[Lemma 5 of JPS07]. A process is defined by the solution of
Show that is a continuous strict local martingale, with the last element
.
Proof: Let
For any , because
,
is a well-defined Martingale on
by Proposition 3.2.10 of [KS98].
Next, one can show that, on
is a strong solution of (1), and the uniquness follows by Theorem IX.3.5(ii) of [RY99]. Thus, is local martingale on
.
With function of
, consider time-changed process
and
One can check and
is a continuous local martingale. By Levy’s characterization theorem,
is a Brownian motion w.r.t.
. Note that
as by Law of Iterated Logarithm. This implies
is strict local martingale with
i don’t know, what it is, mabye it’s math.
I guess you are right.