  We show that the problem of existence of equilibrium in Kyle's continuous
time insider trading model can be tackled by considering a forward-backward
system coupled via an optimal transport type constraint at maturity. The
forward component is a stochastic differential equation representing an
endogenously determined state variable and the backward component is a
quasilinear parabolic equation representing the pricing function. By obtaining
a stochastic representation for the solution of such a system, we show the
well-posedness of solutions and study the properties of the equilibrium
obtained for small enough risk aversion parameter. In our model, the insider
has exponential type utility and the belief of the market maker on the
distribution of the price at final time can be non-Gaussian.
