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Let $\{B_t:t\geq 0\}$ be a standard Brownian Motion and let $\{\mathcal{F}_t\}_{t\geq 0}$ be the natural filtration associated to Brownian Motion (that is, $\mathcal{F}_t=\sigma(B_s:0\leq s\leq t)$). Fix $T>0$. Let $u:[0,T]\times\Omega\rightarrow\mathbb{R}$ be a measurable function such that $u\in L_{a,T}^2$, that is:

  1. $u_t$ is $\mathcal{F}_t$ measurable for all $t\in [0,T]$.
  2. $$E\left[\int_0^Tu_t^2dt\right]=\int_{[0,T]\times\Omega} u_t^2(\omega)dt\times dP(\omega)<\infty.$$

Under these hypotheses, one can define $$ \int_0^Tu_tdB_t $$ as a limit in $L^2(\Omega)$ of Itô's integrals of step processes.

Is there a way to generalize this concept in order to define integrals such as $$ \int_{-\infty}^t u_sdB_s, \; \int_{t}^{\infty}u_sdB_s\,? $$

user311684
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  • Actually, they can be defined exactly the same way. E.g. if $$\mathbb{E} \left( \int_0^{\infty} u_t^2 , dt \right)<\infty,$$ then we can define the integral $$\int_t^{\infty} u_s , dB_s$$ for any $t \geq 0$. – saz Feb 08 '16 at 15:17

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