No abstract

In this paper we will study the approximation of arbitrary law invariant risk measures. As a starting point, we approximate the average value at risk using stochastic gradient Langevin dynamics, which can be seen as a variant of the stochastic gradient descent algorithm. Further, the Kusuoka's spectral representation allows us to bootstrap the estimation of the average value at risk to extend the algorithm to general law invariant risk measures. We will present both theoretical, non-asymptotic convergence rates of the approximation algorithm and numerical simulations.Keywords Convex risk measure • Stochastic Optimization • Risk minimization • Average value at risk • Stochastic gradient Langevin Mathematics Subject Classification 91G70 • 90C90 Statements and Declarations The authors gratefully acknowledge support from the NSF grant DMS-2005832. The authors have no competing interests to declare.