We consider a backward stochastic differential equation, whose data (the final condition and the coefficient) are given functions of a jump-diffusion process. We prove that under mild conditions the solution of the BSDE provides a viscosity solution of a system of parabolic integral-partial differential equations. Under an additional assumption, that system of equations is proved to have a unique solution, in a given class of continuous functions

Backward stochastic differential equations and integral-partial differential equations

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

This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate, continuous dependence on a parameter) of forward–backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations. We use a purely probabilistic approach, and allow the forward equation to be degenerate.

Forward-backward stochastic differential equations and quasilinear parabolic PDES

Backward stochastic differential equations with jumps and related non-linear expectations

We prove results on bounded solutions to backward stochastic equations driven by random measures. Those bounded BSDE solutions are then applied to solve different stochastic optimization problems with exponential utility in models where the underlying filtration is noncontinuous. This includes results on portfolio optimization under an additional liability and on dynamic utility indifference valuation and partial hedging in incomplete financial markets which are exposed to risk from unpredictable events. In particular, we characterize the limiting behavior of the utility indifference hedging strategy and of the indifference value process for vanishing risk aversion.

Bounded solutions to backward SDEs with jumps for utility optimization and indifference hedging

A maximum principle is proved for optimal controls of stochastic systems with random jumps. The control is allowed to enter into both diffusion and jump terms. The form of the maximum principle turns out to be quite different from the one corresponding to the pure diffusion system (the word "pure" here means the absence of the jump term). In calculating the first-order coefficient for the cost variation, only a property for Lebesgue integrals of scalar-valued functions in the real number space ${\Cal R}$ is used. This shows that there is no essential difference between deterministic and stochastic systems as far as the derivation of maximum principles is concerned.

Xunjing Li

Papers

Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps