November  2019, 24(11): 6209-6238. doi: 10.3934/dcdsb.2019136

Portfolio optimization and model predictive control: A kinetic approach

1. 

Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany

2. 

Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, I-44121 Ferrara, Italy

3. 

Karlsruhe Institute of Technology, Steinbuch Center for Computing, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany

Corresponding author: trimborn@igpm.rwth-aachen.de

Received  June 2018 Revised  January 2019 Published  July 2019

In this paper, we introduce a large system of interacting financial agents in which all agents are faced with the decision of how to allocate their capital between a risky stock or a risk-less bond. The investment decision of investors, derived through an optimization, drives the stock price. The model has been inspired by the econophysical Levy-Levy-Solomon model [30]. The goal of this work is to gain insights into the stock price and wealth distribution. We especially want to discover the causes for the appearance of power-laws in financial data. We follow a kinetic approach similar to [33] and derive the mean field limit of the microscopic agent dynamics. The novelty in our approach is that the financial agents apply model predictive control (MPC) to approximate and solve the optimization of their utility function. Interestingly, the MPC approach gives a mathematical connection between the two opposing economic concepts of modeling financial agents to be rational or boundedly rational. Furthermore, this is to our knowledge the first kinetic portfolio model which considers a wealth and stock price distribution simultaneously. Due to the kinetic approach, we can study the wealth and price distribution on a mesoscopic level. The wealth distribution is characterized by a log-normal law. For the stock price distribution, we can either observe a log-normal behavior in the case of long-term investors or a power-law in the case of high-frequency trader. Furthermore, the stock return data exhibit a fat-tail, which is a well known characteristic of real financial data.

Citation: Torsten Trimborn, Lorenzo Pareschi, Martin Frank. Portfolio optimization and model predictive control: A kinetic approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6209-6238. doi: 10.3934/dcdsb.2019136
References:
[1]

G. AlbiM. Herty and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Communications in Mathematical Sciences, 13 (2015), 1407-1429. doi: 10.4310/CMS.2015.v13.n6.a3. Google Scholar

[2]

G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20140138, 18pp. doi: 10.1098/rsta.2014.0138. Google Scholar

[3]

A. Beja and M. B. Goldman, On the dynamic behavior of prices in disequilibrium, The Journal of Finance, 35 (1980), 235-248. Google Scholar

[4]

D. Bertsimas and D. Pachamanova, Robust multiperiod portfolio management in the presence of transaction costs, Computers & Operations Research, 35 (2008), 3-17. doi: 10.1016/j.cor.2006.02.011. Google Scholar

[5]

M. BisiG. Spiga and G. Toscani, Kinetic models of conservative economies with wealth redistribution, Communications in Mathematical Sciences, 7 (2009), 901-916. doi: 10.4310/CMS.2009.v7.n4.a5. Google Scholar

[6]

J.-P. Bouchaud and M. Mézard, Wealth condensation in a simple model of economy, Physica A: Statistical Mechanics and its Applications, 282 (2000), 536-545. doi: 10.1016/S0378-4371(00)00205-3. Google Scholar

[7]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Communications in Mathematical Physics, 56 (1977), 101-113. doi: 10.1007/BF01611497. Google Scholar

[8]

W. A. Brock and C. H. Hommes, Heterogeneous beliefs and routes to chaos in a simple asset pricing model, Journal of Economic Dynamics and Control, 22 (1998), 1235-1274. doi: 10.1016/S0165-1889(98)00011-6. Google Scholar

[9]

M. Burger, L. Caffarelli, P. A. Markowich and M.-T. Wolfram, On a Boltzmann-type price formation model, Proc. R. Soc. A, 469 (2013), 20130126, 20pp. doi: 10.1098/rspa.2013.0126. Google Scholar

[10]

E. Camacho and C. Bordons, Model Predictive Control, Springer, USA, 2004.Google Scholar

[11]

A. Chatterjee and B. K. Chakrabarti, Kinetic exchange models for income and wealth distributions, The European Physical Journal B-Condensed Matter and Complex Systems, 60 (2007), 135-149. doi: 10.1140/epjb/e2007-00343-8. Google Scholar

[12]

