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 stochastic differential equations. 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 optimization 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.