A new model of security price change and volume behavior over transaction data

A Gaussian loglikelihood of stock market returns is represented as a stochastic Lagrangian. That is as a function whose optimal solution paths in price and quantity space for a given set of parameters can be found by introduction into the Euler-Lagrange equations. These equations for appropriate initial conditions form a system of equations in price and quantity which may be used as prediction equations for price and quantity. These equations are regressed on transations data and compared to simple ad hoc equations representing the findings in the literature on transaction data. Restrictions on the form of the Lagrangian are introduced in consequence.;The theory is then extended to a partial differential equation representation analogous to the Fokker-Planck equation and data is presented to show that implied variance structure of the restricted model from the initial regression results is inadequate to explain the observed return and quantity variances. This result suggests the reformation of the theory into two densities--a buyer density and a seller density whose product represents the observed price transactions. The new model is still dependent upon the Lagrangian formulation but describes well the time pattern of return and quantity variances and the relation between price and quantity variance.;The resulting theory bears a great resemblance to elementary Quantum Theory and may be formulated in terms of Hermetian operators.