  In this paper we propose the notion of dynamic deviation measure, as a
dynamic time-consistent extension of the (static) notion of deviation measure.
To achieve time-consistency we require that a dynamic deviation measures
satisfies a generalised conditional variance formula. We show that, under a
domination condition, dynamic deviation measures are characterised as the
solutions to a certain class of backward SDEs. We establish for any dynamic
deviation measure an integral representation, and derive a dual
characterisation result in terms of additively $m$-stable dual sets. Using this
notion of dynamic deviation measure we formulate a dynamic mean-deviation
portfolio optimisation problem in a jump-diffusion setting and identify a
subgame-perfect Nash equilibrium strategy that is linear as function of wealth
by deriving and solving an associated extended HJB equation.
