Consistent pricing process

A consistent pricing process (CPP) is any representation of (frictionless) "prices" of assets in a market. It is a stochastic process in a filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t=0}^{T},P)}$such that at time ${\displaystyle t}$the ${\displaystyle i^{th}}$component can be thought of as a price for the ${\displaystyle i^{th}}$asset.

Mathematically, a CPP ${\displaystyle Z=(Z_{t})_{t=0}^{T}}$in a market with d-assets is an adapted process in ${\displaystyle \mathbb {R} ^{d}}$if Z is a martingale with respect to the physical probability measure ${\displaystyle P}$, and if ${\displaystyle Z_{t}\in K_{t}^{+}\backslash \{0\}}$at all times ${\displaystyle t}$such that ${\displaystyle K_{t}}$is the solvency cone for the market at time ${\displaystyle t}$.

The CPP plays the role of an equivalent martingale measure in markets with transaction costs. In particular, there exists a 1-to-1 correspondence between the CPP ${\displaystyle Z}$and the EMM ${\displaystyle Q}$.