 Solution

Conditionally on past observations, adding the independent state increment gives the predictionThe new observation then has conditional distribution $N({\hat{S}}_{t-1},R_t+V)$, and its conditional covariance with $S_t$ is $R_t$. Equivalently, multiplying the state prediction probability density function by the observation likelihood and completing the square givesFor positive variances, rearrangement yields the scalar Gaussian Kalman recursionHere the innovation is the observation minus its predicted mean. The displayed covariance formula also handles zero variances whenever its denominator is nonzero; a completely deterministic prediction and observation require no probabilistic update. Independence of the new noises from the past and the initial state is the usual state-space assumption needed for this conditional calculation.