Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs.

*(English)*Zbl 1035.91027This paper deals with an infinite horizon investment-consumption model that captures the effects of intertemporal substitution and possible jumps in the multidimensional stock market. Let us denote by \(X_0(t)\) the amount of money having the investor in the bank and \(X_{i}(t)\) the amount of money having the investor in the \(i\)th stock, \(i=1\ldots,n\). The authors assume that the investor holding have the following dynamics
\[
X_0(t)=x_0-C(t)+\int _{0}^{t}rX_0(s)\,ds -\sum_{i=1}^{n}[(1+\lambda_{i})L_{i}(t)- (1-\mu_{i})M_{i}(t)],
\]

\[ X_{i}(t)=x_{i}+ \int _{0}^{t}a_{i}X_{i}(s)\,ds+ \int _{0}^{t}\sigma_{i}X_{i}(s)\,dW(s) +\int _{0}^{t}\int _{R\setminus\{0\}} \eta_{i}(z)X_{i}(s-)\,\widetilde N_{i} (dz,ds)+ \]

\[ +L_{i}(t)-M_{i}(t),\quad i=1,\ldots,n, \] where \(a_{i},\;\sigma_{i}>0\) are constants, \(C(t)\) is the cumulative consumption up to time \(t\), \(L_{i}(t)\) is the cumulative value of the shares bought up to time \(t\) from the \(i\)th stock, \( M_{i}(t)\) is the cumulative value of the shares sold up to time \(t\) from the \(i\)th stock, and \(\mu_{i}\in[0,1]\) and \(\lambda_{i}\geq0\) are the proportional transaction costs of, respectively, selling and buying shares from the \(i\)th stock, \(\lambda_{i}+\mu_{i}>0\) for all \(i\), \(W_{i}(t)\) is the standard Brownian motion and \(N_{i}\) is a Poisson random measure. The investor’s consumption and transactions of wealth between the assets are understood as cumulative processes which may be singular with respect to the Lebesgue measure. The problem of maximizing the investor’s expected utility over these controls is therefore a singular stochastic control problem. The market assumption of no borrowing of money nor short-selling of stocks imposes restrictions on the set of admissible consumption and transaction policies. To investigate the investment-consumption model the authors use Bellman’s dynamic programming method. The main result of this paper is a characterization of the value function as the unique constrained viscosity solution of the dynamic programming equation associated with the considered singular problem.

\[ X_{i}(t)=x_{i}+ \int _{0}^{t}a_{i}X_{i}(s)\,ds+ \int _{0}^{t}\sigma_{i}X_{i}(s)\,dW(s) +\int _{0}^{t}\int _{R\setminus\{0\}} \eta_{i}(z)X_{i}(s-)\,\widetilde N_{i} (dz,ds)+ \]

\[ +L_{i}(t)-M_{i}(t),\quad i=1,\ldots,n, \] where \(a_{i},\;\sigma_{i}>0\) are constants, \(C(t)\) is the cumulative consumption up to time \(t\), \(L_{i}(t)\) is the cumulative value of the shares bought up to time \(t\) from the \(i\)th stock, \( M_{i}(t)\) is the cumulative value of the shares sold up to time \(t\) from the \(i\)th stock, and \(\mu_{i}\in[0,1]\) and \(\lambda_{i}\geq0\) are the proportional transaction costs of, respectively, selling and buying shares from the \(i\)th stock, \(\lambda_{i}+\mu_{i}>0\) for all \(i\), \(W_{i}(t)\) is the standard Brownian motion and \(N_{i}\) is a Poisson random measure. The investor’s consumption and transactions of wealth between the assets are understood as cumulative processes which may be singular with respect to the Lebesgue measure. The problem of maximizing the investor’s expected utility over these controls is therefore a singular stochastic control problem. The market assumption of no borrowing of money nor short-selling of stocks imposes restrictions on the set of admissible consumption and transaction policies. To investigate the investment-consumption model the authors use Bellman’s dynamic programming method. The main result of this paper is a characterization of the value function as the unique constrained viscosity solution of the dynamic programming equation associated with the considered singular problem.

Reviewer: Aleksandr D. Borisenko (Kyïv)

##### MSC:

91G10 | Portfolio theory |

49J55 | Existence of optimal solutions to problems involving randomness |

49L20 | Dynamic programming in optimal control and differential games |

49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |

60G51 | Processes with independent increments; Lévy processes |

93E20 | Optimal stochastic control |