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October  2016, 12(4): 1323-1331. doi: 10.3934/jimo.2016.12.1323

## Merton problem in an infinite horizon and a discrete time with frictions

 1 Paris School of Economics, University of Paris 1, Panthéon Sorbonne, France 2 Paris School of Economics, University of Paris 1, Panthéon Sorbonne, CNRS, CES. M.S.E. 106 Boulevard de l'Hôpital, 75647 Paris cedex 13, France 3 King Saud University, College of Science, Department of Mathematics, Box 2455, Riyadh 11451, Saudi Arabia 4 Department of Mathematics, Swiss Federal Institute of Technology (ETH) Zurich and Swiss Finance Institute, Switzerland

Received  February 2015 Revised  October 2015 Published  January 2016

We investigate the problem of optimal investment and consumption of Merton in the case of discrete markets in an infinite horizon. We suppose that there is frictions in the markets due to loss in trading. These frictions are modeled through nonlinear penalty functions and the classical transaction cost and liquidity models are included in this formulation. In this context, the solvency region is defined taking into account this penalty function and every investigator have to maximize his utility, that is derived from consumption, in this region. We give the dynamic programming of the model and we prove the existence and uniqueness of the value function.
Citation: Senda Ounaies, Jean-Marc Bonnisseau, Souhail Chebbi, Halil Mete Soner. Merton problem in an infinite horizon and a discrete time with frictions. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1323-1331. doi: 10.3934/jimo.2016.12.1323
##### References:
 [1] U. Çetin, R. Jarrow and P. Protter, Liquidity risk and arbitrage pricing theory, Finance and Stochastics 8 (2004), 311-341. doi: 10.1007/s00780-004-0123-x. [2] U. Çetin and L. C. G. Rogers, Modeling liquidity effects in discrete time, Mathematical Finance 17 (2007), 15-29. doi: 10.1111/j.1467-9965.2007.00292.x. [3] U. Çetin, H. M. Soner and N. Touzi, Option hedging for small investors under liquidity costs, Finance and Stochastics, 14 (2010), 317-341. doi: 10.1007/s00780-009-0116-x. [4] S. Chebbi and H. M. Soner, Merton problem in a discrete market with frictions, Nonlinear Analysis: Real World Applications, 14 (2013), 179-187. doi: 10.1016/j.nonrwa.2012.05.011. [5] G. M. Constantinides, Capital market equilibrium with transaction costs, Journal of Political Economy, 94 (1986), 842-862. [6] M. H. A. Davis and A. R. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1990), 676-713. doi: 10.1287/moor.15.4.676. [7] Y. Dolinsky and H. M. Soner, Duality and convergence for binomial markets with friction, Finance and Stochastics, 17 (2013), 447-475. doi: 10.1007/s00780-012-0192-1. [8] B. Dumas and E. Luciano, An exact solution to a dynamic portfolio choice problem under transaction costs, Journal of Finance, 46 (1991), 577-595. doi: 10.1111/j.1540-6261.1991.tb02675.x. [9] S. Goekey and H. M. Soner, Liquidity in a binomial market, Mathematical Finance,22 (2012), 250-276. doi: 10.1111/j.1467-9965.2010.00462.x. [10] E. Jouini and E. Kallal, Martingales and arbitrage in securities markets with transaction costs, Journal of Economic Theory, 66 (1995), 178-197. doi: 10.1006/jeth.1995.1037. [11] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, 1998. doi: 10.1007/b98840. [12] C. Le Van and R.-A. Dana, Dynamic Programming in Economics, Kluer Academic Publishers, 2003. [13] M. J. P. Magill and G. M. Constantinides, Portfolio selection with transaction costs, Journal of Economic Theory, 13 (1976), 254-263. doi: 10.1016/0022-0531(76)90018-1. [14] R. C. Merton, Optimum consumption and portfolio rules in a continuous time case, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X. [15] S. E. Shreve and H. M. Soner, Optimal investment and consumption with transaction costs, The Annals of Applied Probability, 4 (1994), 609-692. doi: 10.1214/aoap/1177004966.

show all references

##### References:
 [1] U. Çetin, R. Jarrow and P. Protter, Liquidity risk and arbitrage pricing theory, Finance and Stochastics 8 (2004), 311-341. doi: 10.1007/s00780-004-0123-x. [2] U. Çetin and L. C. G. Rogers, Modeling liquidity effects in discrete time, Mathematical Finance 17 (2007), 15-29. doi: 10.1111/j.1467-9965.2007.00292.x. [3] U. Çetin, H. M. Soner and N. Touzi, Option hedging for small investors under liquidity costs, Finance and Stochastics, 14 (2010), 317-341. doi: 10.1007/s00780-009-0116-x. [4] S. Chebbi and H. M. Soner, Merton problem in a discrete market with frictions, Nonlinear Analysis: Real World Applications, 14 (2013), 179-187. doi: 10.1016/j.nonrwa.2012.05.011. [5] G. M. Constantinides, Capital market equilibrium with transaction costs, Journal of Political Economy, 94 (1986), 842-862. [6] M. H. A. Davis and A. R. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1990), 676-713. doi: 10.1287/moor.15.4.676. [7] Y. Dolinsky and H. M. Soner, Duality and convergence for binomial markets with friction, Finance and Stochastics, 17 (2013), 447-475. doi: 10.1007/s00780-012-0192-1. [8] B. Dumas and E. Luciano, An exact solution to a dynamic portfolio choice problem under transaction costs, Journal of Finance, 46 (1991), 577-595. doi: 10.1111/j.1540-6261.1991.tb02675.x. [9] S. Goekey and H. M. Soner, Liquidity in a binomial market, Mathematical Finance,22 (2012), 250-276. doi: 10.1111/j.1467-9965.2010.00462.x. [10] E. Jouini and E. Kallal, Martingales and arbitrage in securities markets with transaction costs, Journal of Economic Theory, 66 (1995), 178-197. doi: 10.1006/jeth.1995.1037. [11] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, 1998. doi: 10.1007/b98840. [12] C. Le Van and R.-A. Dana, Dynamic Programming in Economics, Kluer Academic Publishers, 2003. [13] M. J. P. Magill and G. M. Constantinides, Portfolio selection with transaction costs, Journal of Economic Theory, 13 (1976), 254-263. doi: 10.1016/0022-0531(76)90018-1. [14] R. C. Merton, Optimum consumption and portfolio rules in a continuous time case, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X. [15] S. E. Shreve and H. M. Soner, Optimal investment and consumption with transaction costs, The Annals of Applied Probability, 4 (1994), 609-692. doi: 10.1214/aoap/1177004966.
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