Investment in CD

One may define probabilities $p(y^j)$ of discrete values of wealth $y^j,\ j=1,2,....$ by exact expressions. For example,

$$p(y^0) = \prod_{i} q_i,$$

$$p(y^1) = p_1 \prod_{i \ne 1} q_i,$$

$$p(y^2) = p_2 \prod_{i \ne 2} q_i,$$

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$$p(y^n) = p_n \prod_{i \ne n} q_i ,$$

$$p(y^{n+1}) = p_1 p_2 \prod_{i \ne 1, i \ne 2} q_i ,$$

$$p(y^{n+2}) = p_1 p_3 \prod_{i \ne 1, i \ne 3} q_i$$ (5)

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Here
$y^0=0,\ y^1=a_1 x_1,\ y^2=a_2 x_2,\ y^n= a_n x_n,\\ y^{n+1}=a_1 x_1+a_2 x_2,\ y^{n+2}=a_1 x_1 + a_3 x_3$. From expression (5)

$$U(x)=\sum_{k=1}^M u(y^k) p(y^k).$$ (6)

Here $M$ is the number of different values of wealth $y$.

One determines $U(x)$ approximately by the Monte Carlo approach:

$$U_K(x)= 1/K \sum_{k=1}^K u(y^k).$$ (7)

Here

$$y^k= \sum_{i=1}^n y_i^k ,$$ (8)

where

$$y_i^k= \left\{ \begin{array}{ll} c_i x_i, & \mbox{if $\eta_i^k \in [0,p_i]$} \\ 0, & \mbox{otherwise} \end{array}\right.$$ (9)

Here $K$ is the number of Monte Carlo samples. $\eta_i^k$ is a random number uniformly distributed on the unit interval. In this case

$$U(x)= \lim_{K \to \infty} U_K(x).$$