Malliavin derivative under change of measure

Question

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$.

Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative for which it holds $\mathbb{Q}(d\omega)=\mathbb{P}(d\omega)Z_\theta$. Then by Girsanov, $B_t = \widetilde{B}_t + \int_0^t \theta_s ds$ is a Brownian Motion under $\mathbb{Q}$ .

Let $$d\widetilde{F}_t=\mu(t,\widetilde{F}_t)dt+\sigma(t,\widetilde{F}_t)d\widetilde{B}_t$$ be an Ito process with deterministic (but not constant) drift and diffusion, satisfying Malliavin derivability conditions. Denote by $D^{\widetilde{B}}_s \widetilde{F}_t$ the Malliavin derivative at time $s$ of the stochastic process $\widetilde{F}$ at time $t$ with respect to the Brownian Motion $\widetilde{B}$.

Then the process $\widetilde{F}$ under change of measure is: $$ dF_t=\big[\mu(t,F_t) - \theta_t\cdot\sigma(t,F_t) \big]dt+\sigma(t,F_t)dB_t $$

Let $W$ be a Brownian Motion under $\mathbb{Q}$ correlated with $B$, correlation coefficient being $\rho$.

One wishes to compute $D^{{W}}_s {F}_t$, having close form expression for $D^{\widetilde{B}}_s \widetilde{F}_t$.

$\cdot$ Does an explicit relation between $D^{\widetilde{B}}_s \widetilde{F}_t$ and $D^{{W}}_s {F}_t$ exist? More specifically, can we express $D^{{W}}_s {F}_t$ as a function of $D^{\widetilde{B}}_s \widetilde{F}_t$ ?

(Where indeed $D^{{W}}_s {F}_t$ is the Malliavin derivative of the process $\widetilde{F}$ expressed under $\mathbb{Q}$, with respect to $W$).

My guess is that it should depend on the Malliavin derivative of $\theta$, but with respect to what Brownian Motion? I have spent a week on this but did not manage to arrive at satisfying results. I tried to leverage the Clark–Ocone Formula under Change of Measure and Malliavin derivative operator under change of variable. The second seems to be particularly close to what I am looking for. However its notation is not clear to me (with respect to which BM the Malliavin derivatives are taken and under which measure the random variable are expressed) and was not able to actually use the result.

EDIT: In my particular case an Ito process with deterministic drift and diffusion is considered. It would be nice to have such result for a general Malliavin-derivable stochastic process though.

Answer 1

Here an answer for the case with determinist drift as mentioned in the edit. (Note: I fail to see why to use different notations for $F$ and $\tilde{F}$ as it is the same process)

As $$ dF_t = \mu_t \, dt + \sigma_t d\tilde{B}_t$$ we have $$ D_s^{\tilde{B}}F_t = \sigma_s$$.

Furthermore, writing $B_t = \rho W_t + \sqrt{1-\rho^2} W_t^\perp$ for $W^\perp$ a Brownian motion orthogonal to $W$ and noting that $D_s^{W} W^\perp_t =0$, we have from

$$ dF_t = \bigl(\mu_t - \theta_t \sigma_t\bigr) \, dt + \sigma_t dB_t = \bigl(\mu_t - \theta_t \sigma_t\bigr) \, dt + \rho \sigma_t dB_t + \sqrt{1-\rho^2}\sigma_t dW^\perp_t$$

that

$$D^W_sF_t = \rho\sigma_s - \int_s^t \sigma_u D_s^W\theta_u \, du$$

and can thus conclude

$$D^\tilde{B}_sF_t = \frac{D^W_sF_t + \int_s^t \sigma_u D_s^W\theta_u \, du}{\rho}$$.