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 ${\displaystyle (\mathrm{\Omega },{ℱ},\{{{ℱ}}_t{\}}_{t=0}^T,P)}$ 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 ${\displaystyle Z_t\in K_t^{+}\mathrm{\setminus }\{0\}}$ at all times ${\displaystyle t}$ such that ${\displaystyle K_t}$ is the solvency cone for the market at time ${\displaystyle t}$.^[1][2]

The CPP plays the role of an equivalent martingale measure in markets with transaction costs.^[3] In particular, there exists a 1-to-1 correspondence between the CPP ${\displaystyle Z}$ and the EMM ${\displaystyle Q}$.^{[citation needed]}

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