A fundamental theorem of asset pricing in continuous time with square integrable portfolios

This paper studies a continuous-time market with multiple stocks whose prices are governed by geometric Brownian motions, and admissible investment portfolios are defined via certain square integrability condition. It is proved that, when the investment opportunity set is deterministic (albeit possibly time varying), such a market being arbitrage free is equivalent to the existence of a square integrable (in time) market price of risk, and as a result equivalent to the existence of an equivalent martingale measure. Counterexamples are given to show that these equivalent results are no longer true in a market with a stochastic investment opportunity set.