[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.