Abstract
We establish necessary and sufficient conditions of near-optimality for nonlinear systems governed by forward-backward stochastic differential equations with controlled jump processes (FBSDEJs in short). The set of controls under consideration is necessarily convex. The proof of our result is based on Ekeland’s variational principle and continuity in some sense of the state and adjoint processes with respect to the control variable. We prove that under an additional hypothesis, the near-maximum condition on the Hamiltonian function is a sufficient condition for near-optimality. At the end, as an application to finance, mean-variance portfolio selection mixed with a recursive utility optimization problem is given. Mokhtar Hafay
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The first author was partially supported by Algerian PNR project grant 08/u07/857, ATRST-ANDRU 2011-2013. The second author was supported by the Czech CTU grant SGS12/197/OHK4/3T/14 and MSMT grant INGO II INFRA LG12020.
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Hafayed, M., Veverka, P. & Abbas, S. On near-optimal necessary and sufficient conditions for forward-backward stochastic systems with jumps, with applications to finance. Appl Math 59, 407–440 (2014). https://doi.org/10.1007/s10492-014-0064-4
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DOI: https://doi.org/10.1007/s10492-014-0064-4
Keywords
- stochastic near-optimal controls
- jump processes
- forward-backward stochastic systems with jumps
- necessary and sufficient conditions for near-optimality
- Ekeland’s variational principle