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Bayesian dividend optimization and finite time ruin probabilities. (English) Zbl 1292.91182

The paper considers the valuation problem of a company, where the firm value is given by \[ X_t=x+\int_0^t \theta ds +\sigma B_t -L_t, \] \(x\) (\(>0\)) being the initial capital, \(\theta\) the constant unobservable drift, \(\sigma\) the constant known volatility, \((B_t)_{t\geq 0}\) a standard Brownian motion, and \((L_t)_{t\geq 0}\) the accumulated dividend process. Aim of the paper is the solution of the dividend maximization problem.
First of all, since the drift parameter is supposed to be unknown, on the basis of filtering theory, an estimator is derived, so overcoming incomplete information. Then, the Hamilton-Jacobi-Bellman equation for the stochastic optimal control problem is derived and studied, and successively a numerical method for estimating the optimal dividend strategy is presented. The case of threshold strategies is analyzed; finally, the finite-time ruin probabilities are investigated.

MSC:

91G50 Corporate finance (dividends, real options, etc.)
91B30 Risk theory, insurance (MSC2010)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
49L20 Dynamic programming in optimal control and differential games
93E11 Filtering in stochastic control theory
93E20 Optimal stochastic control
91G60 Numerical methods (including Monte Carlo methods)
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