What is the relation between the solutions of the optimal reinsurance problem for a diffusion approximation and the optimal reinsurance problem for the classical risk model?
Apparently this result in the appendix of 'Optimal Control Problems in Insurance' by Shmidli answerx the question, but I dont know why:
The following result motivates us to approximate a control problem by a diffusion approximation:
Let $m:\mathbb{R}\rightarrow \mathbb{R}$ be a Lipshitz continuous function, $\lbrace X^{n}:n\in\mathbb{N}\rbrace$ be semimartingales such that $X_{0}^{n}=0$, and $X$ be a semimartingale such that $X_{0}=0$. Suppose that $Z^{n}$ fulfils the stochastic integral equation
$$Z_{t}^{n}=x + X_{t}^{n} + \int_{0}^{t}m(Z_{s}^{n})ds$$
and $Z$ fulfils
$$Z_{t}=x + X_{t} + \int_{0}^{t}m(Z_{s})ds$$
Then $Z^{n}_{t}$ converges weakly to $Z$ if and only if $X^{n}$ converges weakly to $X$.