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 ${\displaystyle t}$.[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 ${\displaystyle Q}$.

References

1. Schachermayer, Walter (November 15, 2002). "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time". Cite journal requires |journal= (help)
2. Yuri M. Kabanov; Mher Safarian (2010). Markets with Transaction Costs: Mathematical Theory. Springer. p. 114. ISBN 978-3-540-68120-5.
3. Jacka, Saul; Berkaoui, Abdelkarem; Warren, Jon (2008). "No arbitrage and closure results for trading cones with transaction costs". Finance and Stochastics. 12 (4): 583–600. arXiv:math/0602178. doi:10.1007/s00780-008-0075-7.