A Dual Method For Evaluation of Dynamic Risk in Diffusion Processes Andrzej Ruszczy ´ nski * Jianing Yao † Abstract We propose a numerical recipe for risk evaluation defined by a backward stochastic differ- ential equation. Using dual representation of the risk measure, we convert the risk evaluation to a simple stochastic control problem where the control is a certain Radon-Nikodym derivative process. By exploring the maximum principle, we show that a piecewise-constant dual control provides a good approximation on a short interval. A dynamic programming algorithm extends the approximation to a finite time horizon. Finally, we illustrate the application of the procedure to financial risk management in conjunction with nested simulation and on an multidimensional portfolio valuation problem. Subject Classification: 60J60,60H35,49L20,49M25,49M29 Keywords: Dynamic Risk Measures, Forward–Backward Stochastic Differential Equations, Stochas- tic Maximum Principle, Financial Risk Management Introduction The main objective of this paper is to present a simple and efficient numerical method for solving backward stochastic differential equations with convex and homogeneous drivers. Such equations are fundamental modeling tools for continuous-time dynamic risk measures with Brownian filtra- tion, but may also arise in other applications. The key property of dynamic risk measures is time-consistency, which allows for dynamic pro- gramming formulations. The discrete time case was extensively explored by Detlfsen and Scandolo [12], Bion-Nadal [5], Cheridito et al. [8, 9], Föllmer and Penner [13], Fritelli and Rosazza Gianin [16], Frittelli and Scandolo [17], Riedel [38], and Ruszczy ´ nski and Shapiro [43]. For the continuous-time case, Coquet, Hu, Mémin and Peng [10] discovered that time-consistent dynamic risk measures, with Brownian filtration, can be represented as solutions of Backward Stochastic Differential Equations (BSDE) [36]; under mild growth conditions, this is the only form possible. Specifically, the y-part solution of one-dimensional BSDE, defined below, measures the risk of a variable ξ T at the current time t : Y t = ξ T +  T t g(s, Z s ) ds −  T t Z s dW s , 0 ≤ t ≤ T, (1) * Rutgers University, Department of Management Science and Information Systems, Piscataway, NJ 08854, USA, Email: rusz@rutgers.edu † RBC Capital Markets, New York, NY 10281 Email: yaojn_1@hotmail.com 1 arXiv:1701.06234v3 [math.OC] 29 Mar 2020