Mean-variance hedging in an illiquid market

Consider a market consisting of two correlated assets: one liquidly traded asset and one illiquid asset that can only be traded at time 0. For a European derivative written on the illiquid asset, we find a hedging strategy consisting of a constant (time 0) holding in the illiquid asset and dynamic trading strategies in the liquid asset and a riskless bank account that minimizes the expected square replication error at maturity. This mean-variance optimal strategy is first found when the liquidly traded asset is a local martingale under the real world probability measure through an application of the Kunita-Watanabe projection onto the space of attainable claims. The result is then extended to the case where the liquidly traded asset is a continuous square integrable semimartingale, and we again use the Kunita-Watanabe decomposition, now under the variance optimal martingale measure, to find the mean-variance optimal strategy in feedback form. In an example, we consider the case where the two assets are driven by correlated Brownian motions and the derivative is a call option on the illiquid asset. We use this example to compare the terminal hedging profit and loss of the optimal strategy to a corresponding strategy that does not use the static hedge in the illiquid asset and conclude that the use of the static hedge reduces the expected square replication error significantly (by up to 90% in some cases). We also give closed form expressions for the expected square replication error in terms of integrals of well-known special functions