By Steve Barr
A section too uncomplicated. in case you particularly can't draw anything...ok. differently, a section too easy.
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Extra info for 1-2-3 draw cool cartoon stuff: a step-by-step guide
Oper. Res. 28, 801–835 (2003) 3. : Optimal Control and Viscosity Solutions of Hamilton-JacobiBellman Equations. Birkhauser, Boston (1997) 4. : Stochastic games for N players. J. Optim. Theory Appl. 105, 543–565 (2000) 5. : Stochastic games with risk sensitive pay offs for N players. Le Matematiche 55(Suppl. 2), 5–54 (2000) 6. : Ergodic control for constrained diffusions: characterization using HJB equations. SIAM J. Control Optim. 43, 1467–1492 (2004/2005) 7. : Stochastic differential games: occupation measure based approach.
4 the solution to the Skorokhod problem gives a Nash equilibrium; this also follows from the argument given below. Let λ1 ≥ 0, 0 ≤ λ2 ≤ 1 be such that λ2 + R21 λ1 = 1. Put yˆ1 (t) = λ1 (t), yˆ2 (t) = λ2 t. Fix yˆ1 (·). Let 0 ≤ y2 (t) < λ2 t for some t. Corresponding to yˆ1 (·), y2 (·) note that z1 (t) ≥ 0 but z2 (t) = −t + y2 (t) + R21 yˆ1 (t) < 0. So with yˆ1 (·) fixed, y2 (·) cannot be feasible unless y2 (t) ≥ λ2 t, ∀t. In an entirely analogous manner with yˆ2 (·) fixed, y1 (·) cannot be feasible unless y1 (t) ≥ λ1 t for all t.
But (0, 0) cannot be a feasible control. Hence there is no utopian equilibrium even for a single t > 0. References 1. : Singular control with state constraints on unbounded domain. Ann. Probab. 34, 1864–1909 (2006) 2. : An escape-time criterion for queueing networks: asymptotic risksensitive control via differential games. Math. Oper. Res. 28, 801–835 (2003) 3. : Optimal Control and Viscosity Solutions of Hamilton-JacobiBellman Equations. Birkhauser, Boston (1997) 4. : Stochastic games for N players.