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We study relaxed stochastic control problems where the state equation is a one dimensional linear stochastic differential equation with random and unbounded coefficients. The two main results are existence ofan optimal relaxed control and necessary conditions for optimality in the form of a relaxed maximum principle. The main motivation is an optimal bond portfolio problem in a market where there exists a continuum of bonds and the portfolio weights are modeled as measure-valued processes on the set of times to maturity.

A maximum principle for relaxed stochastic control of linear SDEs with application to bond portfolio optimization