Two price economy provides a new approach to describe incomplete markets. Unlike the classical economy theory, in which the law of one price prevails, a two price economy determines prices by the directions of the trades. Static one period and discrete time two price economies are described and applied in a number of papers.

Following the static and discrete time models, continuous time two price economies are studied in this thesis. Dynamically consistent nonlinear pricing functionals are generated from backward stochastic dierential equations (BSDEs) on continuous time Markov chains (CTMCs) and G-expectations. This thesis also includes

a convergence theorem of BSDEs on CTMCs, and the existence and uniqueness of solution to the distorted partial integro-dierential equation coming from the G-expectation approach.

The continuous time models for two price economies are illustrated through three examples. BSDEs on CTMCs are used to generate bid and ask prices for option spreads. Then G-expectation theory is applied to produce credit capital commitments for derivatives with bilateral counterparty risk and give bid and ask interest rate swap rates and swaption prices.