# Arithmetic Brownian Motion in Market Making papers

We often consider high-frequency market maker and suppose that the reference price is the arithmetic Brownian Motion:

$$dS_{t} = \sigma d W_t$$

What is the difference $$t_n - t_{n-1}$$ in this case? Is is one day or one second? Estimation in those two cases based on datasets would be different, so what is the case here?

My question is based on the paper: Dealing with inventory risk - a solution to the marker making model by Gueant, Lehalle and Tapia.

• @Lipa_FNTE After a closer look I think t is a continuous time variable. Your paper begins with "Market makers continuously set bid and ask quotes." Your paper also references a paper by Avellaneda and Stoikov that integrates and differentiates relative to time. In section 3.1 they use a PDE that's continuous in t. All of this strongly implies that t is continuous... I imagine that practical applications with discrete time steps are an approximation of the continuous theory (as is fairly usual). – sfmiller940 Oct 8 '19 at 18:41
• @Lipa_FNTE Also in that Avellaneda and Stoikov paper they mention their "continuous-time model" towards the end of section 2.1... Anyway for your question about $t_n - t_{n-1}$ I think $dS_t$ is the derivative in theory, so in an approximation you would use something like $\frac{S_{t_n} - S_{t_{n-1}}}{t_n - t_{n-1}}$. Of course these estimates will vary depending on the underlying data. – sfmiller940 Oct 8 '19 at 18:55