The Multiplicity of an Increasing Family of $\Sigma$-Fields

Abstract

Let $(\Omega, \mathscr{F}, P)$ be a probability space and let $\mathscr{F}_t, t \in R_+$, be an increasing family of sub-$\sigma$-fields of $\mathscr{F}$ such that $\mathscr{F}_0$ is trivial and $\mathscr{F} = V_t\mathscr{F}_t$. Let $\mathscr{M}^2$ be the family of all square-integrable martingales $m_t$ with $m_0 = 0$. Suppose that $L^2(\Omega, \mathscr{F}, P)$ is separable. Then there exists a finite or countable sequence in $\mathscr{M}^2, m_t^1, m_t^2, \cdots$, such that (i) the stable subspaces generated by $m_t^i, m_t^j$ are orthogonal for $i \neq j$; (ii) $\langle m^1\rangle \succ \langle m^2\rangle \succ\cdots$ where $\langle m^i\rangle$ is the nonnegative measure on the predictable $\sigma$-field on $\Omega \times R_+$ induced by the quadratic variation process $\langle m^i\rangle$ of $m^i$, and (iii) every $m$ in $\mathscr{M}^2$ has a representation $m_t = \sum_i \int^t_0 \phi_i(s) dm_s^i$ a.s. for some predictable integrands $\phi_i$. Furthermore, if $n_t^1, n_t^2, \cdots$ is another such sequence, then $\langle n^i\rangle \sim \langle m^i\rangle$ for all $i$.