Optional decompositions under constraints. (English) Zbl 0882.60063

Föllmer, H.; Kramkov, D.

Probab. Theory Relat. Fields 109, No. 1, 1-25 (1997).

Summary: Motivated by a hedging problem in mathematical finance, N. El Karoui and M.-C. Quenez [SIAM J. Control Optimization 33, No. 1, 29–66 (1995; Zbl 0831.90010)] and the second author [Probab. Theory Relat. Fields 105, No. 4, 459–479 (1996; Zbl 0853.60041)] have developed optional versions of the Doob-Meyer decomposition which hold simultaneously for all equivalent martingale measures. We investigate the general structure of such optional decompositions, both in additive and in multiplicative form, and under constraints corresponding to different classes of equivalent measures. As an application, we extend results of I. Karatzas and J. Cvitanić [Ann. Appl. Probab. 3, No. 3, 652–681 (1993)] on hedging problems with constrained portfolios.

MSC:

60H30	Applications of stochastic analysis (to PDEs, etc.)
60H05	Stochastic integrals
60G44	Martingales with continuous parameter
91G10	Portfolio theory
93E20	Optimal stochastic control

Keywords:

hedging problem in mathematical finance; optional decompositions; additive and multiplicative form