Abstract

We consider three problems motivated by mathematical and computational finance which utilize forward-backward stochastic differential equations (FBSDEs) and other techniques from stochastic control. Firstly, we review the case of post-retirement annuitization with labor income in framework of optimal stochastic control and optimal stopping. We apply the martingale approach to a Cobb–Douglas type utility maximization problem. We have proved the theoretical existence and uniqueness of an optimal solution. Several analyses are made based on the simulations for the optimal stopping choice and strategies. Secondly, We review the convolution method in backward stochastic differential equations (BSDEs) framework and study the application of convolution method to Heston model. We provide an easy representation of the Heston characteristic function that avoids the discontinuities caused by branch rotations in the logarithm of complex functions and is able to be applied in calibration. We proposed two convolution schemes to the Heston model and provide the error analysis that shows the error orders of discretization and truncation. We review two error control methods and improve the accuracy on the boundaries. Numerical results comparing to a Fourier method and an integration method is provided. Thirdly, we review the forecasting problem in bond markets. Our data include both U.S. Treasuries and coupon bonds from twelve corporate issuers. We apply the arbitrage-free model in predicting the yields and the prices of coupon bonds in a sequential model with the Kalman filter, the extended Kalman filter and the particle filter. We implement the arbitrage penalty and obtain the optimal dynamic parameterization using deep neural networks. The purpose of the prediction is to examine the effect of arbitrage penalty and the forecasting performance on different time horizons. Our result shows that the arbitrage-free penalty has improving performance on short time period but downgrading performance on long time period. We provide analysis on the prediction errors, the distribution of errors, and the average excess return. The predicted bond prices shows the prediction errors have non-Gaussian distribution, excess kurtosis, and fat tails. Future works will be from two aspects, refine the importance sampling by non-parametric distribution and refine the term structure model with jump process and credit risk.