期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2003
卷号：65
期号：04
出版社：Indian Statistical Institute
摘要：Let $\mathcal{X}$ be the set of real functions on $[0,\infty)$ which are c\`{a}dl\`{a}g or c\`{a}gl\`{a}d, $\mathcal{B}$ the Borel $\sigma$-algebra on $\mathcal{X}$ (with respect to the topology of uniform convergence on compacts), and $BV=\{x\in\mathcal{X}: x$ has bounded variation on compacts$\}$. Fix a probability $P$ on $\mathcal{B}$ such that $P(C[0,\infty))=1$ and the coordinate process $X(t,x)=x(t)$ is a local martingale under $P$, and define the Ito integral $I(t)=\int_0^t Y dX$ for a suitable process $Y:[0,\infty)\times\mathcal{X}\rightarrow\mathbb{R}$. For $x\in BV$, define also the Lebesgue-Stieltjes integral $J(x)(t)=\int_{[0,t]} Y(s,x) dx(s)$. A finitely additive probability $Q$ on $\mathcal{B}$ is constructed such that $Q(BV)=1$, and, for a large class of integrands $Y$, $P\{I\in A\}=Q\{J\in A\}$ whenever $A\in\mathcal{B}$ and $P\{I\in\partial A\}=0$. In particular, $P\{I(t_1)\leq a_1,\ldots,I(t_k)\leq a_k\}=Q\{J(t_1)\leq a_1,\ldots,J(t_k)\leq a_k\}$ provided $P\{I(t_i)= a_i\}=0$ for all $i$. Such $Q$ is useful for connecting Ito integral $I$ with pathwise (Lebesgue-Stieltjes) integral $J$.
关键词：Finite additivity, pathwise integral, stochastic integral, weak convergence of probability measures.
