In this paper a problem of consumption and investment is presented as a model of a discounted Markov decision process with discrete-time. In this problem, it is assumed that the wealth is affected by a production function. This assumption gives the investor a chance to increase his wealth before the investment. For the solution of the problem there is established a suitable version of the Euler Equation (EE) which characterizes its optimal policy completely, that is, there are provided conditions...

We propose to test the homogeneity of a Poisson process observed on a finite interval. In this framework, we first provide lower bounds for the uniform separation rates in -norm over classical Besov bodies and weak Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies coincide with the minimax estimation rates over such classes. Then we construct non-asymptotic and non-parametric testing procedures that are adaptive in the sense that they achieve, up to a possible logarithmic...

We consider a diffusion process $X_t$ smoothed with (small) sampling parameter $\epsilon$. As in Berzin, León and Ortega (2001), we consider a kernel estimate ${\widehat{\alpha }}_{\epsilon }$ with window $h\left(\epsilon \right)$ of a function $\alpha$ of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the $L^p$ deviations such as$$\frac{1}{\sqrt{h}}{\left(\frac{h}{\epsilon }\right)}^{\frac{p}{2}}\left({∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p-𝔼{∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p\right).$$

We consider a diffusion process Xt smoothed with (small) sampling parameter ε. As in Berzin, León and Ortega (2001), we consider a kernel estimate ${\widehat{\alpha }}_{\epsilon }$ with window h(ε) of a function α of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the Lp deviations such as $$\frac{1}{\sqrt{h}}{\left(\frac{h}{\epsilon }\right)}^{\frac{p}{2}}\left({∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p-\text{I}\text{E}{∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p\right).$$