Necessary conditions are proved for certain problems
of optimal control of diffusions where hard end constraints occur. The main
results apply to several dimensional problems, where some of the state
equations involve Brownian motions, but not the equations corresponding to
states being hard restricted at the terminal time. The necessary conditions are
stated in terms of weak variations. Two versions of necessary conditions are
given, one version involving solutions of variational equations, the other one
involving first order adjoint equations.
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