Tesi di LAUREA SPECIALISTICA Titolo Stochastic calculus for optimal bankruptcy and retirement in a consumption, leisure rate and investment problem Data 2012-12-23 Autore/i Mercuri, Emanuele Relatore Marazzina, D. Correlatore Barucci, E. Full text non disponibile Abstract In this work, we mix stopping problem and stochastic control to optimally choose multiple stopping times (bankruptcy and retirement dates), in order to maximize the expected discounted individual s utility function, which derives from consumption, leisure and investment opportunities, and that is subject to borrowing and budget constraints. Henceforth, we deal with a risk-averse and infinitely lived economic agent, who could receive a stream of labor income, might have the duty to repay a debt, and faces liquidity constraints and a mixture of portfolio and consumption-leisure choice. Relying on the familiar principle of smooth-fit conditions, we cast the primal constraint optimization problem as a variational inequality, which refers to an individual s dual shadow prices problem. To this end, we apply convex analysis, through the Legendre-Fenchel conjugation transform, and a suitable Hamilton-Jacobi-Bellman equation, to deal with a free-boundaries problem, which we explicitly solve. Therefore, this leads us to obtain the value function, a closed form solution to wealth, consumption, leisure and portfolio stochastic dynamics and to determine optimal strategies to declare bankruptcy and even retirement. Afterwards, at the end of each chapter, through simulations of different scenarios via Montecarlo method, we implement the analytic solution and we outline its sensitivity with respect to small changes in parameter values, which are comparable to those provided in literature. These numerical experiments confirm some theoretical results established in different settings and provide examples of some empirical evidence. The rationale of each aspect is widely outlined and addressed with the aid of tables and figures, that are the synthesis of several numerical experiments, in order to sum up the main features of each model. Henceforth, we deal with a bankruptcy problem with no wage rate, a retirement problem with debt repayment, a bankruptcy problem with wage rate and finally a problem with both optimal bankruptcy and retirement dates