关于局部平方可积鞅的重对数律
THE LIL FOR LOCALLY SQUARE INTEGRABLE MARTINGALE
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摘要: 设 \left(M_t, \mathcalF_t, t \geqslant 0\right) 为 (\Omega, \mathcalF, P) 上的局部平方可积鞙, (简记 M \in m_i o f^2 ), M_0=0,\left\\mathscrF_\\right\满足通常条件, \mathscrF_0=\\phi, \Omega\,\langle M\rangle 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, \lim _t \rightarrow \infty k_t=0 a.s., 使得\left.\left|\Delta M_t\right| \leqslant k_k_0 \cdot\langle M\rangle_1^1 / 2 /\left2 \lg _2 k^2 V\langle M\rangle_f\right)\right^1 / 2 \quad \text a.s., \langle M\rangle_m=\infty,则\varlimsup_t \rightarrow \infty M_1 /\left2\langle M\rangle, \lg _2\left(e^2 V\langle M\rangle_0\right)\right^1 / 2=1 \text a.s. ii) 若存在可料过程 K_t, t>0 和常数 0Abstract: 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 disorete-time locally square integrable martingale.
Let (\Omega, F, P) be a probability space with a filtration, \left\F_t, t \geqslant 0\right\ F_0=\\phi, \Omega\; Let M=\left(M_t, t \geqslant 0\right) be a locally square integrable martingale, M_0=0,\langle M\rangle be the compensator of M^2, we have the following Theorem 1. Let \left\k_t, t \geqslant 0\right\ be \left\F_t\right\ predicatable, sappose for some constant 0
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