×

The Föllmer-Schweizer decomposition: comparison and description. (English) Zbl 1196.60077

The authors elaborate upon the relationship between the Föllmer-Schweizer decomposition and the Galtchouk-Kunita-Watanbe decomposition under the minimal martingale measure. The key difference between these two decompositions is exhibited through an example.
The Föllmer-Schweizer decomposition relates to the local risk-minimization concept. The authors cite many references describing the FS-decomposition, including one by J. P. Ansel and C. Stricker [Ann. Inst. Henri Poincaré, Probab. Stat. 28, 375–392 (1992; Zbl 0772.60033)]. If the price of the discounted risky asset is a martingale, the decomposition coincides with the Galtchouk-Kunita-Watanbe decomposition. The FS-decomposition and GKW-decomposition coincide under minimal martingale measure. This relationship breaks down when jumps are involved. The main difference between the two decompositions relates to the orthogonality and local martingales.
Section 2 presents the preliminary material while the third section compares the descriptions. The final section provides a description of the FS-decomposition in terms of a discounted stock price process.

MSC:

60G46 Martingales and classical analysis

Citations:

Zbl 0772.60033
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albert, A., Regression and the Moore-Penrose pseudoinverse, (Mathematics in Science and Engineering, vol. 94 (1972), Academic Press) · Zbl 0253.62030
[2] Ansel, J.-P.; Stricker, C., Lois de martingale, densités et décomposition de Föllmer-Schweizer, Annales de l’institut Henri-Poincaré-Probabilités et Statistiques, 28, 3, 375-392 (1992) · Zbl 0772.60033
[3] Ansel, J.-P.; Stricker, C., Décomposition de Kunita-Watanabe, Seminaire de Probabilités de Strasbourg, 27, 30-32 (1993) · Zbl 0788.60057
[4] Benth, F.; Meyer-Brandis, T., The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps, Finance and Stochastics, 9, 563-575 (2005) · Zbl 1092.91020
[5] Biagini, F.; Cretarola, A., Quadratic hedging methods for defaultable claims, Applied Mathematics and Optimization, 56, 3, 425-443 (2007) · Zbl 1142.91028
[6] Biagini, F.; Cretarola, A., Local risk-minimization for defaultable markets, Mathematical Finance, 19, 4, 669-689 (2009) · Zbl 1185.91092
[7] F. Biagini, A. Cretarola, Local risk-minimization for defaultable claims with recovery process, Tech. Rep., LMU University of München and University of Bologna, 2006.; F. Biagini, A. Cretarola, Local risk-minimization for defaultable claims with recovery process, Tech. Rep., LMU University of München and University of Bologna, 2006. · Zbl 1244.93152
[8] Choulli, T.; Krawczyk, L.; Stricker, C., \(E\)-martingales and their applications in mathematical finance, Annals of Probability, 26, 2, 853-876 (1998) · Zbl 0938.60032
[9] Choulli, T.; Stricker, C., Deux applications de la décomposition Galtchouk-Kunita-Watanabe, Séminaire de Probabilités de Strasbourg, 30, 12-23 (1996) · Zbl 0877.60028
[10] Choulli, T.; Stricker, C., Minimal entropy-Hellinger martingale measure in incomplete markets, Mathematical Finance, 15, 3, 465-490 (2005) · Zbl 1136.91419
[11] Choulli, T.; Stricker, C., More on minimal entropy-Hellinger martingale measure, Mathematical Finance, 16, 1, 1-19 (2006) · Zbl 1136.91420
[12] Choulli, T.; Stricker, C.; Li, J., Minimal Hellinger martingale measures of order \(q\), Finance and Stochastics, 11, 3, 399-427 (2007) · Zbl 1164.60035
