By Ramasubramanian S.
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Stochastic Processes for Insurance and Finance. Wiley, New York (2001) 23. : A lemma of variational distance between maximal functions with application to the Skorokhod problem in a nonnegative orthant with state dependent reflection directions. Stoch. Stoch. Rep. 48, 161–194 (1994) 24. : Optimal control with state space constraints I. SIAM J. Control Optim.
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. Therefore it follows that for any λ1 , λ2 as above (uˆ 1 (·), uˆ 2 (·)) ≡ (λ1 , λ2 ) gives a Nash equilibrium for each t ≥ 0. So even Nash equilibrium serving for all t need not be unique. Next note that (λ1 , λ2 ) = (0, 1) as well as (λ1 , λ2 ) = ( R121 , 0) give feasible controls (in fact both are Nash equilibria).
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