Obstacle problem for semilinear PIDEs

02 Nov 2015 NUS mathematicians discovered a probabilistic interpretation for the weak Sobolev solution of the obstacle problem for semilinear parabolic partial integro-differential equations (PIDEs).

The results of Léandre [1] concerning the homeomorphic property for the solution of SDEs with jumps are used to construct random test functions for the variational equation for such PIDEs. This results in the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDEs), namely Feynman–Kac’s formula for the solution of the PIDEs.

It gives an application to the pricing and hedging of contingent claims with constraints in the wealth or portfolio processes in financial markets including jumps.

The next step is to consider the obstacle problem for stochastic PDEs.

A team led by Prof ZHOU Chao from the Department of Mathematics in NUS solved the problem by developing a stochastic flow method based on the results of Léandre [1] about the homeomorphic property for the solution of SDEs with jumps. The stochastic flow method is used in [2] to transforms the variational formulation of the PDEs to the associated BSDEs. Thus it plays the same role as Ito’s formula in the case of the classical solution of PDEs. Note that more recently, in [3] based on stochastic flow arguments, the author shows that the probabilistic equivalent formulation of Dupire’s PDE is the Put-Call duality equality in local volatility models including exponential Levy jumps. Also in [4] and [5], the inversion of stochastic flow techniques are used for building a family of forward utilities for a given optimal portfolio.

The obstacle problem for PIDEs [Equation (4) in the published article by World Scientific]

References

1. Léandre R. “Flot d’une équation différentielle stochastique avec semimartingale directrice discontinue.” Sém. Probab. (Strasbourg) 19 (1985) 271.

2. Bally V, Matoussi A. Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theor. Probab. 14 (2001) 125.

3. Jourdain B. Stochastic flows approach to Dupire’s formula.Fin. Stoch. 11 (2007) 521.

4. El Karoui N, Mrad M. Stochastic utilities with a given optimal portfolio: Approach by stochastic flows. arXiv:1004.5192.

5. El Karoui N, Mrad M. An exact connection between two solvable SDEs and a nonlinear utility stochastic PDE.SIAM J. Finan. Math 4(1) 697.