§ 04 / localization

Localization and Local Martingales

Extending the Itô integral beyond L² and the language of local martingales.

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Section 1 / 4

Localization and Local Martingales — Overview

The Itô integral defined in the previous chapter still leaves room for improvement. One weakness is that it is not necessarily defined even for some very friendly integrands. For example, one would like

∫0Tf(Bt) dBt,f:R→R continuous,\int_0^T f(B_t)\,dB_t,\qquad f:\R\to\R\text{ continuous},

to be well-defined. With the previous definition this requires f(Bt(ω))∈H2[0,T]f(B_t(\omega))\in H^2[0,T], which fails, for example, for f(x)=exp⁡(x2)f(x)=\exp(x^2). Such integrals occur frequently in applications.

We therefore extend the domain of the Itô integral by localization. Localization also leads to the important concept of a local martingale.