It is a well known results that if $f$ is deterministic and Ito integrable, then $\int_0^t f(s) dB(s)$ is Gaussian. Is the converse true? If $\int_0^t f(s) dB(s)$ is Gaussian, then is $f$ deterministic?
2026-03-26 12:07:03.1774526823
Is it true that an Ito integral is Gaussian if and only if the integrand is deterministic?
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No. By the Lévy criterion, $$ W(t) = \int_0^t \operatorname{sign} B(s) dB(s) $$ is a standard Wiener process.