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On Stochastic Control in Finance

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 134))

Abstract

Stochastic control/optimization problems arise in various applications in finance where the control is usually given by an investment strategy. The purpose of this paper is to review some of these applications together with appropriate solution methodologies and also to discuss the latter in comparison with one another.

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References

  1. M. Akian, A. Sulem and M. Taksar, Dynamic optimization of a long term growth rate for a mixed portfolio with transaction costs, Mathematiccd Finance 11 (2001), pp. 153–188.

    Article  MathSciNet  MATH  Google Scholar 

  2. J.P. Ansel and C. Srticker, Couverture des actifs contingents et prix maximum, Annales Institut Henri Poincaré 30 (1994), pp. 303–315.

    MATH  Google Scholar 

  3. S. Asmussen, B. Hojgaard and M. Taksar, Optimal risk and dividend distribution policies. Example of excess-of-loss reinsurance for an insurance corporation, Finance and Stochastics 4 (2000), pp. 299–342.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Avellaneda, A. Levy and A. Paras, Pricing and Hedging Derivative Securities in Markets With Uncertain Volatilities, Applied Mathematical Finance 1 (1995), pp. 73–88.

    Article  Google Scholar 

  5. O.E. Barndorff-Nielsen, T. Mikosch and S. Resnick (eds). Levy Processes: Theory and Applications, Birkhäuser Verlag 2001.

    MATH  Google Scholar 

  6. T.R. Bielecki and S.K. Pliska, Risk sensitive dynamic asset management. Applied Mathematics and Optimization 39 (1999), pp. 337–360.

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Black and M. Scholes, The pricing of options and corporate liabilities. Journal of Political Economics 81 (1973), pp. 637–659.

    Article  Google Scholar 

  8. B. Bouchard, Utility maximization on the real line under proportional transaction costs. Finance and Stochastics 6 (2002), to appear.

    Google Scholar 

  9. S. Browne, Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastics 3 (1999), pp. 275–294.

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Browne and W. Whitt, Portfolio choice and the Bayesian Kelly criterion, Advances in Applied Probability 28 (1996), pp. 1145–1176.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Cadenillas and F. Zapatero, Classical and impulse stochastic control of the exchange rate using interest rates and reserves. Mathematical Finance 10 (2000), pp. 141–156.

    Article  MathSciNet  MATH  Google Scholar 

  12. T. Choulli, M. Taksar and X.Y. Zhou, Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quantitative Finance 1 (2001), pp. 573–596.

    Article  MathSciNet  Google Scholar 

  13. J.C. Cox, S.A. Ross and M. Rubinstein, Option pricing: a simplified approach, Journal of Financial Economics 7 (1979), pp. 229–263.

    Article  MATH  Google Scholar 

  14. J. Cvitanic, Optimal trading under constraints, In: Financial Mathematics (W.J. Runggaldier, ed.). Lecture Notes in Mathematics Vol. 1656, Springer Verlag, Berlin-Heidelberg 1997, pp. 123–190.

    Chapter  Google Scholar 

  15. J. Cvitanic, Minimizing expected loss of hedging in incomplete and constrained markets, SIAM Journal on Control and Optimization 38 (2000), pp. 1050–1066.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Cvitanic and I. Karatzas, Convex duality in constrained portfolio optimization, Annals of Applied Probability 2 (1992), pp. 767–818.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Cvitanic and I. Karatzas, Hedging and portfolio optimization under transaction costs: a martingale approach, Mathematical Finance 6 (1996), pp. 133–165.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Cvitanic and I. Karatzas, On dynamic measures of risk. Finance and Stochastics 4 (1999), pp. 451–482.

    MathSciNet  Google Scholar 

  19. J. Cvitanic, W. Schachermayer and H. Wang, Utility maximization in incomplete markets with random endowment Preprint, Institute of Financial and Actuarial Mathematics, Vienna Univ. of Technology, 1999.

    Google Scholar 

  20. P. Dai Pra, G.B. Di Masi and B. Trivellato, Pathwise optimality in stochastic control, SIAM Journal on Control & Optimization 39 (2000), pp. 1540–1557.

    Article  Google Scholar 

  21. M.H.A. Davis and A.R. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research 16 (1990), pp. 676–713.

    Article  MathSciNet  Google Scholar 

  22. G. Deelstra, H. Pham and N. Touzi, Dual formulation of the utility maximization problem under transaction costs. Annals of Applied Probability, 11 (2001), pp. 1353–1383.

    Article  MathSciNet  MATH  Google Scholar 

  23. F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300 (1994), pp. 463–520.

    Article  MathSciNet  MATH  Google Scholar 

  24. G.B. Di Masi, E. Platen and W.J. Runggaldier, Hedging of options under discrete observation on assets with stochastic volatility. In: Seminar on Stochastic Analysis, Random Fields and Applications (E. Holthausen, M. Dozzi, and F. Russo, eds.). Progress in Probabihty, Vol. 36, Birkhäuser Verlag, 1995, pp. 359–364.

    Google Scholar 

  25. D. Duffie and H.R. Richardson, Mean variance hedging in continuous time. Annals of Applied Probability 1 (1991), pp. 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  26. T.E. Duncan, Y.Z. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion, I. Theory, SIAM Journal on Control and Optimization 38 (2000), 582–612.

