连续两指标速降过程的Snell包的构造

THE STRUCTURE OF SNELL'S ENVELOPE FOR CONTINUOUS TWO INDEX RAPIDLY DECREASING PROCESSES

本文讨论了连续两指标速降过程X＝（Xz，Fz，zεR2）的Snell包的构造问题．令γ'z＝esssupE（Xo|Fz），其中Σ为（Fz）zεR2有限停点全体．本文首先证明了Γ'＝（γ'z,Fz,zεR2+）有连续适应修正J＝（Jz，Fz；zεR2+）．然后，利用上鞅收敛定理与walsh可选样本定理，证明了γz＝JzvXz,zεR2+是控制X的最小正则上鞅，即X的Snell包．

In this paper, the structure of Snell's envelope for continuous two index rapidlydecreasing processes is discussed. Set γ'z = esssup E(Xo | Fz), where Σ is all (Fx), z ε R2+finite stopping points. First, it is showed that г'= (γ'z,Fz,zε R2+) has adaptive rightcontinuous modification J = (Jz, Fz, z ε R2+). Then it is proved that γz = Jz VXz， zεR2+is minimum regular supermartingale of controlling X by using supermartingale convergencetheorem and Walsh's optional sample theorem i.e. Snell's envelope of X.