[13] Colwell, D.; Elliott, R., Discontinuous asset prices and non-attainable contingent claims, Mathematical Finance, 3, 3, 295-308 (1993) · Zbl 0884.90021
[14] Cont, R.; Tankov, P., Financial Modelling with Jump Processes (2004), Chapman & Hall · Zbl 1052.91043
[15] Delbaen, F.; Monat, P.; Stricker, C.; Schachermayer, W.; Schweizer, M., Weighted norm inequalities and hedging in incomplete markets, Finance and Stochastics, 1, 3, 181-227 (1997) · Zbl 0916.90016
[16] Delbaen, F.; Schachermayer, W., The Mathematics of Arbitrage, Finance (2006), Springer · Zbl 1106.91031
[17] Dellacherie, C.; Meyer, P.-A., Théorie des Martingales, (Probabilités et potentiel (1980), Hermann), (Chapters V-VIII)
[18] Föllmer, H.; Schweizer, M., Hedging of contingent claims under incomplete information, (Davis, M.; Elliot, R., Applied Stochastic Analysis. Applied Stochastic Analysis, Stochastic Monographs, vol. 5 (1991), Gordon and Breach), 389-414 · Zbl 0738.90007
[19] Föllmer, H.; Sondermann, D., Hedging of non-redundant contingent claims, (Hildenbrand, W.; Mas-Colell, A., Contributions to Mathematical Economics (1986), North-Holland: North-Holland Elsevier), 205-223 · Zbl 0663.90006
[20] Fujiwara, T.; Miyahara, Y., The minimal entropy martingale measure for geometric Lévy processes, Finance and Stochastics, 7, 509-531 (2003) · Zbl 1035.60040
[21] Jacod, J., Calcul stochastique et problèmes de martingales, (Lecture Notes in Mathematics, vol. 714 (1979), Springer-Verlag) · Zbl 0414.60053
[22] Jacod, J.; Shiryaev, A., Limit Theorems for Stochastic Processes (2002), Springer
[23] Jeanblanc, M.; Klöppel, S.; Miyahara, Y., Minimal \(f^q\)-martingale measures for exponential Lévy processes, Annals of Applied Probability, 17, 5-6, 1615-1638 (2007) · Zbl 1140.60026
[24] S. Kassberger, T. Liebmann, \(q\); S. Kassberger, T. Liebmann, \(q\) · Zbl 1302.60073
[25] Riesner, M., Hedging life insurance contracts in a Lévy process financial market, Insurance: Mathematics and Economics, 38, 599-608 (2006) · Zbl 1168.91419
[26] Rheinländer, T.; Steiger, G., The minimal entropy martingale measure for general Barndorff-Nielsen/Shephard models, Annals of Applied Probability, 16, 3, 1319-1351 (2006) · Zbl 1154.28305
[27] Schweizer, M., Risk-minimality and orthogonality of martingales, Stochastics and Stochastics Reports, 30, 1, 123-131 (1990) · Zbl 0702.60049
[28] Schweizer, M., Option hedging for semimartingales, Stochastic Processes and their Applications, 37, 339-363 (1991) · Zbl 0735.90028
[29] Schweizer, M., Approximating random variables by stochastic integrals, Annals of Probability, 22, 3, 1536-1575 (1994) · Zbl 0814.60041
[30] Schweizer, M., On the minimal martingale measure and the Föllmer-Schweizer decomposition, Stochastic Analysis and Applications, 13, 573-599 (1995) · Zbl 0837.60042
[31] Schweizer, M., Approximation pricing and the variance-optimal martingale measure, Annals of Probability, 24, 1, 206-236 (1996) · Zbl 0854.60045
[32] Schweizer, M., A guided tour through quadratic hedging approaches, (Jouini, E.; Cvitanić, J.; Musiela, M., Option Pricing, Interest Rates and Risk Management (2001), Cambridge University Press), 538-574 · Zbl 0992.91036
[33] Vandaele, N.; Vanmaele, M., A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a Lévy process financial market, Insurance: Mathematics and Economics, 42, 3, 1128-1137 (2008) · Zbl 1141.91549
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.