×

zbMATH — the first resource for mathematics

Mean-variance hedging and forward-backward stochastic differential filtering equations. (English) Zbl 1229.91327
Summary: This paper is concerned with a mean-variance hedging problem with partial information, where the initial endowment of an agent may be a decision and the contingent claim is a random variable. This problem is explicitly solved by studying a linear-quadratic optimal control problem with non-Markov control systems and partial information. Then, we use the result as well as filtering to solve some examples in stochastic control and finance. Also, we establish backward and forward-backward stochastic differential filtering equations which are different from the classical filtering theory introduced by R. S. Liptser and A. N. Shiryayev [Statistics of random processes. I. General theory. Translated by A. B. Aries. Applications of Mathematics. 5. New York etc.: Springer- Verlag (1977; Zbl 0364.60004)], J. Xiong [An introduction to stochastic filtering theory. Oxford Graduate Texts in Mathematics 18. Oxford: Oxford University Press (2008; Zbl 1144.93003)], and so forth.

MSC:
91G20 Derivative securities (option pricing, hedging, etc.)
49N10 Linear-quadratic optimal control problems
60H30 Applications of stochastic analysis (to PDEs, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Kohlmann and X. Zhou, “Relationship between backward stochastic differential equations and stochastic controls: a linear-quadratic approach,” SIAM Journal on Control and Optimization, vol. 38, no. 5, pp. 1392-1407, 2000. · Zbl 0960.60052
[2] H. Pham, “Mean-variance hedging for partially observed drift processes,” International Journal of Theoretical and Applied Finance, vol. 4, no. 2, pp. 263-284, 2001. · Zbl 1153.91554
[3] J. Xiong and X. Zhou, “Mean-variance portfolio selection under partial information,” SIAM Journal on Control and Optimization, vol. 46, no. 1, pp. 156-175, 2007. · Zbl 1142.91007
[4] Y. Hu and B. Øksendal, “Partial information linear quadratic control for jump diffusions,” SIAM Journal on Control and Optimization, vol. 47, no. 4, pp. 1744-1761, 2008. · Zbl 1165.93037
[5] R. S. Liptser and A. N. Shiryayev, Statistics of random processes, Springer, New York, NY, USA, 1977. · Zbl 0364.60004
[6] J. Xiong, An introduction to stochastic filtering theory, vol. 18, Oxford University Press, Oxford, UK, 2008. · Zbl 1144.93003
[7] G. Wang and Z. Wu, “Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 1280-1296, 2008. · Zbl 1141.93070
[8] G. Wang and Z. Wu, “The maximum principles for stochastic recursive optimal control problems under partial information,” IEEE Transactions on Automatic Control, vol. 54, no. 6, pp. 1230-1242, 2009. · Zbl 1367.93725
[9] J. Huang, G. Wang, and J. Xiong, “A maximum principle for partial information backward stochastic control problems with applications,” SIAM Journal on Control and Optimization, vol. 48, no. 4, pp. 2106-2117, 2009. · Zbl 1203.49037
[10] A. Bensoussan, Stochastic control of partially observable systems, Cambridge University Press, Cambridge, UK, 1992. · Zbl 0795.35008
[11] R. Merton, “Optimum consumption and portfolio rules in a continuous-time model,” Journal of Economic Theory, vol. 3, no. 4, pp. 373-413, 1971. · Zbl 1011.91502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.