1. Introduction


1.1. Searching the Mechanism of Evaluations of Risky Assets


1.2. Axiomatic Assumptions for Evaluations of Derivatives


1.3. Organization of the Lecture






2. Brownian Filtration Consistent Evaluations and Expectations


2.1. Main Notations and Definitions


2.2. Open image in new window -Consistent Nonlinear Expectations


2.3. Open image in new window -Consistent Nonlinear Evaluations






3. Backward Stochastic Differential Equations: g-Evaluations and g-Expectations


3.1. BSDE: Existence, Uniqueness and Basic Estimates


3.2. 1-Dimensional BSDE


3.3. A Monotonic Limit Theorem of BSDE


3.4. g-Martingales and (Nonlinear) g-Supermartingale Decomposition Theorem






4. Finding the Mechanism: Is an Open image in new window -Expectation a g-Expectation?


4.1. Open image in new window -Dominated Open image in new window -Expectations


4.2. Open image in new window -Consistent Martingales


4.3. BSDE under Open image in new window -Consistent Nonlinear Expectations


4.4. Decomposition Theorem for Open image in new window -Supermartingales


4.5. Representation Theorem of an Open image in new window -Expectation by a g-Expectation


4.6. How to Test and Find g?


4.7. A General Situation: Open image in new window -Evaluation Representation Theorem






5. Dynamic Risk Measures


6. Numerical Solution of BSDEs: Euler’s Approximation


7. Appendix


7.1. Martingale Representation Theorem


7.2. A Monotonic Limit Theorem of Ito’s Processes


7.3. Optional Stopping Theorem for Open image in new window -Supermartingale






References


References on BSDE and Nonlinear Expectations

/pdf/nonlinear-expectations-nonlinear-evaluations-and-risk-1df84s74m8.pdf

Nonlinear Expectations, Nonlinear Evaluations and Risk Measures

A new algorithm, which is based on the splitting-step idea and the penalization method, for reflected stochastic differential equation (RSDE) in the upper half-space R"+^1 is presented in this paper. After some important estimates about RSDEs and penalization ODEs are obtained, the local pathwise error of the suggested algorithm are considered, and an estimate of local error in the 2pth moment is proved under the assumptions of Lipschitz continuity and linear growth. Some numerical experiments based on MATLAB programs are given to support the suggested algorithm.

A splitting-step algorithm for reflected stochastic differential equations in R+1

We introduce a new class of control problems in which the gain depends on the solu- tion of a stochastic diﬀerential equation (SDE) reﬂected at the boundary of a bounded domain, along directions which are controlled by a bounded variation process. We provide a PDE characterization of the associated value function. This study is motivated by applications in mathematical ﬁnance, where such equations are related to the pricing of barrier options under portfolio constraints.

Optimal reflection of diffusions and barrier options pricing under constraints

Editors and Authors

Solutions of quasilinear mixed boundary problems for the some parabolic and elliptic partial differential equations are interpreted as solutions of a kind of backward stochastic differential equations, which are associated with the classical Ito forward stochastic differential equations with reflecting boundary conditions.

A note on probabilistic interpretation for quasilinear mixed boundary problems

Stochastic optimal control problems for a class of reflected diffusion with Poisson jumps in a half-space are considered. The nonlinear Nisio's semigroup for such optimal control problems was constructed. A Hamilton-Jacobi-Bellman equation with the Neumann boundary condition associated with this semigroup was obtained. Then, viscosity solutions of this equation were defined and discussed, and various uniqueness of this equation was also considered. Finally, the value function was such optimal control problems is shown to be a viscosity solution of this equation.

https://www.amm.shu.edu.cn/CN/article/downloadArticleFile.do?attachType=PDF&id=12508

Ding Deng

Papers

A note on probabilistic interpretation for quasilinear mixed boundary problems

A note on stochastic optimal control of reflected diffusions with jumps