Completeness of securities market models – an operator point of view. (English) Zbl 0941.91019

The aim of this paper is to propose a novel approach to a securities market model, based on the notion of market completeness that is invariant under change to an equivalent probability measure.
Section 2 of the paper discusses the new approach to the market completeness, analyzing the second fundamental theorem of asset pricing from an operator theoretic point of view. The simple setting of a securities market model on a finite probability space allows one to point out the fundamental connection between completeness, operators and equivalent martingale measures. The market completeness means that an operator \(T\) acting on stopping time simple trading strategies has dense range in the weak* topology on bounded random variables.
In the exposed approach of Section 3, the space of claims which can be approximated by attainable ones has the codimension equal to the dimension of the kernel of the adjoint operator \(T^*\) acting on signed measures, which in most cases is equal to the dimension of the space of martingale measures.
Section 4 contains the Artzaer-Heath example illustrating the problems that arise when there are an infinite number of discontinuous price processes. In the new market completeness approach, the Artzner-Heath is no longer paradoxical since the space of attainable claims has the codimension one.
The final Sections 5 and 6 show how to check for the injectivity of \(T^*\), hence for the completeness in the case of price processes on a Brownian filtration, and price processes driven by a multivariate point process.


91B02 Fundamental topics (basic mathematics, methodology; applicable to economics in general)
60H30 Applications of stochastic analysis (to PDEs, etc.)
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[1] ANSEL, J.-P. and STRICKER, C. 1992. Lois de martingales, densities, et decomposition de \' \' Follmer Schweizer, Ann. Inst. H. Poincare Probab. Statist. 28 375 392. Ź. · Zbl 0772.60033
[2] ANSEL, J.-P. and STRICKER, C. 1994. Couverture des actifs contingents et prix maximum. Ann. Inst. H. Poincare Probab. Statist. 30 303 315. Ź. · Zbl 0796.60056
[3] ARTZNER, P. and HEATH, D. 1995. Approximate completeness with multiple martingale measures. Math. Finance 5 1 11. Z. · Zbl 0872.60032
[4] BJORK, T., DI MASI, G., KABANOV, Y. and RUNGGALDIER, W. 1997. Towards a general theory of \" bond markets. Finance and Stochastics. To appear. Z.
[5] BJORK, T., KABANOV, Y. and RUNGGALDIER, W. 1996. Bond market structure in the presence of \" marked point processes. Preprint. Z.
[6] DALANG, R. C., MORTON, A. and WILLINGER, W. 1990. Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stochastics Stochastic Rep. 29 185 201. Z. · Zbl 0694.90037
[7] DELBAEN, F. and SCHACHERMAy ER, W. 1994. A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463 520. Z. · Zbl 0865.90014
[8] DELLACHERIE, C. and MEy ER, P.-A. 1975. Probabilites et Potentiel, 1 3. Hermann, Paris. Ź.
[9] GROTHENDIECK, A. 1973. Topological Vector Spaces. Gordon and Breanch, Montreux. · Zbl 0275.46001
[10] HARRISON, J. M. and KREPS, D. M. 1979. Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20 381 408. Z. · Zbl 0431.90019
[11] HARRISON, J. M. and PLISKA, S. R. 1981. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11 215 260. Z. · Zbl 0482.60097
[12] HE, S.-W., WANG, J.-G. and YAN, J.-A. 1992. Semimartingale Theory and Stochastic Calculus. Science Press, Beijing. Z. · Zbl 0781.60002
[13] HEATH, D., JARROW, R. and MORTON, A. 1992. Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 60 77 105. Z. · Zbl 0751.90009
[14] JACOD, J. 1975. Multivariate point processes: predictable projection, Radon Nikody m derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235 253. Z. · Zbl 0302.60032
[15] JACOD, J. 1979. Calcul Stochastique et Problemes de Martingales. Lecture Notes in Math. 714. Springer, Berlin. Z. · Zbl 0414.60053
[16] JACOD, J. 1980. Integrales stochastiques par rapport a une semimartingale vectorielle et changments de filtration. Seminaire de Probabilite XIV. Lecture Notes in Math. 784 \' \' 161 172. Springer, Berlin. Z. · Zbl 0429.60054
[17] JACOD, J. and SHIRy AEV, A. N. 1987. Limit Theorems for Stochastic Processes. Springer, Berlin. Z.
[18] JARROW, R. A. and MADAN, D. B. 1997. Hedging contingent claims on semimartingales. Finance and Stochastics. To appear. Z. JEANBLANC-PICQUE, M. and PONTIER, M. 1990. Optimal portfolio for a small investor in a ḿarket model with discontinuous prices. Appl. Math. Optim. 22 287 310. Z.
[19] LAKNER, P. 1993. Martingale measures for a class of right-continuous processes. Math. Finance 3 43 53. Z. · Zbl 0884.90039
[20] MERCURIO, F. and RUNGGALDIER, W. J. 1993. Option pricing for jump diffusions: approximations and their interpretation. Math. Finance 3 191 200. Z. · Zbl 0884.90043
[21] MERTON, R. C. 1976. Option pricing when underlying stock returns are discontinuous. J. Financial Econom. 3 125 144. Z. · Zbl 1131.91344
[22] REVUZ, D. and YOR, M. 1991. Continuous Martingales and Brownian Motion. Springer, Berlin. Z. · Zbl 0731.60002
[23] RUDIN, W. 1991. Functional Analy sis, 2nd ed. McGraw-Hill, New York. Z.
[24] SCHACHERMAy ER, W. 1994. Martingale measures for discrete-time processes with infinite horizon. Math. Finance 4 25 55. Z. · Zbl 0893.90017
[25] TAQQU, M. S. and WILLINGER, W. 1987. The analysis of finite security markets using martingales. Adv. Appl. Probab. 19 1 25. JSTOR: · Zbl 0618.60047
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