摘要
设(M_t,f_t,t≥0)为(Ω,f,P)上的局部平方可积鞅,(简记M ∈m_100~2),M_=0,{f_t}满足通常条件,f_0={Φ,Ω},<M>为M^2的可料补偿,本文证明了如下结论: ⅰ)若存在可料过程k_t t≥0,k_t=0 a.s.,使得 |△M_t|≤k_t·<M>_t^(1/2)/[2lg_2 k^2V<M>_t)]^(1/2) a.s.,<M>_∞=∞, 则 M_t/[2<M>_t lg_2(e^2V<M>_t)]^(1/2)=1 a.s. ⅱ)若存在可料过程K_t,t≥0和常数0<K≤1/2,使K_t<K △M_t≤k_t<M>_t^(1/2)/[21g_2(e^2V<M>_t)]^(1/2) a.s. 则存在0<8(K)<1,↓8(K)=0,使在<M>_∞=∞上, M_t/[2<M>_t lg_2(e^2V<M>_t)]^(1/2)≤1+8(K) a.s.
In this paper, we derive the LIL for continuous-time locally square integrable martingale whose jumps grow at a controlled rate. The results generalize Stout's LIL for discrete-time locally square integrable martingale. Let (Ω, F, P) be a probability space with a filtration, {F_t, t≥0} F_0={φ, Ω}; Let M=(M_t, t≥0) be a locally square integrable martingale, M_0=0, <M> be the compensator of M^2, we have the following Theorem 1. Let {k_t, t≥0} be {F_t} predicatable, suppose for some constant 0<k≤1/2 such that k_t<k, a. s. and ΔM_t≤k_t<M>_t^(1/2)/[2lg_2(e^2∨<M>_t)]^(1/2), a. s. for all t≥0. Then there exists a funotion ε(·) which satisfies ε(k)<1, ε(k)↓0 (k↓0) such that M_t/[2<M>_tlg_2(e^2∨<M>_t)]^(1/2)≤1+ε(k) a. s. on {<M>_∞=∞}. Theorem 2, Suppose <M>_∞=∞ a. s., Let {k_t, t≥0}be {F_t}predictable, k_t=0 a. s. and |ΔM_t|≤k_t<M>_t^(1/2)/[2lg_2(e^2∨<M>_t)]^(1/2) a. s. for all t≥0. Then M_t/[2<M>_tlg_2(e^2∨<M>_t)]^(1/2)=1, a. s.
出处
《应用概率统计》
CSCD
北大核心
1990年第3期290-301,共12页
Chinese Journal of Applied Probability and Statistics
