  Managing risk in dynamic decision problems is of cardinal importance in many
fields such as finance and process control. The most common approach to
defining risk is through various variance related criteria such as the Sharpe
Ratio or the standard deviation adjusted reward. It is known that optimizing
many of the variance related risk criteria is NP-hard. In this paper we devise
a framework for local policy gradient style algorithms for reinforcement
learning for variance related criteria. Our starting point is a new formula for
the variance of the cost-to-go in episodic tasks. Using this formula we develop
policy gradient algorithms for criteria that involve both the expected cost and
the variance of the cost. We prove the convergence of these algorithms to local
minima and demonstrate their applicability in a portfolio planning problem.
