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Stochastic Differential Games: A Sampling Approach via FBSDEs

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Abstract

The aim of this work is to present a sampling-based algorithm designed to solve various classes of stochastic differential games. The foundation of the proposed approach lies in the formulation of the game solution in terms of a decoupled pair of forward and backward stochastic differential equations (FBSDEs). In light of the nonlinear version of the Feynman–Kac lemma, probabilistic representations of solutions to the nonlinear Hamilton–Jacobi–Isaacs equations that arise for each class are obtained. These representations are in form of decoupled systems of FBSDEs, which may be solved numerically.

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Notes

  1. A process \(H_s\) is called square-integrable if \( \mathbb {E}\big [\int _{t}^{T}H_s^2\mathrm {d}s\big ] < \infty \) for any \(T>t\).

  2. The Isaacs condition renders the viscosity solutions of the upper and lower value functions equal, thus making the order of maximization/minimization inconsequential.

  3. While X is a function of s and \(\omega \), we shall use \(X_s\) for notational brevity.

  4. Here, \(Y_i^m\) denotes the quantity \(Y^m_{i+1} +\varDelta t_ih(t_{i+1},X^m_{i+1},Y^m_{i+1},Z^m_{i+1})\), which is the \(Y^m_i\) sample value before the conditional expectation operator has been applied.

  5. Whenever the m index is not present, the entirety with respect to this index is to be understood.

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Acknowledgements

Funding was provided by Army Research Office (W911NF-16-1-0390) and National Science Foundation (CMMI-1662523).

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  1. Department of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0150, USA

    Ioannis Exarchos, Evangelos Theodorou & Panagiotis Tsiotras

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  1. Ioannis Exarchos
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  2. Evangelos Theodorou
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Correspondence to Ioannis Exarchos.

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Exarchos, I., Theodorou, E. & Tsiotras, P. Stochastic Differential Games: A Sampling Approach via FBSDEs. Dyn Games Appl 9, 486–505 (2019). https://doi.org/10.1007/s13235-018-0268-4

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