We extend the study of [B. Bouchard, G. Loeper, and Y. Zou, SIAM J. Control Optim., 55 (2017), pp. 3319-3348; G. Loeper, Ann. Appl. Probab., 28 (2018), pp. 2664-2726] stochastic target problems with general market impacts. Namely, we consider a general abstract model which can be associated to a fully nonlinear parabolic equation. Unlike the earlier articles, the equation is not concave, and the regularization/verifcation approach of our 2017 cannot be applied. We also relax the gamma constraint of the 2017 article. Instead, we need to generalize the a priori estimates of Loeper's article and exhibit smooth solutions from the classical parabolic equations theory. Up to an additional approximating argument, this allows us to show that the superhedging price solves the parabolic equation and that a perfect hedging strategy can be constructed when the coefcients are smooth enough. This representation leads to a general dual formulation. We fnally provide an asymptotic expansion around a model without impact.
All Science Journal Classification (ASJC) codes
- Control and Optimization
- Applied Mathematics
- Asymptotic expansion
- Fully nonlinear parabolic equation
- Generalized market impact
- Second-order stochastic target