  The classical maximum principle for optimal stochastic control states that if
a control $\hat{u}$ is optimal, then the corresponding Hamiltonian has a
maximum at $u=\hat{u}$. The first proofs for this result assumed that the
control did not enter the diffusion coefficient. Moreover, it was assumed that
there were no jumps in the system. Subsequently it was discovered by Shige Peng
(still assuming no jumps) that one could also allow the diffusion coefficient
to depend on the control, provided that the corresponding adjoint backward
stochastic differential equation (BSDE) for the first order derivative was
extended to include an extra BSDE for the second order derivatives.
  In this paper we present an alternative approach based on Hida-Malliavin
calculus and white noise theory. This enables us to handle the general case
with jumps, allowing both the diffusion coefficient and the jump coefficient to
depend on the control, and we do not need the extra BSDE with second order
derivatives.
  The result is illustrated by an example of a constrained mean-variance
portfolio problem.
