Arbitrage et lois de martingale. (Arbitrage and martingale laws). (French) Zbl 0704.60045

Stricker, Christophe

Ann. Inst. Henri Poincaré, Probab. Stat. 26, No. 3, 451-460 (1990).

Let \(X=(X_ t;\;0\leq t\leq 1)\) be a vector-valued process on some filtered probability space (\(\Omega,\underline F,(\underline F_ t;0\leq t\leq 1),P)\). What are the necessary and sufficient conditions (NSC) on the stochastic integrals \((H\cdot X)_ 1\) for the existence of a probability measure Q, equivalent to P, such that X is a martingale with respect to Q?
Let \(K=\{(H\cdot X)_ 1\); H predictable, elementary and bounded\(\}\) and let \(L^ p_+\) be the set of nonnegative elements in \(L^ p(\Omega,\underline F,P)\) \((1\leq p<\infty)\). By means of a theorem of J.-A. Yan [Séminaire de Probabilités XIV, 1978/79, Lect. Notes Math. 784, 220-222 (1980; Zbl 0429.60004)], the author shows that \(\bar K\cap L^ p_+=\{0\}\) is an NSC for \(X\subset L^ p(\Omega,\underline F,P)\) with continuous trajectories. Moreover, the author shows that the condition is sufficient for processes X such that X is càdlàg and bounded, the jumps of X are predictable and there is only a finite number of them \((p=1).\)
The second result is a partial extension of a recent result of R. C. Dalang, A. Morton and W. Willinger [Stochastics Stochastics Rep. 29, No. 2, 185-201 (1990; Zbl 0694.90037)], who obtained an NSC in the discrete case. The problem of transforming processes into martingales by an equivalent change of measure is of particular interest in the analysis of stochastic models of securities markets.

Reviewer: M.Dozzi

MSC:

60G44

Martingales with continuous parameter

Keywords:

stochastic integrals; martingale; equivalent change of measure; stochastic models of securities markets

Citations:

Zbl 0429.60004; Zbl 0694.90037