## LECTURER : Gechun LIANG, King´s College

**1. BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS IN FINANCIAL MATHEMATICS**

Backward stochastic differential equations (BSDEs) constitute one of the most active areas in stochastic analysis and have wide applicability, for they provide a general framework to describe the evolution of stochastic systems (controlled or not) given information about their terminal state. In this mini-course, we will give a short introduction of BSDEs with emphasis on their financial application. A basic understanding of Ito’s calculus and stochastic differential equations is required.

**Lecture 1:** An introduction to BSDEs : We will introduce two different approaches to solve a standard BSDE with Lipschitz driver. One is via the standard fixed point argument as in [1]; the other is via a reformulation of the equation as a functional differential equation as recently established in [2].

[1] El Karoui, N., Peng, S., & Quenez, M. C. (1997). Backward stochastic differential equations in finance. Mathematical Finance, 7(1), 1-71.

[2] Liang, G., Lyons, T., & Qian, Z. (2011). Backward stochastic dynamics on a filtered probability space. The Annals of Probability, 39(4), 1422-1448.

**Lecture 2**: BSDEs and stochastic control: We will show how BSDEs can be used solve stochastic control problems and optimal stopping problems. Several verification results will be established for a variety of optimal control/stopping problems via BSDEs.

[1] El Karoui, N., Hamadne, S., & Matoussi, A. (2008). Backward stochastic differential equations and applications. Indifference pricing: theory and applications, edited by Carmona, R., Princeton University Press, 267-320.

[2] Liang, G. (2015). Stochastic control representations for penalized backward stochastic differential equations. SIAM Journal on Control and Optimization, 53(3), 1440-1463.

**Lecture 3:** Quadratic BSDEs and utility maximisation: BSDEs with quadratic growth (Quadratic BSDEs) are one of the most important class of equations as they have natural applications in Financial Mathematics, in particular in utility maximisation. We will review how to solve a quadratic BSDE by using Tevzadze’s fixed point argument [2] and how to use quadratic BSDEs to solve utility maximisation problems.

[1] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. The Annals of Probability, 558-602.

[2] Tevzadze, R. (2008). Solvability of backward stochastic differential equations with quadratic growth. Stochastic processes and their Applications, 118(3), 503-515.

[3] Hu, Y., Imkeller, P., & Mller, M. (2005). Utility maximization in incomplete markets. The Annals of Applied Probability, 15(3), 1691-1712.

[4] Henderson, V., & Liang, G (2014). Pseudo linear pricing rule for utility indifference valuation. Finance and Stochastics, 18(3), 593-615.

**Lecture 4:** Ergodic BSDEs and forward performance processes: Ergodic BSDEs run over an infinite horizon and have natural applications in ergodic stochastic control. In the final lecture, we will discuss a class of ergodic BSDEs with quadratic growth and show how they can be used to construct forward performance processes as introduced by Musiela and Zariphopoulou.

[1] Musiela, M., & Zariphopoulou, T. (2009). Portfolio choice under dynamic investment performance criteria. Quantitative Finance, 9(2), 161-170.

[2] Musiela, M., & Zariphopoulou, T. (2010). Portfolio choice under space-time monotone performance criteria. SIAM Journal on Financial Mathematics,1(1), 326-365.

[3] Fuhrman, M., Hu, Y., & Tessitore, G. (2009). Ergodic BSDEs and optimal ergodic control in Banach spaces. SIAM journal on control and optimization,48(3), 1542-1566.

[4] Liang, G., & Zariphopoulou, T. (2015). Representation of homothetic forward performance processes via ergodic and infinite horizon quadratic BSDE in stochastic factor models. arXiv preprint arXiv:1511.04863.