Investors who borrow in order to buy securities in the stock market are required to maintain a minimum amount of cash or securities with their broker-dealer. Such a requirement is called a margin requirement. In the paper the authors study the optimal consumption and investment choices and the cost of hedging European continget claims, given certain margin requirements. This problem without the margin rquirement constraint has been studied ealier by many authors [see, for example, I. Karatzas, J. P. Lehoczky, S. E. Shreve
and G.-L. Xu
, SIAM J. Control Optim. 29, 702-730 (1987; Zbl 0733.93085
)]. Martingale techniques, similar to those employed in the reference cited above have been used to show the existence of optimal policies in the presence of margin requirements. Duality results are used to provide a characterization of optimal policies. An explicit solution is derived for an agent with a logarithmic utility.