  The conservative wealth-exchange process derived from trade interactions is
modeled as a multiplicative stochastic transference of value, where each
interaction multiplies the wealth of the poorest of the two intervening agents
by a random gain eta=(1+kappa), with kappa a random return. Analyzing the
kinetic equation for the wealth distribution P(w,t), general properties are
derived for arbitrary return distributions pi(kappa). If the geometrical
average of the gain is larger than one, i.e. if <ln eta> >0, in the long time
limit a nontrivial equilibrium wealth distribution P(w) is attained. Whenever
<ln eta> <0, on the other hand, Wealth Condensation occurs, meaning that a
single agent gets the whole wealth in the long run. This concentration
phenomenon happens even if the average return <kappa> of the poor agent is
positive. In the stable phase, P(w) behaves as w^{(T-1)} for w -> 0, and we
find T exactly. This exponent is nonzero in the stable phase but goes to zero
on approach to the condensation interface. The exact wealth distribution can be
obtained analytically for the particular case of Kelly betting, and it turns
out to be exponential. We show, however, that our model is never reversible, no
matter what pi(kappa) is. In the condensing phase, the wealth of an agent with
relative rank x is found to be w(x,t) \sim e^{x t <ln eta>} for finite times t.
The wealth distribution is consequently P(w) \sim 1/w for finite times, while
all wealth ends up in the hands of a single agent for large times. Numerical
simulations are carried out, and found to satisfactorily compare with the above
mentioned analytic results.

General Finance;Statistical Mechanics;Statistical Finance