See Lesson 05 about the discrete-time S_i+1 concept.
See Lesson 15 about TVRV.
I feel in general any variable dependent on a random variable is also a random variable, such as the S in
S_i+1 – S_i = deltaS = a * deltaT + b * deltaW
The dependency is signified by the ordinary-looking “+” operator. To me this addition operator means “superimpose”. The deltaS or stepsize is a combination of deterministic shift superimposed on a non-deterministic noise. That makes S itself a time-varying random variable which can follow a trillion possible paths from last-observation to Expiry.
The addition doesn’t mean the stepsize_i+1 will be known once both components i.e. (a * deltaT) and (b * deltaW) are known. In fact, deltaW can take a trillion possible values, so the stepsize in S is not exactly predictable i.e. non-deterministic. This stepsize is random. Therefore S itself is a TVRV.