I am doing a first pass of Stochastic Calculus from the text, Stochastic calculus for finance, Volume I by Steven Shreve. My aim is to learn computational skill, rather than deriving each result rigorously at this point.
In section 2.5 on page 45, Shreve first gives the following definition.
Definition (Markov Process). Consider the binomial market model. Let $(X_n)$ be a discrete stochastic process. $(X_n)$ is said to be Markov, if
\begin{align*} (\forall n \in \mathbf{N})(\forall f(x))(\exists g(x))(\text{s.t.}\mathbb{E}_n[f(X_{n+1})]=g(X_n)) \end{align*}
He then proves that the stock-price process $(S_t)$, where $t \in \mathbf{Z}_{+}$ is a discrete Markov process, which I follow.
He then derives a recursive algorithm for computing the price of a derivative whose payoff only depends on the terminal stock price $S_N$, as below.
I don't quite understand, how is the Markov property of $(S_n)$ being applied to the step in yellow. Any hints/suggestions here would be extremely helpful.
I understand, that derivative price process is a martingale, so I could use that for the deduction in the yellow step. But, that's besides the point. I don't get, how the Markov property is being applied.