## 1 Introduction

## 2 Risk averse multistage problem

^{1}we assume that

## 3 Nested distance

^{2}is a transportation from \({\mathbb {P}}[\xi _{t}\in \bullet |\overline{\xi }_{t-1}]\) into \({\mathbb {P}}[\varsigma _{t}\in \bullet |\overline{\varsigma }_{t-1}]\).

^{3}

## 4 Smoothed quantization

^{4}Thus, by Proposition 4, the Lipschitz properties of (18)–(20) hold and, consequently, Theorem 2 holds once the tails of each \(\sum _{\tau =1}^{t}a^{t-\tau }\epsilon _{t}\), \(1\le t\le T,\) are \(O(x^{-\alpha })\) for some \(\alpha >1\), which is true in most cases because the converse would imply infinite second moments.

## 5 Approximation of the multistage problem

1. | Determine suitable rectangular sets \((C_{t,i})_{1\le i\le k_{t}1\le ,t\le T}\) |

2. | For each \(t=1\) to \(T-1\) |

3. | For each \(i=1\) to p |

4. | For each \(j=1\) to \(k_{t,i}\) |

5. | Put \(e_{t,i}^{j}=\textrm{median}({\omega }^i_{t,j})\) |

6. | End For |

7. | End For |

8. | End For |

9. | Construct the smoothed quantization \(\varsigma\) of \(\xi\) defined by \(C_\bullet\) and \(e_\bullet\). |

10. | If \(\varsigma\) is suitable |

11. | |

12. | Else |

13. | Refine \(\varsigma\) to get a suitable approximation \(\chi\) |

14. | |

15. | End If |

1. | Let c be such that \({\mathbb {P}}[\xi _t\le c]\) is small for each \(1\le t\le T\) |

2. | Let e be slightly smaller than c |

3. | For each \(t=1\) to T |

4. | Add c to the collection \(c_{t,\bullet }\) |

5. | Add e to the collection \(e_{t,\bullet }\) |

6. | Determine \(\omega '_{t,\bullet }\) according to (37) |

7. | Put \(p_{t,i,j}=\mu _t(C_{t,j}|e_{t-1,i})\), \(1\le j\le k_t\), \(1\le i\le k_{t-1}\) |

8. | End For |

9. | For each \(t=2\) to T |

10. | For each \(i=1\) to \(k_{t-1}\) |

11. | If \(\varrho \int x \varpi _{t}(dx|e_{t-1,i})> e_{t-1,i}\) |

12. | Assign \(p_{t,i,\bullet }\) the optimal solution of Problem (34) |

13. | End If |

14. | End For |

15. | End For |

## 6 Application

^{5}and subsitute it into the approximate policy to get the corresponding incomes \(Z=(Z_1,Z_2){\mathop {=}\limits ^{\text {def}}}(\varrho z_1, \varrho ^2 z_2)\). As a result of this procedure, we have a sample of 100 values of \(Z_1\) and 100 values of \(Z_2\) for each \(Z_1\). The mean part of the criterion has been estimated by the sample means of Z, each \(\textrm{CVaR}(Z_2|Z_1=z)\) as a mean of the 20 highest values of \(Z_2\) with \(Z_1=z\), and the outer \(\textrm{CVaR}\) as the average of the highest 20 inner CVaRs. Finally, we estimate \(\rho\) by the average of our 20 estimators, denoting it by \(\tilde{v}(=\tilde{v}_{k_1,k_2})\).

\(k_1\) | \(k_2\) | \(\tilde{v}_{k_1,k_2}\) | t | \(k_1\) | \(k_2\) | \(\tilde{v}_{k_1,k_2}\) | t |
---|---|---|---|---|---|---|---|

no opt | 54.63 (0.4) | ||||||

3 | 3 | 60.48 (0.12) | 61 | 6 | 7 | 60.72 (0.08) | 116 |

3 | 4 | 60.43 (0.13) | 64 | 6 | 8 | 60.65 (0.09) | 121 |

3 | 5 | 60.47 (0.14) | 67 | 6 | 9 | 60.69 (0.09) | 127 |

4 | 4 | 60.47 (0.09) | 76 | 7 | 11 | 60.64 (0.09) | 152 |

4 | 5 | 60.59 (0.11) | 80 | 8 | 12 | 60.69 (0.07) | 173 |

4 | 6 | 60.51 (0.1) | 84 | 9 | 14 | 60.78 (0.07) | 203 |

5 | 5 | 60.62 (0.07) | 93 | 10 | 15 | 60.78 (0.08) | 225 |

5 | 6 | 60.48 (0.1) | 98 | 11 | 17 | 60.84 (0.08) | 256 |

5 | 7 | 60.65 (0.08) | 102 | 12 | 16 | 60.75 (0.09) | 261 |

5 | 8 | 60.5 (0.11) | 108 | 12 | 18 | 60.92 (0.07) | 281 |

6 | 6 | 60.76 (0.07) | 110 | 13 | 19 | 60.9 (0.06) | 306 |