# American Institute of Mathematical Sciences

January  2016, 3(1): 1-24. doi: 10.3934/jdg.2016001

## Uncertainty and inside information

 1 Department of Statistics, Athens University of Economics and Business, GR10434 Athens-Greece

Received  March 2015 Revised  June 2015 Published  March 2016

In this paper, we study a robust-entropic optimal control problem in the presence of inside information. To be more precise, we consider an economic agent who is allowed to invest her wealth in a classical Black-Scholes type financial market. From the beginning of the trading interval, the agent exclusively possesses some inside information concerning the future realization of the stock price process. However, we assume that she is uncertain as to the validity of this information, thus introducing in this way robust aspects to our model. The aim of the economic agent is to solve an expected utility maximization problem under the worst-case scenario, taking into account her enlarged information set. By formulating this problem as a two-player, zero sum stochastic differential game, we are able to provide closed form solutions for the optimal robust strategies and the robust value function, in the case of the exponential and the power utility functions.
Citation: Ioannis D. Baltas, Athanasios N. Yannacopoulos. Uncertainty and inside information. Journal of Dynamics and Games, 2016, 3 (1) : 1-24. doi: 10.3934/jdg.2016001
##### References:
 [1] E. Anderson, L. Hansen and T. Sargent, A quartet of semigroups for model specification, robustness, prices of risk, and model detection, Journal of the European Economic Association, 1 (2003), 68-123. doi: 10.1162/154247603322256774. [2] E. Anderson, E. Ghysels and J. Juergens, The impact of risk and uncertainty on expected returns, Journal of Financial Economics, 94 (2009), 233-263. doi: 10.1016/j.jfineco.2008.11.001. [3] I. D. Baltas, N. E. Frangos and A. N. Yannacopoulos, Optimal investment and reinsurance policies in insurance markets under the effect of inside information, Applied Stochastic Models in Business and Industry, 28 (2012), 506-528. doi: 10.1002/asmb.925. [4] N. Branger, L. Larsen and C. Munk, Robust portfolio choice with ambiguity and learning about return predictability, Journal of Banking and Finance, 37 (2013), 1397-1411. doi: 10.1016/j.jbankfin.2012.05.009. [5] W. Brock, A. Xepapadeas and A. N. Yannacopoulos, Robust control of a spatially distributed commercial fishery, in Dynamic Optimization in Environmental Economics, (eds. E. Moser, W. Semmler, G. Tragler, V. Veliov) Springer-Verlag, Heidelberg, 15 (2014), 215-241. doi: 10.1007/978-3-642-54086-8_10. [6] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958. doi: 10.1287/moor.20.4.937. [7] A. Cairns, A discussion of parameter and model uncertainty in insurance, Insurance: Mathematics and Economics, 27 (2000), 313-330. doi: 10.1016/S0167-6687(00)00055-X. [8] R. Cont, Model uncertainty and its impact on the pricing of derivative instruments, Mathematical Finance, 16 (2006), 519-547. doi: 