2003, Volume 65, Pt. 4 , 715--732

INTEGRATION WITH RESPECT TO BROWNIAN-LIKE PROCESSES

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PATRIZIA BERTI, Universit\`{a} di Modena e Reggio-Emilia, Modena, Italy                               EUGENIO REGAZZINI, Universit\`{a} di Pavia, via Ferrata 1, 27100 Pavia, Italy and              PIETRO RIGO, Universit\`{a} di Pavia, via S. Felice 5, 27100 Pavia, Italy

SUMMARY. 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 $J$.

AMS (1991) subject classification. 60H05, 60A05, 60B10.

Key words and phrases. Finite additivity, pathwise integral, stochastic integral, weak convergence of probability measures.

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