No-arbitrage And Market Completeness

On discrete finite horizon finance market, several equivalence forms of self financing strategies are given, and any d-dimension adapted processes can be modified to a self financing strategy. A new idea about no-arbitrage is obtained, i.e. . a positive process will never be negative. The market is arbitrage-free iff equivalent martingale measure exists. An attainable contingent claim can be predictable represented, all contingent claim attainable when the market is complete. The market is complete iff there exists a martingale with the predictable representation property, and the martingale can only take several values, then the filter is finite-generated and finite-generated filter lead to complete market. Therefore we have four different forms of statements about market completeness.On continuous finite horizon finance market especially the It(o|^)'s market, an equivalence forms of self financing strategy is obtained, and any d-dimension adapted processes can be modified to a self financing strategy, too. An attempt is made to prove the law of the iterated logarithm, and by the means of random time change we discuss the relationship between the Brownian motion and a continuous local martingale which resulting in a criterion of the uniformly integrability of the Doleans-Dade exponential martingale and to the classic Girsanov theorem, a limitation to finite time interval is needed. The Brownian motion is the martingale with the predictable representation property with respect to the equivalent martingale measure, by using the It(o|^) formula we connect the deflated basic asset process with the Q-Brownian motion under the equivalent martingale measure, so in some case the deflated basic asset process became the process with the predictable representation property, finally we get a statement equivalent to market completeness.