10.1111/j.1467-9965.2006.00281.x. [9] D. David and Y. Okur, Optimal consumption and portfolio for an insider in a market with jumps, Communications on Stochastic Analysis, 3 (2009), 101-117. [10] W. Fleming and P. Souganidis, On the existence of value functions of two player zero sum stochastic differential games, Indiana University Mathematics Journal, 38 (1989), 293-314. doi: 10.1512/iumj.1989.38.38015. [11] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edition, Springer-New York, 2006. doi: 10.1007/0-387-31071-1. [12] C. Flor and L. Larsen, Robust portfolio choice with stochastic interest rates, Annals of Finance, 10 (2014), 243-265. doi: 10.1007/s10436-013-0234-5. [13] L. Hansen and T. Sargent, Robust control and model uncertainty, American Economic Review, 91 (2001), 60-66. [14] P. Imkeller, M. Pontier and F. Weisz, Free lunch and arbitrage possibilities in a financial market with an insider, Stochastic Processes and their Applications, 92 (2001), 103-130. doi: 10.1016/S0304-4149(00)00071-5. [15] K. Itô, Extension of stochastic integrals, Proceedings of international symposium in stochastic differential equations, Kyoto, 1976. [16] J. Jacod, Grossissement initial, hypothese (H'), et theoreme de Girsanov, in Grossissement de Filtrations: Exemples et Applications, (eds. Th. Jeulin, M. Yor) Springer-Verlag, 1118 (1985), 15-35. doi: 10.1007/BFb0075768. [17] M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods in Financial Markets, Springer-Verlag, 2009. doi: 10.1007/978-1-84628-737-4. [18] T. Jeulin, Semi-martingales et Grossissement D'unde Filtration, Lecture Notes in Mathematics, 833, Springer-Berlin, 1980. doi: 10.1007/BFb0093539. [19] R. Korn, Worst case scenario investment for insurers, Insurance: Mathematics and economics, 36 (2005), 1-11. doi: 10.1016/j.insmatheco.2004.10.004. [20] A. Lioui and P. Pocet, On model ambiguity and money neutrality, Journal of Macroeconomics, 34 (2012), 1020-1033. doi: 10.1016/j.jmacro.2012.08.003. [21] R. Lipster and A. Shiryaev, Statistics of Random Processes: I. General Theory, 2nd edition, Springer-Verlag, 2001. doi: 10.1007/978-3-662-13043-8. [22] H. Liu, Robust consumption and portfolio choice for time varying investment, Annals of Finance, 6 (2010), 435-454. doi: 10.1007/s10436-010-0164-4. [23] S. Mataramvura and B. Øksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics: An International Journal of Probability and Stochastic Processes, 80 (2008), 317-337. doi: 10.1080/17442500701655408. [24] P. Maenhout, Robust portfolio rules and asset pricing, The Review of Financial Studies, 17 (2004), 951-983. doi: 10.1093/rfs/hhh003. [25] I. Pikovsky and I. Karatzas, Anticipative portfolio optimization, Advances of Applied Probability, 28 (1996), 1095-1122. doi: 10.2307/1428166. [26] U. Rieder and C. Wopperer, Robust consumption-investment problems with random market coefficients, Math Finan Econ, 6 (2012), 295-311. doi: 10.1007/s11579-012-0073-6. [27] C. Skiadas, Robust control and recursive utility, Finance and Stochastics, 7 (2003), 475-489. doi: 10.1007/s007800300100.