L. ChayesM. del Mar GonzálezM. P. Gualdani and I. Kim, Global existence and uniqueness of solutions to a model of price formation, SIAM Journal on Mathematical Analysis, 41 (2009), 2107-2135. doi: 10.1137/090753346. Google Scholar

[13]

J. Che, A kinetic model on portfolio in finance, Communications in Mathematical Sciences, 9 (2011), 1073-1096. doi: 10.4310/CMS.2011.v9.n4.a7. Google Scholar

[14]

C. ChiarellaR. Dieci and X.-Z. He, Heterogeneous expectations and speculative behavior in a dynamic multi-asset framework, Journal of Economic Behavior & Organization, 62 (2007), 408-427. Google Scholar

[15]

D. Colander, H. Föllmer, A. Haas, M. D. Goldberg, K. Juselius, A. Kirman, T. Lux and B. Sloth, The financial crisis and the systemic failure of academic economics, 2009.Google Scholar

[16]

R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, 2001.Google Scholar

[17]

S. CordierL. Pareschi and C. Piatecki, Mesoscopic modelling of financial markets, Journal of Statistical Physics, 134 (2009), 161-184. doi: 10.1007/s10955-008-9667-z. Google Scholar

[18]

S. CordierL. Pareschi and G. Toscani, On a kinetic model for a simple market economy, Journal of Statistical Physics, 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0. Google Scholar

[19]

R. CrossM. GrinfeldH. Lamba and T. Seaman, A threshold model of investor psychology, Physica A: Statistical Mechanics and its Applications, 354 (2005), 463-478. doi: 10.1016/j.physa.2005.02.029. Google Scholar

[20]

M. Delitala and T. Lorenzi, A mathematical model for value estimation with public information and herding, Kinetic & Related Models, 7 (2014), 29-44. doi: 10.3934/krm.2014.7.29. Google Scholar

[21]

R. L. Dobrushin, Vlasov equations, Functional Analysis and Its Applications, 13 (1979), 115-123. Google Scholar

[22]

B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Physical Review E, 78 (2008), 056103, 12pp. doi: 10.1103/PhysRevE.78.056103. Google Scholar

[23]

E. EgenterT. Lux and D. Stauffer, Finite-size effects in Monte Carlo simulations of two stock market models, Physica A: Statistical Mechanics and its Applications, 268 (1999), 250-256. doi: 10.1016/S0378-4371(99)00059-X. Google Scholar

[24]

J. D. Farmer and D. Foley, The economy needs agent-based modelling, Nature, 460 (2009), 685-686. doi: 10.1038/460685a. Google Scholar

[25]

T. Hellthaler, The influence of investor number on a microscopic market model, International Journal of Modern Physics C, 6 (1996), 845-852. doi: 10.1142/S0129183195000691. Google Scholar

[26]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica: Journal of the Econometric Society, 47 (1979), 263-291. doi: 10.2307/1914185. Google Scholar

[27]

K. Kanazawa, T. Sueshige, H. Takayasu and M. Takayasu, Derivation of the boltzmann equation for financial brownian motion: direct observation of the collective motion of high-frequency traders, Physical Review Letters, 120 (2018), 138301. doi: 10.1103/PhysRevLett.120.138301. Google Scholar

[28]

K. Kanazawa, T. Sueshige, H. Takayasu and M. Takayasu, Kinetic theory for finance brownian motion from microscopic dynamics, arXiv: 1802.05993, 2018.Google Scholar

[29]

R. Kohl, The influence of the number of different stocks on the Levy–Levy–Solomon model, International Journal of Modern Physics C, 8 (1997), 1309-1316. doi: 10.1142/S0129183197001168. Google Scholar

[30]

M. LevyH. Levy and S. Solomon, A microscopic model of the stock market: Cycles, booms, and crashes, Economics Letters, 45 (1994), 103-111. Google Scholar

[31]

T. Lux et al., Stochastic Behavioral Asset Pricing Models and the Stylized Facts, Technical report, Economics working paper/Christian-Albrechts-Universität Kiel, Department of Economics, 2008.Google Scholar

[32]

T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500. doi: 10.1038/17290. Google Scholar

[33]

