, 825–851.Google Scholar
Tikanmäki, H. J. and Mishura, Y. (2011). Fractional Lévy processes as a result of compact interval integral transformation. Stoch. Anal. Appl. 29, 1081–1101.CrossRefGoogle Scholar
Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282.CrossRefGoogle Scholar
Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Prob. Theory Relat. Fields 111, 333–374.Google Scholar
Zähle, M. (2001). Integration with respect to fractal functions and stochastic calculus. II. Math. Nachr. 225, 145–183.Google Scholar