Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T20:20:11.425Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffusion approximations for randomly arriving expert opinions in a financial market with Gaussian drift

Part of: Mathematical finance Stochastic systems and control Limit theorems

Published online by Cambridge University Press:  25 February 2021

Jörn Sass ,
Dorothee Westphal  and
Ralf Wunderlich
Jörn Sass*
Affiliation:
Technische Universität Kaiserslautern
Dorothee Westphal*
Affiliation:
Technische Universität Kaiserslautern
Ralf Wunderlich*
Affiliation:
Brandenburg University of Technology Cottbus-Senftenberg
*
*Postal address: Department of Mathematics, Technische Universität Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany.
*Postal address: Department of Mathematics, Technische Universität Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany.
****Postal address: Institute of Mathematics, Brandenburg University of Technology Cottbus-Senftenberg, P.O. Box 101344, 03013 Cottbus, Germany. Email address: ralf.wunderlich@b-tu.de
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper investigates a financial market where stock returns depend on an unobservable Gaussian mean reverting drift process. Information on the drift is obtained from returns and randomly arriving discrete-time expert opinions. Drift estimates are based on Kalman filter techniques. We study the asymptotic behavior of the filter for high-frequency experts with variances that grow linearly with the arrival intensity. The derived limit theorems state that the information provided by discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process. These diffusion approximations are extremely helpful for deriving simplified approximate solutions of utility maximization problems.

Keywords

Diffusion approximationsKalman filterOrnstein–Uhlenbeck processexpert opinionsportfolio optimizationpartial information

MSC classification

Primary: 91G10: Portfolio theory
Secondary: 93E11: Filtering 93E20: Optimal stochastic control 60F25: $L^p$-limit theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 58 , Issue 1 , March 2021 , pp. 197 - 216
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aalto, A. (2016). Convergence of discrete-time Kalman filter estimate to continuous time estimate. Internat. J. Control 89, 668–679.CrossRefGoogle Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 14). World Scientific, Singapore.Google Scholar
Black, F. and Litterman, R. (1992). Global portfolio optimization. Finan. Analysts J. 48, 28–43.CrossRefGoogle Scholar
Brendle, S. (2006). Portfolio selection under incomplete information. Stoch. Process. Appl. 116, 701–723.CrossRefGoogle Scholar
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman and Hall/CRC, London.Google Scholar
Coquet, F., Mémin, J. and Słominski, L. (2001). On weak convergence of filtrations. Séminaire prob. Strasbourg 35, 306–328.Google Scholar
Davis, M. H. A. and Lleo, S. (2013). Black–Litterman in continuous time: The case for filtering. Quant. Finance Lett. 1, 30–35.CrossRefGoogle Scholar
Frey, R., Gabih, A. and Wunderlich, R. (2012). Portfolio optimization under partial information with expert opinions. Internat. J. Theoret. Appl. Finance 15, DOI: 10.1142/S0219024911006486.CrossRefGoogle Scholar
Frey, R., Gabih, A. and Wunderlich, R. (2014). Portfolio optimization under partial information with expert opinions: A dynamic programming approach. Commun. Stoch. Anal. 8, 49–79.Google Scholar
Gabih, A., Kondakji, H., Sass, J. and Wunderlich, R. (2014). Expert opinions and logarithmic utility maximization in a market with Gaussian drift. Commun. Stoch. Anal. 8, 27–47.Google Scholar
Gabih, A., Kondakji, H. and Wunderlich, R. (2020). Asymptotic filter behavior for high-frequency expert opinions in a market with Gaussian drift. Stoch. Models, DOI: 10.1080/15326349.2020.1758567.CrossRefGoogle Scholar
Glynn, P. W. (1990). Diffusion approximations. In Stochastic Models, eds. D. P. Heyman and M. J. Sobel (Handbooks Operat. Res. Manag. Sci. 2), Elsevier, Amsterdam, pp. 145–198.Google Scholar
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.CrossRefGoogle Scholar
Iglehart, D. L. (1969). Diffusion approximations in collective risk theory. J. Appl. Prob. 6, 285–292.CrossRefGoogle Scholar
Karatzas, I. and Zhao, X. (2001). Bayesian Adaptive Portfolio Optimization. In Handbooks in Mathematical Finance: Option Pricing, Interest Rates and Risk Management, eds. E. Jouini, J. Cvitanic and M. Musiela, Cambridge University Press, pp. 632–669.CrossRefGoogle Scholar
Kondakji, H. (2019). Optimale Portfolios für partiell informierte Investoren in einem Finanzmarkt mit Gaußscher Drift und Expertenmeinungen. PhD thesis. Brandenburg University of Technology Cottbus-Senftenberg.Google Scholar
Kučera, V. (1973). A review of the matrix Riccati equation. Kybernetika 9, 42–61.Google Scholar
Liptser, R. S. and Shiryaev, A. N. (1974). Statistics of Random Processes I: General Theory. Springer, New York.Google Scholar
Pachpatte, B. G. (1997). Inequalities for Differential and Integral Equations. Academic Press, New York.Google Scholar
Rosiński, J. and Suchanecki, Z. (1980). On the space of vector-valued functions integrable with respect to the white noise. Colloq. Math. 43, 183–201.CrossRefGoogle Scholar
Salgado, M., Middleton, R. and Goodwin, G. C. (1988). Connection between continuous and discrete Riccati equations with applications to Kalman filtering. IEEE Proc. D Control Theory Appl. 135, 28–34.CrossRefGoogle Scholar
Sass, J., Westphal, D. and Wunderlich, R. (2017). Expert opinions and logarithmic utility maximization for multivariate stock returns with Gaussian drift. Int. J. Theoret. Appl. Finance 20, DOI: 10.1142/S0219024917500224.CrossRefGoogle Scholar
Schmidli, H. (2017). Risk Theory. Springer, New York.CrossRefGoogle Scholar
Westphal, D. (2019). Model uncertainty and expert opinions in continuous-time financial markets. PhD thesis. Technische Universität Kaiserslautern.Google Scholar