D. Maldarella and L. Pareschi, Kinetic models for socio-economic dynamics of speculative markets, Physica A: Statistical Mechanics and its Applications, 391 (2012), 715-730. doi: 10.1016/j.physa.2011.08.013. Google Scholar

[34]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. Google Scholar

[35]

D. Matthes and G. Toscani, Analysis of a model for wealth redistribution, Kinetic and Related Models, 1 (2008), 1-22. doi: 10.3934/krm.2008.1.1. Google Scholar

[36]

D. Q. Mayne and H. Michalska, Receding horizon control of nonlinear systems, IEEE Trans. Automat. Control, 35 (1990), 814-824. doi: 10.1109/9.57020. Google Scholar

[37]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The review of Economics and Statistics, 51 (1969), 247-257. doi: 10.2307/1926560. Google Scholar

[38]

J. E. Mitchell and S. Braun, Rebalancing an investment portfolio in the presence of convex transaction costs, including market impact costs, Optimization Methods and Software, 28 (2013), 523-542. doi: 10.1080/10556788.2012.717940. Google Scholar

[39]

H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles, Trans. Fluid Dynamics, 18 (1977), 663-678. Google Scholar

[40]

A. Pagan, The econometrics of financial markets, Journal of Empirical Finance, 3 (1996), 15-102. doi: 10.1016/0927-5398(95)00020-8. Google Scholar

[41]

L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models, Journal of statistical physics, 124 (2006), 747-779. doi: 10.1007/s10955-006-9025-y. Google Scholar

[42] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013. Google Scholar
[43]

H. A. Simon, A behavioral model of rational choice, The Quarterly Journal of Economics, 69 (1955), 99-118. doi: 10.2307/1884852. Google Scholar

[44]

D. Sornette, Physics and financial economics (1776–2014): Puzzles, Ising and agent-based models, Reports on Progress in Physics, 77 (2014), 062001, 28pp. doi: 10.1088/0034-4885/77/6/062001. Google Scholar

[45]

A.-S. Sznitman, Topics in propagation of chaos, In Ecole d'Eté de Probabilités de Saint-Flour XIX-1989, 165–251, Lecture Notes in Math., 1464, Springer, Berlin, 1991. doi: 10.1007/BFb0085169. Google Scholar

[46]

T. TrimbornM. Frank and S. Martin, Mean field limit of a behavioral financial market model, Physica A: Statistical Mechanics and its Applications, 505 (2018), 613-631. doi: 10.1016/j.physa.2018.03.079. Google Scholar

[47]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Archive for Rational Mechanics and Analysis, 143 (1998), 273-307. doi: 10.1007/s002050050106. Google Scholar

[48]

L. Walras., Études D'économie Politique Appliquée: (Théorie de la Production de la Richesse Sociale), F. Rouge, 1898.Google Scholar

[49]

E. Zschischang and T. Lux, Some new results on the Levy, Levy and Solomon microscopic stock market model, Physica A: Statistical Mechanics and its Applications, 291 (2001), 563-573. doi: 10.1016/S0378-4371(00)00609-9. Google Scholar

show all references

References:
[1]

G. AlbiM. Herty and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Communications in Mathematical Sciences, 13 (2015), 1407-1429. doi: 10.4310/CMS.2015.v13.n6.a3. Google Scholar

[2]

G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20140138, 18pp. doi: 10.1098/rsta.2014.0138. Google Scholar

[3]

A. Beja and M. B. Goldman, On the dynamic behavior of prices in disequilibrium, The Journal of Finance, 35 (1980), 235-248. Google Scholar

[4]

D. Bertsimas and D. Pachamanova, Robust multiperiod portfolio management in the presence of transaction costs, Computers & Operations Research, 35 (2008), 3-17. doi: 10.1016/j.cor.2006.02.011. Google Scholar

[5]

M. BisiG. Spiga and G. Toscani, Kinetic models of conservative economies with wealth redistribution, Communications in Mathematical Sciences, 7 (2009), 901-916. doi: 10.4310/CMS.2009.v7.n4.a5. Google Scholar

[6]

J.-P. Bouchaud and M. Mézard, Wealth condensation in a simple model of economy, Physica A: Statistical Mechanics and its Applications, 282 (2000), 536-545. doi: 10.1016/S0378-4371(00)00205-3. Google Scholar

