On martingale measures for stochastic processes with discrete time

• Autorzy: Czkwianianc E. , Paszkiewicz A.
• Języki publikacji: EN

Abstrakty

EN

Let (X(t); t ϵ N⁺) be a random sequence adopted to a filtration (ℱ_t) in (Ω, ℱ, P ) satisfying some natural assumption. If none of the events (X (t + 1) > X (t)), (X (t + 1) < X (t)) can be predicted, i.e, none contains some A € ℱ_t, P (A) > 0, then (X (t), ℱ_t) is a martingale for some probability P* on ℱ. It is a version of the "fundamental theorem of option pricing".

Słowa kluczowe

EN

stochastic process; Banach-Mazur theory; classical Kolmogorov theory; random variable

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 1999
• Tom: Vol. 19, Fasc. 1
• Strony: 203--209
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Czkwianianc E.

• Łódź Technical University, Institute of Mathematics, Żwirki 36, 90-924 Łódź, Poland

autor

Paszkiewicz A.

• Department of Mathematics, Łódź University, Banacha 22, 90-238 Łódź, Poland

Bibliografia

• [1] F. Black and M. Scholes, The pricing of options and corporate liabilities; J. Political Economy 3 (1973), pp. 637-659.
• [2] R. C. Dalang, A. Morton and W. Willinger, Equivalent martingale measures and no-arbitrage in stochastic securities market models, Stochastics and Stochastic Reports 29 (1990), pp. 185-201.
• [3] F. Delbean and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann. 300 (1994), pp. 463-520.
• [4] I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer, Berlin 1972.
• [5] - Theory of Stochastic Processes, Vol. 1, Springer, New York-Berlin 1974-1979.
• [6] J. M. Harrison and D. M. Kreps, Martingales and arbitrage in multiperiod security markets, J. Econom. Theory 20 (1979), pp. 381-408.
• [7] J. M. Harrison and S. R. Pliska, A stochastic calculus modes of continuous trading: complete markets, Stochastic Process. Appl. 15 (1983), pp. 313-316.
• [8] - Martingales and stochastic arbitrage in multiperiod security markets, J. Econom. Theory 20 (1979), pp. 381-408.
• [9] E. Jouini, Market imperfections, equilibrium and arbitrage, in: B. Biais et al., Financial Mathematics, Lecture Notes in Math. 1656 (1997), pp. 247-307.
• [10] I. Karatzas, Lectures on the Mathematics and Finance, Centre de Recherches Mathématiques Université de Montréal, Amer. Math. Soc., CRM Monograph Series 8 (1997).
• [11] - and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Springer, New York 1991.
• [12] M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, Springer, 1997.
• [13] M. M. Rao, Stochastic Processes: General Theory, Kluwer Acad. Publ., Math. Appl. 342 (1995).