  Information relaxation and duality in Markov decision processes have been
studied recently by several researchers with the goal to derive dual bounds on
the value function. In this paper we extend this dual formulation to controlled
Markov diffusions: in a similar way we relax the constraint that the decision
should be made based on the current information and impose penalty to punish
the access to the information in advance. We establish the weak duality, strong
duality and complementary slackness results in a parallel way as those in
Markov decision processes. We explore the structure of the optimal penalties
and expose the connection between Markov decision processes and controlled
Markov diffusions. We demonstrate the use of the dual representation for
controlled Markov diffusions in a classic dynamic portfolio choice problem. We
evaluate the lower bounds on the expected utility by Monte Carlo simulation
under a sub-optimal policy, and we propose a new class of penalties to derive
upper bounds with little extra computation. The small gaps between the lower
bounds and upper bounds indicate that the available policy is near optimal as
well as the effectiveness of our proposed penalty in the dual method.
