# Under what measure is the SABR stochastic differential equations

The SABR Model is a CEV (constant elasticity of variance) Cox asset process with correlated lognormal stochastic volatility. A forward rate $$F(t,T)$$ to time $$T$$, observed at $$t$$, and the instantaneous volatility, $$\sigma(t)$$ follow the stochastic differential equations \begin{align} &dF(t,T)=\sigma(t)F(t,T)^\beta dW_F(t) \label{eq:true_sabr_model1} \\ &d\sigma(t)=\xi\sigma(t)dW_\sigma(t) \label{eq:true_sabr_model2} \end{align} where as the parameter $$\rho$$ represents the instantaneous correlation between the standard Brownian motions $$W_F(t)$$ and $$W_\sigma(t)$$ ($$\langle dW_F(t)dW_\sigma(t)\rangle=\rho dt$$).

My question is, are the Brownian motions in the SABR model under the physical measure $$P$$ or the risk-neutral measure $$Q$$? I can not find anything about it in the original paper. Can anyone help me with a reference to where it is stated explicitly?

The simple forward rate $$F_n(t) = F(t, T_n, T_{n+1})$$ is a martingale under the measure $$Q^{T_{n+1}}$$, which means that the associated numeraire is the zero coupon bond $$P(t, T_{n+1})$$.
In the SABR model, the forward rate $$F_n(t)$$ is assumed to evolve under the associated measure $$Q^{T_{n+1}}$$ according to:
\begin{aligned} dF_n(t) &= \sigma(t) \cdot F_n(t)^{\beta} \cdot dW^{Q^{T_{n+1}}}_n(t),\\ d\sigma(t) &= \xi \cdot \sigma(t) \cdot dZ^{Q^{T_{n+1}}}(t) \end{aligned}