Portfolio optimization and model predictive control: A kinetic approach

Figure 1.  Sketch of the modelling process

Figure 2.  Example of the value function $ U_{\gamma} $ with different reference points

Figure 3.  Stock price evolution in the long-term investor case with a constant fundamental price $ s^f $ (left figure) and a time varying fundamental price (right figure). In both figures one obtains that the average stock price is above the funcamental value

Figure 4.  Quantile-quantile plot of logarithmic stock return distribution (left-hand side) and logarithmic return of fundamental prices (right-hand side). The simulation has been performed in the case of long-term investors and a stochastic fundamental price. The risk tolerance has been set to $ \gamma = 0.9 $, the scale to $ \rho = \frac{5}{8} $ and the random seed is chosen to be $\texttt{rng(767)}$. All further parameters are chosen as reported in section A.4 of the Appendix

Figure 5.  Stock price distribution in the long-term investor case. The solid lines are analytical solution, whereas the circles are the numerical result

Figure 6.  Distribution of the wealth invested in stocks with a Gaussian fit (solid line). Left figure has a linear scale, whereas the right figure shows the distribution in log-log scale

Figure 7.  Distribution of the wealth invested in bonds in the special case $ K>0 $. The numerical results (circles) are plotted with the corresponding log-normal analytic self-similar solution (solid lines)

Figure 8.  Stock price distribution in the high-frequency case (red circles). The fit by the inverse-gamma distribution (solid line) clearly underestimates the tail. This reveals that the full model can create heavier tails than the inverse-gamma distribution

Figure 9.  Marginal wealth distributions in the high-frequency investor case. The left hand side illustrates the distribution of investments in stocks and the right-hand side the wealth invested in bonds at $ t = 1 $

Figure 10.  Steady state stock price distribution in the high-frequency investor case (circles) together with the analytically computed steady state of inverse-gamma type (solid line)