[7]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Communications in Mathematical Physics, 56 (1977), 101-113. doi: 10.1007/BF01611497. Google Scholar

[8]

W. A. Brock and C. H. Hommes, Heterogeneous beliefs and routes to chaos in a simple asset pricing model, Journal of Economic Dynamics and Control, 22 (1998), 1235-1274. doi: 10.1016/S0165-1889(98)00011-6. Google Scholar

[9]

M. Burger, L. Caffarelli, P. A. Markowich and M.-T. Wolfram, On a Boltzmann-type price formation model, Proc. R. Soc. A, 469 (2013), 20130126, 20pp. doi: 10.1098/rspa.2013.0126. Google Scholar

[10]

E. Camacho and C. Bordons, Model Predictive Control, Springer, USA, 2004.Google Scholar

[11]

A. Chatterjee and B. K. Chakrabarti, Kinetic exchange models for income and wealth distributions, The European Physical Journal B-Condensed Matter and Complex Systems, 60 (2007), 135-149. doi: 10.1140/epjb/e2007-00343-8. Google Scholar

[12]

L. ChayesM. del Mar GonzálezM. P. Gualdani and I. Kim, Global existence and uniqueness of solutions to a model of price formation, SIAM Journal on Mathematical Analysis, 41 (2009), 2107-2135. doi: 10.1137/090753346. Google Scholar

[13]

J. Che, A kinetic model on portfolio in finance, Communications in Mathematical Sciences, 9 (2011), 1073-1096. doi: 10.4310/CMS.2011.v9.n4.a7. Google Scholar

[14]

C. ChiarellaR. Dieci and X.-Z. He, Heterogeneous expectations and speculative behavior in a dynamic multi-asset framework, Journal of Economic Behavior & Organization, 62 (2007), 408-427. Google Scholar

[15]

D. Colander, H. Föllmer, A. Haas, M. D. Goldberg, K. Juselius, A. Kirman, T. Lux and B. Sloth, The financial crisis and the systemic failure of academic economics, 2009.Google Scholar

[16]

R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, 2001.Google Scholar

[17]

S. CordierL. Pareschi and C. Piatecki, Mesoscopic modelling of financial markets, Journal of Statistical Physics, 134 (2009), 161-184. doi: 10.1007/s10955-008-9667-z. Google Scholar

[18]

S. CordierL. Pareschi and G. Toscani, On a kinetic model for a simple market economy, Journal of Statistical Physics, 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0. Google Scholar

[19]

R. CrossM. GrinfeldH. Lamba and T. Seaman, A threshold model of investor psychology, Physica A: Statistical Mechanics and its Applications, 354 (2005), 463-478. doi: 10.1016/j.physa.2005.02.029. Google Scholar

[20]

M. Delitala and T. Lorenzi, A mathematical model for value estimation with public information and herding, Kinetic & Related Models, 7 (2014), 29-44. doi: 10.3934/krm.2014.7.29. Google Scholar

[21]

R. L. Dobrushin, Vlasov equations, Functional Analysis and Its Applications, 13 (1979), 115-123. Google Scholar

[22]

B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Physical Review E, 78 (2008), 056103, 12pp. doi: 10.1103/PhysRevE.78.056103. Google Scholar

[23]

E. EgenterT. Lux and D. Stauffer, Finite-size effects in Monte Carlo simulations of two stock market models, Physica A: Statistical Mechanics and its Applications, 268 (1999), 250-256. doi: 10.1016/S0378-4371(99)00059-X. Google Scholar

[24]

J. D. Farmer and D. Foley, The economy needs agent-based modelling, Nature, 460 (2009), 685-686. doi: 10.1038/460685a. Google Scholar

[25]

T. Hellthaler, The influence of investor number on a microscopic market model, International Journal of Modern Physics C, 6 (1996), 845-852. doi: 10.1142/S0129183195000691. Google Scholar

[26]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica: Journal of the Econometric Society, 47 (1979), 263-291. doi: 10.2307/1914185. Google Scholar

[27]

