1. The problem is to verify the derivation that the discounted stock price $\tilde{S}_t = e^{-rt}S_t$ is a martingale under the risk-neutral measure $Q$ given the SDE for $S_t$: $$dS_t = rS_t dt + \sigma S_t dW_t^Q.$$\n\n2. First, apply Itô's product rule correctly to $\tilde{S}_t = e^{-rt}S_t$: $$d\tilde{S}_t = d(e^{-rt})S_t + e^{-rt} dS_t + d(e^{-rt}) dS_t.$$\n\n3. Calculate each term:\n- $d(e^{-rt}) = -r e^{-rt} dt$ (deterministic exponential function).\n- $dS_t = r S_t dt + \sigma S_t dW_t^Q$ by given SDE.\n- The product of differentials $d(e^{-rt}) dS_t$ is zero because $d(e^{-rt})$ is order $dt$ and $dS_t$ contains $dt$ and $dW_t^Q$, their product is of higher order infinitesimal and neglectable.\n\n4. Substitute these into Itô's product rule: $$d\tilde{S}_t = (-r e^{-rt} dt) S_t + e^{-rt} (r S_t dt + \sigma S_t dW_t^Q) + 0 = -r e^{-rt} S_t dt + r e^{-rt} S_t dt + \sigma e^{-rt} S_t dW_t^Q.$$\n\n5. Simplify the drift terms: $$-r e^{-rt} S_t dt + r e^{-rt} S_t dt = 0,$$ so the SDE for $\tilde{S}_t$ reduces to: $$d\tilde{S}_t = \sigma \tilde{S}_t dW_t^Q.$$\n\n6. Since the drift term in the SDE for $\tilde{S}_t$ is zero, $\tilde{S}_t$ is a local martingale. Under standard conditions (e.g., integrability, non-explosiveness), $\tilde{S}_t$ is indeed a true martingale, meaning: $$\mathbb{E}^Q [\tilde{S}_t | \mathcal{F}_s] = \tilde{S}_s,$$ for $s \le t$.\n\n7. Therefore, the workings and derivation provided are correct and consistent with stochastic calculus theory and risk-neutral pricing models.