The stopping theorem of strengthened form of two parameter right continuous strong martingale (strong super martingale,strong sub martingale) on stopping line under normal F 1-F 3 conditions is proved and generalized.

Submartingale Analysis of Convergence Rate Genetic Algorithm;

The paper studied the convergence problems on partial sums of adapted stochastic sequence with submartingale convergence theorem, stopping time and conditional three series theorem.

Assuming that the discount sequence {Yi} and sequence {Zi} are dependent,where Zi is the difference between premium and claim of term i,the upbound of ruin probability is obtained by sub-martingale method.

Under the condition of changing premium, the upbound of ruin probability was obtained by sub-martingale property.

The present paper presents the definition of the stopping for two-parameter strong martingale and deals with the stopping properties for two-parameter strong martingale.

The stopping theorem of strengthened form of two parameter right continuous strong martingale (strong super martingale,strong sub martingale) on stopping line under normal F 1-F 3 conditions is proved and generalized.

The principal results include stopping theorem of weak martingale and strong martingale on stopping point and strong stopping point.

Theorem 2 gives the sufficient and necessary condition of continuous parameter set-valued submartingale existing Doob-Meyer decomposition.