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# 金融代写|随机分析代写STOCHASTIC ANALYSIS代考|STATS217 Numerical results

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## 金融代写|随机分析代写STOCHASTIC ANALYSIS代考|Numerical results

In this section, we explain the setup of the simulation and exhibit the main results. We have used Tensorflow 2 and deep learning techniques for Python developped in [10]. We consider $d=3$ risky assets and a riskless asset whose return is assumed to be 0, on a 1-year investment horizon for the sake of simplicity. We consider 24 portfolio rebalancing during the 1-year period, i.e., one every two weeks. This means that we have $N=24$ steps in the training of our neural networks. The parameters used in the simulation are detailed in Table $2 .$

First, we show the numerical results for the learning and the non-learning strategies by presenting a performance and an allocation analysis in Subsection 5.3.1. Then, we add the admissible constrained EW to the two previous ones and use this neutral strategy as a benchmark in Subsection 5.3.2. Ultimately, in Subsection 5.3.3, we illustrate numerically the convergence of the non-learning strategy to the constrained Merton problem when the loss aversion parameter $q$ vanishes.

## 金融代写|随机分析代写STOCHASTIC ANALYSIS代考|Learning and non-learning strategies

We simulate $\tilde{N}=1000$ trajectories for each strategy and exhibit the performance results with an initial wealth $x_{0}=1$. Figures 3 illustrates the average historical level of the learning and non-learning strategies with a $95 \%$ confidence interval. Learning outperforms significantly Non-Learning with a narrower confidence interval revealing that less uncertainty surrounds Learning performance, thus yielding less risk.

An interesting phenomenon, visible in Fig. 3, is the nearly flat curve for Learning between time 0 and time 1. Indeed, whereas Non-Learning starts investing immediately, Learning adopts a safer approach and needs a first time step before allocating a significant proportion of wealth. Given the level of uncertainty surrounding $b_{0}$, this first step allows Learning to fine-tune its allocation by updating the prior belief with the first return available at time 1. On the contrary, Non-Learning, which cannot update its prior, starts investing at time 0 .

Fig. 4 shows the ratio of Learning over Non-Learning. A ratio greater than one means that Learning outperforms Non-Learning and underperforms when less than one. It shows the significant outperformance of Learning over Non-Learning except during the first period where Learning was not significantly invested and NonLearning had a positive return. Moreover, this graph reveals the typical increasing concave curve of the value of information described in [17], in the context of investment decisions and costs of data analytics, and in [6] in the resolution of the Markowitz portfolio selection problem using a Bayesian learning approach.

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