# Are changes in the asset price a Markov process?

I'm studying the book "The Mathematics of Financial Derivatives - A Student Introduction" and at the start of the second chapter it says that thanks to the effcient market hypothesis (1-the past history is fully reflected in the present price, 2-markets respond immediately to any new information about an asset) the changes in the asset price are a Markov process (full text below). After this, the book introduces the SDE that model the return on the asset: $\frac{dS}{S}=\mu dt + \sigma dW$.

A stochastic process is said to be a Markov process if it satisfies the Markov property: the next state of the process depends solely on the present one, not on the sequence of events that preceded it.

So I'm a bit confused, how can the changes in the price be a Markov process if the present price fully reflects the past history?

Or maybe I misinterpreted it, and the Markov property just says that all past information about the stock price process is incorporated in the current price and therefore only the current price is relevant ?

Let us define a stock process $(S_t)_{t \geq 0}$. We assume the value of the process $S_t$ at $t$ captures all past information which might affect the stock price (its "past history"). Let us consider two times $t_1 < t_2$.
If we make the (realistic) assumption that information accumulates and increases over time then, given the stock price $S_t$ captures all relevant information up to $t$, the stock price $S_{t_2}$ encapsulates all relevant information up to and including $t_2$. Therefore it contains at least the same amount of relevant information than $S_{t_1}$. As this is true for all $t<t_2$, all information affecting the stock process at $t_2$ is contained in $S_{t_2}$ and the history of the process is irrelevant (because it is "encapsulated" in $S_{t_2}$): the stock process is thus a Markov process.
• Indeed. Note that is what your text says: if "the past history is fully reflected in the present price", then any price $S_t$ contains all information up to and including $t$, for any $t$; assuming information accumulates, this implies that the only relevant price is the current one as it contains at least as much information as all other past prices. – Daneel Olivaw Oct 31 '17 at 17:47