Dynamic risk measures and backward stochastic differential equations: From discrete to continuous time

This work analyzes the behavior of dynamic time consistent risk measures in a continuous time framework. While risk measures in discrete-time such as Value at Risk and Average Value at Risk are well understood (because they have long been in use and have a natural interpretation), dynamic risk measures in a continuous framework are much harder to interpret (with the exception of the entropic risk measure).;Our results show that a large class of risk measures in continuous time also has very natural interpretations as limits of discrete time-consistent risk measures which are stable. The discrete-time risk measures are constructed from properly rescaled ('tilted') one-period risk measures using a d-dimensional random walk WN that converges to a Brownian Motion W. We show that stable discrete-time risk measures in this setting satisfy a discrete Backward Stochastic Differential Equation (BSDeltaE), and obtain convergence of the discrete risk measures to the solution of a continuous time Backward Stochastic Differential Equation (BSDE). This allows us to define a continuous time risk measure, whose driver can be viewed as the continuous-time analogue of the discrete 'driver' characterizing the one-period risk.;We derive the limiting drivers for the semi-deviation risk measure, Value at Risk, Average Value at Risk, and the Gini risk measure, in closed form. This motivates an extensive general study of BSDeltaEs (driven by a general random walk and governed by non-Lipschitz driver functions). We first show that, under very mild conditions on the driver, BSDeltaEs always have a solution and that, under certain additional assumptions, the comparison principle is satisfied. Furthermore, we prove some regularity results. It is then shown that BSDeltaEs driven by a Bernoulli random walk with a Lipschitz continuous driver function converge to a process that is the solution of a BSDE driven by a Brownian Motion. Hence, we obtain a Donsker principle for BSDE's, i.e., we can prove the existence of a solution of a continuous time BSDE driven by a Brownian Motion by exclusively looking at sequences of BSDeltaE driven by a random walk. Finally, we show that if we already assume the existence of a solution, then we can also get convergence results in the case of non-Lipschitz drivers.