  This paper shows how the theory of dynamic risk measures provides viscosity
solutions to a family of second-order parabolic partial differential equations,
even in the degenerate case.
  First, motivated by the martingale problem approach of Stroock and Varadhan,
we make use of the time consistency characterization for dynamic risk measures,
to construct time consistent convex Markov processes. This is done in a general
setting in which compacity arguments cannot be used, and for which there does
not always exist an optimal control. Second, we prove that these processes lead
to viscosity solutions for semi linear second-order partial differential
equations with convex generator.
  Finally we give an application to mathematical finance. We show that our
results allow for the construction of a No Arbitrage Pricing Procedure in the
context of stochastic volatility. Within this approach, convexity takes into
account liquidy risk.
