Proposition The convergence time to be close to the risk-sensitive payoff with error tolerance η is at most

max(Gj)1(log[ error(0)η])j

where (Gj)-1 is the inverse of the mapping tGj(t):=0tgj(t)dt and errorj(0):=||r^j,0E1μj(eμjU˜j1)||, error = maxj errorj. In particular for gj = 1 (almost active case), the convergence time is of order of log[ error(0)η].

Proof. We verify that the solution of the ODE is ||r^j,trsE1μj(eμjU˜j1)||=errorje0tgj(s)ds. From the assumptions, the primitive function Gj is a bijection and r^j,trsE1μj(eμjU˜j1)ηiftmaxj(Gj)1(log[ error(0)η]). The last assertion is obtained for gj = 1. This completes the proof.    Explicit Solutions

As promised in our introduction of this Chapter, ...

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