**Proposition 9.3.5.7**. *The convergence time to be close to the risk-sensitive payoff with error tolerance η is at most*

$\underset{j}{\mathrm{max}{\left({G}^{j}\right)}^{-1}\left(\mathrm{log}\left[\frac{error\left(0\right)}{\eta}\right]\right)}$

*where* (*G*^{j})^{-1} *is the inverse of the mapping* $t\mapsto {G}^{j}(t):={\displaystyle {\int}_{0}^{t}{g}_{j}({t}^{\prime})\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{t}^{\prime}}$ *and* $erro{r}_{j}(0):=\left|\right|{\widehat{\mathrm{r}}}_{j,0}-\mathbb{E}\frac{1}{{\mu}_{j}}({e}^{{\mu}_{j}{\tilde{U}}_{j}}-1)\left|\right|$, *error* = max_{j} *error*_{j}. *In particular for g _{j}* = 1

*(almost active case), the convergence time is of order of*$\mathrm{log}\text{\hspace{0.17em}}\left[\frac{error(0)}{\eta}\right]$.

*Proof*. We verify that the solution of the ODE is $\left|\right|{\widehat{\mathrm{r}}}_{j,t}^{rs}-\mathbb{E}\frac{1}{{\mu}_{j}}({e}^{{\mu}_{j}{\tilde{U}}_{j}}-1)\left|\right|=\mathrm{erro}{r}_{j}{e}^{-\text{\hspace{0.17em}}}{{\displaystyle {\int}_{0}^{t}{g}_{j}(s)\text{\hspace{0.17em}}{\text{\hspace{0.17em}}}^{ds}\text{\hspace{0.17em}}\text{\hspace{0.17em}}}}^{}$. From the assumptions, the primitive function *G ^{j}* is a bijection and $\Vert {\widehat{\mathrm{r}}}_{j,t}^{rs}-\mathbb{E}\frac{1}{{\mu}_{j}}\left({e}^{{\mu}_{j}{\tilde{U}}_{j}}-1\right)\Vert \le \eta \text{\hspace{0.17em}}if\text{\hspace{0.17em}}t\ge {\mathrm{max}}_{j}{\left({G}^{j}\right)}^{-1}\left(\mathrm{log}\left[\frac{error\left(0\right)}{\eta}\right]\right)$. The last assertion is obtained for

*g*

_{j}= 1. This completes the proof.

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

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