Stochastic differential equations are fundamental tools for describing dynamics of irregulaly varying functions, and are applied to many areas. This course aims to get students to learn the fundamental theory and computational methods for estimations and controls of stochastic differential equations.
By the end of this course, students will be able to bulid models and compute optimal controls of stochastic differential equations, and moreover to explain the validity, limitation, and development of the methods used there.
Martingales, Stochastic integration, Stochastic differential equations, diffusion processes, Estimation of stochastic processes, Control of stochastic processes, Hamilton-Jacobi-Bellman equations
|Intercultural skills||Communication skills||✔ Specialist skills||Critical thinking skills||✔ Practical and/or problem-solving skills|
|Course schedule||Required learning|
|Class 1||Conditional expectation, Martingales||Explain the definitions of conditional expectations and martingales, and prove its basis properties.|
|Class 2||Wiener processes||Explain and prove basic properties of Wiener processes.|
|Class 3||Stochastic integration||Explain how stochastic integration is constructed, and validate it.|
|Class 4||Stochastic differential equations||Explain the definition, concept, and examples of stochastic differential equations.|
|Class 5||Stochastic differential equations||Explain and prove basic properties of stochastic differential equations.|
|Class 6||Stochastic differential equations||Explain and prove basic properties of stochastic differential equations.|
|Class 7||Estimation of stochastic differential equations: theory||Explain estimation methods for stochastic differential equations.|
|Class 8||Optimal control of stochastic differential equations: theory||Explain optimal control methods for stochastic differential equations.|
|Class 9||Function approximations||Explain methods for function approximations.|
|Class 10||Viscosity solutions||Explain the definition and basis properties of viscosity solutions.|
|Class 11||Numerical analysis of partial differential equations||Explain and implement numerical methods for partial differential equations.|
|Class 12||Estimation of stochastic differential equations: computation||Implement the estimation methods for stochastic differential equations.|
|Class 13||Optimal control of stochastic differential equations||Implement the control methods for stochastic differential equations.|
|Class 14||Applications||Explain applications.|
No specific text
Course materials can be found on OCW-i.
1)B. Oksendal, Stochastic differential equaions: an introduction with applications, Springer
2) W. H. Fleming and H. M. Soner, Controlled Markov processes and viscosity solutions, Springer
3) H. Pham, Continuous-time stochastic control and optimization with financial applications, Springer
It is preferable that students have completed MCS.T212:Fundamentals of Probability and MCS.T312:Markov Analysis.