##
**Optimal dividends and capital injections in the dual model with diffusion.**
*(English)*
Zbl 1242.91089

The study considers the dual model with diffusion, where the insurance company surplus at time \(t\) is depicted by
\[
U(t)=x-ct+S(t)+\sigma W(t), t \geq 0
\]
where \(U(0-)=x \geq 0\) is the initial surplus, \(c>0\) is the expense rate per unit time, \(S(t)\) is a compound Poisson process with intensity \(\lambda\), \(W(t)\) is a standard Brownian motion independent of \(S(t)\) with volatility equal to \(\sigma\).

Within a general optimal control problem – where the main goal is the optimal control strategy maximizing the expected present value of dividends less capital injections until ruin – dividend payments and equity issuance are used as suitable control keys.

In this framework two steps are considered. Firstly, dividends only are considered, obtaining an optimal barrier strategy and a closed form of its value function.

Then another barrier strategy is obtained in the case of forced injections when the surplus is null to prevent ruin.

Moreover the optimal joint strategy is investigated on the basis of the two afore described steps.

Within a general optimal control problem – where the main goal is the optimal control strategy maximizing the expected present value of dividends less capital injections until ruin – dividend payments and equity issuance are used as suitable control keys.

In this framework two steps are considered. Firstly, dividends only are considered, obtaining an optimal barrier strategy and a closed form of its value function.

Then another barrier strategy is obtained in the case of forced injections when the surplus is null to prevent ruin.

Moreover the optimal joint strategy is investigated on the basis of the two afore described steps.

Reviewer: Emilia Di Lorenzo (Napoli)