1,653 research outputs found
A sufficient maximum principle for backward stochastic systems with mixed delays
In this paper, we study the problem of optimal control of backward stochastic differential equations with three delays (discrete delay, moving-average delay and noisy memory). We establish the sufficient optimality condition for the stochastic system. We introduce two kinds of time-advanced stochastic differential equations as the adjoint equations, which involve the partial derivatives of the function and its Malliavin derivatives. We also show that these two kinds of adjoint equations are equivalent. Finally, as applications, we discuss a linear-quadratic backward stochastic system and give an explicit optimal control. In particular, the stochastic differential equations with time delay are simulated by means of discretization techniques, and the effect of time delay on the optimal control result is explained
Large-scale games in large-scale systems
Many real-world problems modeled by stochastic games have huge state and/or
action spaces, leading to the well-known curse of dimensionality. The
complexity of the analysis of large-scale systems is dramatically reduced by
exploiting mean field limit and dynamical system viewpoints. Under regularity
assumptions and specific time-scaling techniques, the evolution of the mean
field limit can be expressed in terms of deterministic or stochastic equation
or inclusion (difference or differential). In this paper, we overview recent
advances of large-scale games in large-scale systems. We focus in particular on
population games, stochastic population games and mean field stochastic games.
Considering long-term payoffs, we characterize the mean field systems using
Bellman and Kolmogorov forward equations.Comment: 30 pages. Notes for the tutorial course on mean field stochastic
games, March 201
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