K. Kanazawa, T. Sueshige, H. Takayasu and M. Takayasu, Derivation of the boltzmann equation for financial brownian motion: direct observation of the collective motion of high-frequency traders, Physical Review Letters, 120 (2018), 138301. doi: 10.1103/PhysRevLett.120.138301. Google Scholar

[28]

K. Kanazawa, T. Sueshige, H. Takayasu and M. Takayasu, Kinetic theory for finance brownian motion from microscopic dynamics, arXiv: 1802.05993, 2018.Google Scholar

[29]

R. Kohl, The influence of the number of different stocks on the Levy–Levy–Solomon model, International Journal of Modern Physics C, 8 (1997), 1309-1316. doi: 10.1142/S0129183197001168. Google Scholar

[30]

M. LevyH. Levy and S. Solomon, A microscopic model of the stock market: Cycles, booms, and crashes, Economics Letters, 45 (1994), 103-111. Google Scholar

[31]

T. Lux et al., Stochastic Behavioral Asset Pricing Models and the Stylized Facts, Technical report, Economics working paper/Christian-Albrechts-Universität Kiel, Department of Economics, 2008.Google Scholar

[32]

T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500. doi: 10.1038/17290. Google Scholar

[33]

D. Maldarella and L. Pareschi, Kinetic models for socio-economic dynamics of speculative markets, Physica A: Statistical Mechanics and its Applications, 391 (2012), 715-730. doi: 10.1016/j.physa.2011.08.013. Google Scholar

[34]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. Google Scholar

[35]

D. Matthes and G. Toscani, Analysis of a model for wealth redistribution, Kinetic and Related Models, 1 (2008), 1-22. doi: 10.3934/krm.2008.1.1. Google Scholar

[36]

D. Q. Mayne and H. Michalska, Receding horizon control of nonlinear systems, IEEE Trans. Automat. Control, 35 (1990), 814-824. doi: 10.1109/9.57020. Google Scholar

[37]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The review of Economics and Statistics, 51 (1969), 247-257. doi: 10.2307/1926560. Google Scholar

[38]

J. E. Mitchell and S. Braun, Rebalancing an investment portfolio in the presence of convex transaction costs, including market impact costs, Optimization Methods and Software, 28 (2013), 523-542. doi: 10.1080/10556788.2012.717940. Google Scholar

[39]

H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles, Trans. Fluid Dynamics, 18 (1977), 663-678. Google Scholar

[40]

A. Pagan, The econometrics of financial markets, Journal of Empirical Finance, 3 (1996), 15-102. doi: 10.1016/0927-5398(95)00020-8. Google Scholar

[41]

L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models, Journal of statistical physics, 124 (2006), 747-779. doi: 10.1007/s10955-006-9025-y. Google Scholar

[42] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013. Google Scholar
[43]

H. A. Simon, A behavioral model of rational choice, The Quarterly Journal of Economics, 69 (1955), 99-118. doi: 10.2307/1884852. Google Scholar

[44]

D. Sornette, Physics and financial economics (1776–2014): Puzzles, Ising and agent-based models, Reports on Progress in Physics, 77 (2014), 062001, 28pp. doi: 10.1088/0034-4885/77/6/062001. Google Scholar

[45]

A.-S. Sznitman, Topics in propagation of chaos, In Ecole d'Eté de Probabilités de Saint-Flour XIX-1989, 165–251, Lecture Notes in Math., 1464, Springer, Berlin, 1991. doi: 10.1007/BFb0085169. Google Scholar

[46]

T. TrimbornM. Frank and S. Martin, Mean field limit of a behavioral financial market model, Physica A: Statistical Mechanics and its Applications, 505 (2018), 613-631. doi: 10.1016/j.physa.2018.03.079. Google Scholar

[47]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Archive for Rational Mechanics and Analysis, 143 (1998), 273-307. doi: 10.1007/s002050050106. Google Scholar

[48]

L. Walras., Études D'économie Politique Appliquée: (Théorie de la Production de la Richesse Sociale), F. Rouge, 1898.Google Scholar

[49]

E. Zschischang and T. Lux, Some new results on the Levy, Levy and Solomon microscopic stock market model, Physica A: Statistical Mechanics and its Applications, 291 (2001), 563-573. doi: 10.1016/S0378-4371(00)00609-9. Google Scholar

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)
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