  To any utility maximization problem under transaction costs one can assign a
frictionless model with a price process $S^*$, lying in the bid/ask price
interval $[\underline S, \bar{S}]$. Such process $S^*$ is called a \emph{shadow
price} if it provides the same optimal utility value as in the original model
with bid-ask spread.
  We call $S^*$ a \emph{generalized shadow price} if the above property is true
for the \emph{relaxed} utility function in the frictionless model. This
relaxation is defined as the lower semicontinuous envelope of the original
utility, considered as a function on the set $[\underline S, \bar{S}]$,
equipped with some natural weak topology. We prove the existence of a
generalized shadow price under rather weak assumptions and mark its relation to
a saddle point of the trader/market zero-sum game, determined by the relaxed
utility function. The relation of the notion of a shadow price to its
generalization is illustrated by several examples. Also, we briefly discuss the
interpretation of shadow prices via Lagrange duality.
