| WANG JIAGANG.[J].数学年刊A辑,1981,2(1):13~20 |
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| THE UNIFORM INTEGRABILITY OF A CLASSOF EXPONENTIAL MARTINGALES |
| Received:December 25, 1979 |
| DOI： |
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| 若说\[(\Omega ,\mathcal{F},P)\]为完备概率空间，\[F = {({\mathcal{F}_t})_{t \in [a,b]}}\]为\[\mathcal{F}\]的递增子\[\sigma \]域族，且满足通常
条件，\[b \leqslant \infty \].又\[W = \{ {W_t},0 \leqslant t \leqslant b\} \]为关于F的Wiener过程，\[X = \{ {X_t},0 \leqslant t < b\} \]为
循序讨测过程，且
\[P\{ \int_0^b {X_t^2} dt < \infty \} = 1\]，
则可定义X关于W的Ito随机积分
\[{(X \cdot W)_t} = \int_0^t {{X_s}} d{W_s},0 \leqslant t \leqslant b\]
这时若记
\[{Z_t} = \exp \{ \int_0^t {{X_s}} d{W_s} - \frac{1}{2}\int_0^t {{X_s}^2} ds\} \]
它便是一个指数（局部）鞅.本文的目的在于证明当X为循序可测正态过程时，只要X关于W的积分存在，\[{\text{\{ }}{Z_t}0 \leqslant {\text{t < b\} }}\]总是一致可积的。
引理1若\[\{ {Z_t},0 \leqslant t < b\} \]为实可测正态过程且
\[\int_0^{\text{b}} {\left\| {{X_t}} \right\|} d{m_t} < \infty \]
其中\[\left\| {{X_t}} \right\| = {(E|{X_t}{|^2})^{1/2}}\],\[{m_t}\]为[0,b)上右连续递增函数，则X的几乎所有样本函数关于\[{m_t}\]可积，且其轨道积分
\[\tilde I = \int_0^{\text{b}} {{X_t}} d{m_t}\]
为正态分布随机变量.
引理2若\[X = \{ {X_t},0 \leqslant t < b\} \]为可测正态过程，其几乎所有样本函数关于右连续增函数\[{m_t}\]可积，即
\[P(\int_0^b {|{X_t}} |d{m_t} < \infty ) = 1\]
则按轨道积分 \[\tilde I = \int_0^{\text{b}} {{X_t}} d{m_t}\]
是正态分布随机变量.
引理3 若\[\{ {\xi _n},n \geqslant 1\} \]为正态分布随机变量序列，则
\[\sum\limits_{j = 1}^\infty {E{\xi _i}^2} \leqslant {[Eexp( - \frac{1}{2}\sum\limits_{j = 1}^\infty {{\xi _i}^2} )]^{ - 2}}\]
进而若\[\sum\limits_{j = 1}^\infty {E{\xi _i}^2} < 1\],则
\[E[exp(\frac{1}{2}\sum\limits_{j = 1}^\infty {{\xi _i}^2} )] \leqslant {(1 - \sum\limits_{j = 1}^\infty {E{\xi _i}^2} )^{ - \frac{1}{2}}}\]
引理4若\[{m_s}\]为[0, b)上右连续增函数，又\[X = \{ X_t^{(i)},0 \leqslant t < b,1 \leqslant i < \infty \} \]为正态
过程，则当\[P\{ \sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_t}} } < \infty \} = 1\]时必有
\[\sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_t}} } < \infty \} = 1\]
进而若;\[\sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_t}} } < 1\]，必有
\[Eexp(\frac{1}{2}\sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_s}} } ) \leqslant {(1 - \sum\limits_{j = 1}^\infty {E\int_0^b {{{({X_t}^{(i)})}^2}d{m_s}} } )^{ - \frac{1}{2}}}\]
定理 若\[W = (W_t^{(1)},...,W_t^{(n)},...)\]为一个具有无限个分量的过程，其分量都是连续
正态独立增量过程且满足
\[\begin{gathered}
E\{ W_t^{(i)} - W_s^{(i)}\} = 0 \hfill \ E\{ (W_t^{(i)} - W_s^{(i)})(W_t^{(j)} - W_s^{(j)})\} = {\delta _{ij}}(m_t^{(i)} - m_s^{(i)}) \hfill \\
\end{gathered} \]
又\[\{ {f_t} = (f_t^{(1)},...,f_t^{(n)},...)\} \]为循序可测正态过程，若
\[P\{ \sum\limits_{i = 1}^\infty {\int_0^b {{{({f_t}^{(i)})}^2}dm_t^{(i)}} } < \infty \} = 1\]
则 \[{Z_t} = \exp \{ \sum\limits_{i = 1}^\infty {\int_0^b {{f_s}^{(i)}dW_s^{(i)} - \frac{1}{2}\int_0^t {{{({f_s}^{(i)})}^2}dm_s^{(i)}} } } \} ,0 \leqslant t < b\]
是一致可积鞅，特别有\[E{Z_0} = 1\]
利用上述结果及正态过程的Hida-Cramer分解，可以象[1]一样方便地讨论正态测
度的等价性问题并求出其Radon-Nikodym导数. |
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