Partial Differential Equations for Time Development of Stock Prices, Properties, etc. and the Inverse Power Law

Assuming that the short time behavior of such quantities as stock prices of each companies is dominated by a probabilistic process and the probability distribution of variation in short time obeys a kind of scaling law, we derive partial differential equations that describe continuous time development of the relevant distributions. The explicit form of the differential equations depends on the choice of the kernel which describes short time development. For the simplest choice of the kernel, we have a second rank partial differential equation which generates inverse power distributions with the exponent −1 and −2 as a stationary solution. An alternative choice of the kernel is also examined. Finally, the drift effect is included to modify the differential equations. One of the two exponents of the stationary solutions now depends on the scaled drift velocity.