Viable Prices in Financial Markets with Solvency Constraints

We study the viability of a securities market model with continuous trading in which agents are required to remain solvent at all times and cannot add funds to their portfolio of securities in excess of an exogenous endowment. We show that viability, a notion introduced by Harrison and Kreps (1979), is equivalent to the existence of a pair of continuous linear functionals, defined on the consumption space and market value space, such that their sum extends the market linear pricing rule. We also show that viability is equivalent to the existence of an adjustment process, such that the adjusted total gains from any asset with positive dividends is a supermartingale. The adjustment process itself is a strictly positive supermartingale. In particular, the price of any asset with positive dividends can be decomposed into two parts: a martingale up to a stopping time and a nonnegative supermartingale which is identically equal to zero at the terminal date. The first part is the value of future cash flows after adjusting for timing and riskiness. The second component is the value due to the solvency services of the asset.