# Do *all* non-dividend paying assets have the risk-free instantaneous return rate under the risk-neutral measure?

For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $$dB_t$$. The risk-free rate is $$r$$ (one may assume it as constant for the time being). I know that, by virtue of Girsanov's theorem, the Brownian motion under the risk-neutral measure is defined by $$dB_t^{\Bbb Q} = \lambda dt + dB_t$$ where $$\lambda$$ is the unique market price of risk, or the so-called Sharpe ratio.

Under the risk-neutral measure, any non-dividend paying stock price process $$S_t$$ thus follows $$\frac{dS_t}{S_t} = rdt + \sigma_SdB_t^{\Bbb Q}.$$

However, in Kerry Back's A Course in Derivative Securities page 220, the author claimed without a proof that the instantaneous rate of return for a call option on the stock price $$C_t$$ is also $$r$$, i.e. $$\frac{dC_t}{C_t} = rdt + \sigma_C d B_t^{\Bbb Q}$$ where $$\sigma_C$$ is some stochastic process that we're not interested in. The author make crucial use of the above formula (i.e. the drift of $$C_t$$ is $$rC_tdt$$) to derive the BS PDE.

Question: is it true that under the risk neutral measure, any non-dividend paying asset price $$X_t$$ must have its instantaneous rate of return equal to $$r$$? If so, what would be a rigorous explanation for this?

Edit: Antoine is spot on. Under the risk neutral measure, any discounted asset price $$Y_t=e^{-rt}X_t$$ must be a martingale or equivalently an Ito integral without drift. Hence $$\frac{dY_t}{Y_t}=\sigma_Y dB_t^{\Bbb Q}.$$ where $$\sigma_Y$$ can be a quite general stochastic process. On the other hand, by the compounding rule of Ito processes, $$\frac{dY_t}{Y_t}=-rdt+\frac{dX_t}{X_t}$$ Therefore it follows $$\frac{dX_t}{X_t}=rdt+\sigma_Y dB_t^{\Bbb Q}.$$