Asset flow and momentum: deterministic and stochastic equations

We use basic conservation and microeconomic identities to derive a nonlinear first-order ordinary differential equation for a market system with a prescribed number of shares and cash supply (including additions in time). The equation incorporates the ideas of the finiteness of assets and preference that is influenced by price momentum and discount from fundamental value. The concept of a 'liquidity value', defined as the total cash in the system divided by the number of shares, emerges as a key price along with the fundamental value. In the absence of a clear focus on fundamentals, the price evolves into the liquidity value. This is consistent with the belief of some market analysts who feel that liquidity, or a large sum of cash available for investment, is a primary factor in moving asset prices higher. These equations can also be derived from the system of equations used in previous work by considering a closed system and taking the limit of short time-scale in the preference or transition function as well as some linearization. Finally, the full system of equations is generalized to include randomness. The resulting stochastic system is studied numerically. In particular, when the deterministic equations are complemented with randomness, the solutions generate a range of stochastic patterns, such as the head and shoulders with certain characteristics in common.