Volume 27, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Number of page(s)||36|
|Published online||01 March 2021|
Probabilistic interpretation of a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations*
School of Mathematics and Statistics, Shandong University,
264209, P.R. China.
2 School of Mathematics and Information Sciences, Yantai University, Yantai 264005, P.R. China.
3 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R. China.
** Corresponding author: email@example.com
Accepted: 19 October 2020
By introducing a stochastic differential game whose dynamics and multi-dimensional cost functionals form a multi-dimensional coupled forward-backward stochastic differential equation with jumps, we give a probabilistic interpretation to a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations. For this, we generalize the definition of the lower value function initially defined only for deterministic times t and states x to stopping times τ and random variables η ∈ L2(Ω, 𝓕τ, P; ℝ). The generalization plays a key role in the proof of a strong dynamic programming principle. This strong dynamic programming principle allows us to show that the lower value function is a viscosity solution of our system of multi-dimensional coupled Hamilton-Jacobi-Bellman-Isaacs equations. The uniqueness is obtained for a particular but important case.
Mathematics Subject Classification: 49N70 / 49L25 / 60H10 / 93E20
Key words: Strong dynamic programming principle / coupled FBSDEs with jumps / stochastic differential games / HJBI equations
The work has been supported in part on one hand by the NSF of P.R. China (No. 11871037, 11971099), National Key R and D Program of China (No. 2018YFA0703900), NSFC-RS (No. 11661130148, NA150344), on the other hand by the Natural Science Foundation of Shandong Province (ZR2017MA015), Doctoral Scientific Research Fund of YantaiUniversity (No. SX17B09), and by the Science and Technology Development Plan Project of Jilin Province (20190103026JH).
© EDP Sciences, SMAI 2021
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