Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T15:08:10.601Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Design of Dynamic Default Risk Measures

Part of: Stochastic analysis Probabilistic methods, simulation and stochastic differential equations Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Leo Shen  and
Robert Elliott
Leo Shen*
Affiliation:
University of Adelaide
Robert Elliott*
Affiliation:
University of Adelaide
*
∗ Current address: WH Bryan Mining and Geology Research Centre, Sustainable Minerals Institute, The University of Queensland, Brisbane, QLD 4072, Australia. Email address: b.shen1@uq.edu.au
∗∗ Postal address: School of Mathematical Sciences, North Terrace Campus, University of Adelaide, SA 5005, Australia. Email address: robert.elliott@adelaide.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the question of an optimal transaction between two investors to minimize their risks. We define a dynamic entropic risk measure using backward stochastic differential equations related to a continuous-time single jump process. The inf-convolution of dynamic entropic risk measures is a key transformation in solving the optimization problem.

Keywords

Inf-convolutiondynamic entropic risk measuresingle jump processbackward stochastic differential equation

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 60G42: Martingales with discrete parameter 65C30: Stochastic differential and integral equations
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 967 - 977
Copyright
© Applied Probability Trust 

References

Barrieu, P. and El Karoui, N. (2002). Optimal design of derivatives in illiquid markets. Quant. Finance 2, 181–188.CrossRefGoogle Scholar
Barrieu, P. and El Karoui, N. (2004). Optimal derivatives design under dynamic risk measures. In Mathematics of Finance (Contemp. Math. 351), eds Yin, G. and Zhang, Q., American Mathematical Society, Providence, RI, pp. 13–25.Google Scholar
Bion-Nadal, J. (2006). Dynamic risk measuring: discrete time in a context of uncertainty, and continuous time on a probability space. CMAP Preprint 596, École Polytechnique.Google Scholar
Cohen, S. N. and Elliott, R. J. (2008). Solutions of backward stochastic differential equations on Markov chains. Commun. Stoch. Anal. 2, 251–262.Google Scholar
Cohen, S. N. and Elliott, R. J. (2010). A general theory of finite state backward stochastic difference equations. Stoch. Process. Appl. 120, 442–466.Google Scholar
Davis, M. H. A. (1976). The representation of martingales of Jump processes. SIAM J. Control Optimization 14, 623–638.CrossRefGoogle Scholar
El Karoui, N. and Huang, S.-J. (1997). A general result of existence and uniqueness of backward stochastic differential equations. In Backward Stochastic Differential Equations (Pitman Res. Notes Math. Ser. 364), Longman, Harlow, pp. 27–36.Google Scholar
Elliott, R. J. (1982). Stochastic Calculus and Applications. Springer, New York.Google Scholar
Föllmer, H. and Schied, A. (2004). Stochastic Finance, 2nd edn. Walter de Gruyter, Berlin.Google Scholar
Frittelli, M. and Scandolo, G. (2006). Risk measures and capital requirements for processes. Math. Finance 16, 589–612.Google Scholar
Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14, 55–61.CrossRefGoogle Scholar
Peng, S. (2004). Nonlinear expectations, nonlinear evaluations and risk measures. In Stochastic Methods in Finance, Springer, Berlin, pp. 165–254.CrossRefGoogle Scholar
Shen, L. and Elliott, R. J. (2011). Backward stochastic differential equations for a single Jump process. Stoch. Anal. Appl. 29, 654–673.CrossRefGoogle Scholar