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##### References:
 [1] E. Anderson, L. Hansen and T. Sargent, A quartet of semigroups for model specification, robustness, prices of risk, and model detection, Journal of the European Economic Association, 1 (2003), 68-123. doi: 10.1162/154247603322256774. [2] E. Anderson, E. Ghysels and J. Juergens, The impact of risk and uncertainty on expected returns, Journal of Financial Economics, 94 (2009), 233-263. doi: 10.1016/j.jfineco.2008.11.001. [3] I. D. Baltas, N. E. Frangos and A. N. Yannacopoulos, Optimal investment and reinsurance policies in insurance markets under the effect of inside information, Applied Stochastic Models in Business and Industry, 28 (2012), 506-528. doi: 10.1002/asmb.925. [4] N. Branger, L. Larsen and C. Munk, Robust portfolio choice with ambiguity and learning about return predictability, Journal of Banking and Finance, 37 (2013), 1397-1411. doi: 10.1016/j.jbankfin.2012.05.009. [5] W. Brock, A. Xepapadeas and A. N. Yannacopoulos, Robust control of a spatially distributed commercial fishery, in Dynamic Optimization in Environmental Economics, (eds. E. Moser, W. Semmler, G. Tragler, V. Veliov) Springer-Verlag, Heidelberg, 15 (2014), 215-241. doi: 10.1007/978-3-642-54086-8_10. [6] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958. doi: 10.1287/moor.20.4.937. [7] A. Cairns, A discussion of parameter and model uncertainty in insurance, Insurance: Mathematics and Economics, 27 (2000), 313-330. doi: 10.1016/S0167-6687(00)00055-X. [8] R. Cont, Model uncertainty and its impact on the pricing of derivative instruments, Mathematical Finance, 16 (2006), 519-547. doi: 10.1111/j.1467-9965.2006.00281.x. [9] D. David and Y. Okur, Optimal consumption and portfolio for an insider in a market with jumps, Communications on Stochastic Analysis, 3 (2009), 101-117. [10] W. Fleming and P. Souganidis, On the existence of value functions of two player zero sum stochastic differential games, Indiana University Mathematics Journal, 38 (1989), 293-314. doi: 10.1512/iumj.1989.38.38015. [11] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edition, Springer-New York, 2006. doi: 10.1007/0-387-31071-1. [12] C. Flor and L. Larsen, Robust portfolio choice with stochastic interest rates, Annals of Finance, 10 (2014), 243-265. doi: 10.1007/s10436-013-0234-5. [13] L. Hansen and T. Sargent, Robust control and model uncertainty, American Economic Review, 91 (2001), 60-66. [14] P. Imkeller, M. Pontier and F. Weisz, Free lunch and arbitrage possibilities in a financial market with an insider, Stochastic Processes and their Applications, 92 (2001), 103-130. doi: 10.1016/S0304-4149(00)00071-5. [15] K. Itô, Extension of stochastic integrals, Proceedings of international symposium in stochastic differential equations, Kyoto, 1976. [16] J. Jacod, Grossissement initial, hypothese (H'), et theoreme de Girsanov, in Grossissement de Filtrations: Exemples et Applications, (eds. Th. Jeulin, M. Yor) Springer-Verlag, 1118 (1985), 15-35. doi: 10.1007/BFb0075768. [17] M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods in Financial Markets, Springer-Verlag, 2009. doi: 10.1007/978-1-84628-737-4. [18] T. Jeulin, Semi-martingales et Grossissement D'unde Filtration, Lecture Notes in Mathematics, 833, Springer-Berlin, 1980. doi: 10.1007/BFb0093539. [19] R. Korn, Worst case scenario investment for insurers, Insurance: Mathematics and economics, 36 (2005), 1-11. doi: 10.1016/j.insmatheco.2004.10.004. [20] A. Lioui and P. Pocet, On model ambiguity and money neutrality, Journal of Macroeconomics, 34 (2012), 1020-1033. doi: 10.1016/j.jmacro.2012.08.003. [21] R. Lipster and A. Shiryaev, Statistics of Random Processes: I. General Theory, 2nd edition, Springer-Verlag, 2001. doi: 10.1007/978-3-662-13043-8. [22] H. Liu, Robust consumption and portfolio choice for time varying investment, Annals of Finance, 6 (2010), 435-454. doi: 10.1007/s10436-010-0164-4. [23] S. Mataramvura and B. Øksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics: An International Journal of Probability and Stochastic Processes, 80 (2008), 317-337. doi: 10.1080/17442500701655408. [24] P. Maenhout, Robust portfolio rules and asset pricing, The Review of Financial Studies, 17 (2004), 951-983. doi: 10.1093/rfs/hhh003. [25] I. Pikovsky and I. Karatzas, Anticipative portfolio optimization, Advances of Applied Probability, 28 (1996), 1095-1122. doi: 10.2307/1428166. [26] U. Rieder and C. Wopperer, Robust consumption-investment problems with random market coefficients, Math Finan Econ, 6 (2012), 295-311. doi: 10.1007/s11579-012-0073-6. [27] C. Skiadas, Robust control and recursive utility, Finance and Stochastics, 7 (2003), 475-489. doi: 10.1007/s007800300100.
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