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

  • Holger Fink (a1)

Abstract

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.

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Copyright

Corresponding author

Postal address: Center for Mathematical Sciences, Technische Universität München, Parkring 13, D-85748 Garching, Germany. Email address: fink@ma.tum.de

References

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[1] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press.
[2] Benassi, A., Cohen, S. and Istas, J. (2002). Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8, 97115.
[3] 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, 391405.
[4] Bender, C. and Marquardt, T. (2008). {Stochastic calculus for convoluted Lévy processes}. Bernoulli 14, 499518.
[5] Bender, C. and Marquardt, T. (2009). {Integrating volatility clustering into exponential Lévy models}. J. Appl. Prob. 46, 609628.
[6] Biagini, F., Fink, H. and Klüppelberg, C. (2013). A fractional credit model with long range dependent default rate. Stoch. Process. Appl. 123, 13191347.
[7] 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. 105132.
[8] Bielecki, T. R. and Rutkowski, M. (2002). Credit Risk: Modelling, Valuation and Hedging. Springer, Berlin.
[9] Blumenthal, R. M. and Getoor, R. K. (1961). Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10, 493516.
[10] Buchmann, B. and Klüppelberg, C. (2006). Fractional integral equations and state space transforms. Bernoulli 12, 431456.
[11] Duffie, D. (2004). Credit risk modeling with affine processes. 2002 Cattedra Galileana Lecture, Stanford University and Scuola Normale Superiore, Pisa.
[12] Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 9841053.
[13] Duncan, T. E. (2006). Prediction for some processes related to a fractional Brownian motion. Statist. Prob. Lett. 76, 128134.
[14] Elliott, R. J. and van der Hoek, J. (2003). A general fractional white noise theory and applications to finance. Math. Finance 13, 301330.
[15] Filipovic, D. (2003). Term-Structure Models. Springer, New York.
[16] Fink, H. and Klüppelberg, C. (2011). Fractional Lévy-driven Ornstein–Uhlenbeck processes and stochastic differential equations. Bernoulli 17, 484506.
[17] Fink, H. and Scherr, C. (2012). Credit risk with long memory. Submitted.
[18] Fink, H., Klüppelberg, C. and Zähle, M. (2013). Conditional distributions of processes related to fractional Brownian motion. J. Appl. Prob. 50, 166183.
[19] Frey, R. and Backhaus, J. (2008). Pricing and hedging of portfolio credit derivatives with interacting default intensities. Internat. J. Theoret. Appl. Finance 11, 611634.
[20] 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. 417438.
[21] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam.
[22] Jost, C. (2006). {Transformation formulas for fractional Brownian motion}. Stoch. Process. Appl. 116, 13411357.
[23] Klüppelberg, C. and Matsui, M. (2010). Generalized fractional Lévy processes with fractional Brownian motion limit. Submitted. Available at www-m4.ma.tum.de/Papers.
[24] Marcus, M. B. and Rosiński, J. (2005). Continuity and boundedness of infinitely divisible processes: a Poisson point process approach. J. Theoret. Prob. 4, 109160.
[25] Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12, 10991126.
[26] Marquardt, T. (2007). {Multivariate fractionally integrated CARMA processes}. J. Multivariate Analysis 98, 17051725.
[27] Monroe, I. (1972). On the γ-variation of processes with stationary independent increments. Ann. Math. Statist. 43, 12131220.
[28] Ohashi, A. (2009). Fractional term structure models: no-arbitrage and consistency. Ann. Appl. Prob. 19, 15531580.
[29] Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Prob. Theory Relat. Fields 118, 251291.
[30] Pipiras, V. and Taqqu, M. S. (2001). Are classes of deterministic integrands for fractional Brownian motion on an interval complete? Bernoulli 7, 873897.
[31] Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451487.
[32] Russo, F. and Vallois, P. (2007). {Elements of stochastic calculus via regularisation}. In Séminaire de Probabilités XL (Lecture Notes Math. 1899), Springer, Berlin, pp. 147185.
[33] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon.
[34] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.
[35] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
[36] Sato, K.-I. (2006). {Additive processes and stochastic integrals}. Illinois J. Math. 50, 825851.
[37] Tikanmäki, H. J. and Mishura, Y. (2011). Fractional Lévy processes as a result of compact interval integral transformation. Stoch. Anal. Appl. 29, 10811101.
[38] Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251282.
[39] Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Prob. Theory Relat. Fields 111, 333374.
[40] Zähle, M. (2001). Integration with respect to fractal functions and stochastic calculus. II. Math. Nachr. 225, 145183.

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

  • Holger Fink (a1)

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