Time-changed squared Brownian motion

Question

To build a deterministic time-changed Brownian motion, we have $$\int_0^t \zeta_s dW_s \sim \int_0^t dW_{\xi_s} = \int_0^{\xi_t} dW_s = W_{\xi_t},$$ where $${\xi_t}=\int_0^t \zeta_s^2 ds.$$ Is there a similar time change that can be built for $\int_0^t \zeta_s^2 dW_s^2=$$\int_0^t \zeta_s^2 (ds+2W_s dW_s)$ ?

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Aside from the abusive notation, is it possible to reconcile the two following (it would work for $\zeta_t=$ constant)?

1. $\zeta_t^2 dW_t^2=\zeta_t^2 (dt+2W_t dW_t)=\zeta_t^2dt+2\zeta_t^2 W_t dW_t =d\xi_t+2\zeta_t W_t dW_{\xi_t}$
2. $dW_{\xi_t}^2=d\xi_t+2W_{\xi_t} dW_{\xi_t}$

Answer 1

Leaving aside the abuse of notations... With $\xi_{t}=\int_{0}^t\zeta_{s}^2ds$, $$W_{\xi_{t}}^{2}=\int_{0}^{\xi_{t}}dW_{s}^{2} \\=\int_{0}^{\xi_{t}}(ds+2W_{s}dW_{s}) \\=\int_{0}^{t}(d\xi_{s}+2W_{\xi_{s}}dW_{\xi_{s}}) \\\overset{\mathrm{law}}{=}\int_{0}^{t}\left(\zeta_{s}^{2}ds+2\sqrt{\frac{\xi_{s}}{\zeta_{s}^{2}s}}\zeta_{s}W_{s}\zeta_{s}dW_{s}\right) \\=\int_{0}^{t}\zeta_{s}^{2}\left(ds+2\sqrt{\frac{\xi_{s}}{\zeta_{s}^{2}s}}W_{s}dW_{s}\right) \\=\int_{0}^{t}\zeta_{s}^{2}\left(\left(1-\sqrt{\frac{\xi_{s}}{\zeta_{s}^{2}s}}\right)ds+\sqrt{\frac{\xi_{s}}{\zeta_{s}^{2}s}}(ds+2W_{s}dW_{s})\right) \\=\int_{0}^{t}\zeta_{s}\left((\zeta_{s}-\sqrt{\xi_{s}/s})ds+\sqrt{\xi_{s}/s}dW_{s}^{2}\right)$$

Answer 2

We know $$\tag{1} W_t^2=t+2\int_0^tW_s\,dW_s\,. $$ It is not an abuse of notation to write this as $$\tag{2} d(W_t^2)=dt+W_t\,dW_t\,. $$ Further, it is not an abuse of notation to plug into (1) any $[0,\infty)$-valued stochastic process you like instead of $t\,:$ $$ W_{\xi t}^2=\xi_t+2\int_0^{\xi t}W_s\,dW_s\,. $$ Since (2) is only a shorthand notation for (1) I think this is as good as it gets. I'd be careful messing around with expressions like $$ dW_{\xi_t}^2=d\xi_t+2W_{\xi_t} dW_{\xi_t}\,. $$ Formulas cannot take the burden away from us to know what we are doing.

If you analyse your attempt 1 in the same fashion (use always the integral form) you should quickly see where you run into problems.