# Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

## Abstract

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model *certainty*, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.

## Keywords

Forward–backward SDEs Stochastic differential games Maximum principle Model uncertainty Robust control Viability Optimal portfolio Optimal consumption Jump diffusions## Notes

### Acknowledgements

We thank Olivier Menoukeu Pamen and Marie-Claire Quenez for helpful comments.

The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no [228087]

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