    Article  MathSciNet  MATH  Google Scholar 

  27. N. El Karoui and M.C. Quenez, Dynamic programming and pricing of contingent claims in an incomplete market, SIAM Journal on Control and Optimization 33 (1995), pp. 29–66.

    Article  MathSciNet  MATH  Google Scholar 

  28. N. El Karoui and M.C. Quenez, Non-linear pricing theory and backward stochastic differential equations. In: Financial Mathematics (W.J. Runggaldier, ed.). Lecture Notes in Mathematics Vol. 1656, Springer Verlag, Berlin-Heidelberg 1997, pp. 191–246.

    Chapter  Google Scholar 

  29. G. Favero, Shortfall risk minimization under model uncertainty in the binomial case: adaptive and robust approaches. Mathematical Methods of Operations Research 53 (2001), pp. 493–503.

    Article  MathSciNet  MATH  Google Scholar 

  30. G. Favero and W.J. Runggaldier, A robustness result for stochastic control, Systems and Control Letters. To appear.

    Google Scholar 

  31. W.H. Fleming and S.J. Sheu, Risk sensitive control and an optimal investment model, Mathematical Finance 10 (2000), pp. 197–213.

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Föllmer and P. Leukert, Efficient hedging: cost versus shortfall risk. Finance and Stochastics 4 (2000), pp. 117–146.

    Article  MATH  Google Scholar 

  33. H. Föllmer and D. Sondermann, Hedging of non-redundant contingemt claims. In: Contributions to Mathematical Economics in Honor of Gérard Debreu (W. Hildenbrand and A. Mas-Colell, eds.), North-Holland, Amsterdam (1986), pp. 205–223.

    Google Scholar 

  34. H. Föllmer and M. Schweizer, Heding of contingent claims under incomplete information. In: Applied Stochastic Analysis (M.H.A. Davis and R.J. Elliott, eds.), Gordon and Breach, London, New York (1991), pp. 389–414.

    Google Scholar 

  35. R. Frey and W.J. Runggaldier, A nonlinear filtering approach to volatility estimation with a view towards high frequency data, International Journal of Theoretical and Applied Finance 4 (2001), pp. 199–210.

    Article  MathSciNet  MATH  Google Scholar 

  36. R. Frey and W.J. Runggaldier, Risk-minimizing hedging strategies under restricted information: the case of stochastic volatility models observed only at discrete random times, Mathematical Methods of Operations Research 50 (1999), pp. 339–350.

    Article  MathSciNet  MATH  Google Scholar 

  37. J. Gaier, P. Grandits and W. Schachermayer, Asymptotic ruin probabilities and optimal investment. Preprint, Institute of Financial and Actuarial Mathematics, Vienna Univ. of Technology, 2002.

    Google Scholar 

  38. P. Guasoni, Risk minimization under transaction costs, Finance and Stochastics 6 (2002), pp. 91–113.

    Article  MathSciNet  MATH  Google Scholar 

  39. C. Hipp and M. Plum, Optimal investment for insurers. Insurance: Mathematics and Economics 27 (2000), pp. 215–228.

    Article  MathSciNet  MATH  Google Scholar 

  40. Y.Z. Hu, B. Oksendal and A. Sulem, Optimal Portfolios in a Black and Scholes Market, Preprint Univ. of Oslo 1999.

    Google Scholar 

  41. M. Jeanblanc-Picque, Impulse control method and exchange rate. Mathematical Finance 2 (1993), pp. 161–177.

    Article  Google Scholar 

  42. Yu. Kabanov and G. Last, Hedging under transaction costs in currency markets: a continuous-time model. Mathematical Finance 12 (2002), pp. 63–70.

    Article  MathSciNet  MATH  Google Scholar 

  43. Yu. Kabanov and C. Stricker, Hedging of contingent claims under transaction costs. Preprint 2002.

    Google Scholar 

  44. J. Kallsen, Optimal portfolios for exponential Levy processes, Mathematical Methods of Operations Research 51 (2000), 357–374.

    Article  MathSciNet  Google Scholar 

  45. I. Karatzas and S.E. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998.

    MATH  Google Scholar 

  46. I. Karatzas and X. Zhao, Bayesian adaptive portfolio optimization, In: Hand book of Mathematical Finance, Cambridge Univ Press, pp. 632–670.

    Google Scholar 

  47. I. Karatzas and G. Zitkovic, Optimal consumption from investment and random endowment in incomplete semimartingale markets. Preprint, Columbia University 2001.

    Google Scholar 

  48. M. Kirch, Efficient hedging in incomplete markets under model uncertainty. PhD thesis, Humboldt University, Berlin, 2002.

    MATH  Google Scholar 

  49. R. Korn, Optimal Portfolios. Stochastic models for optimal investment and risk management in continuous time. World Scientific Publishing Co. Pte. Ltd. 1997.

    Book  MATH  Google Scholar 

  50. R. Korn, Portfolio optimization with strictly positive transaction costs and impulse control, Finance and Stochastics 2 (1998), pp. 85–114.

    Article  MathSciNet  MATH  Google Scholar 

  51. R. Korn, Some applications of impulse control in mathematical finance. Mathematical Methods of Operations Research 50 (1999), pp. 493–518.

    Article  MathSciNet  MATH  Google Scholar 

  52. D. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Theory of Probability and Related Fields 105 (1996), pp. 459–479.

    Article  MathSciNet  MATH  Google Scholar 

  53. D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Annals of Applied Probability 9 (1999), pp. 904–950.

    Article  MathSciNet  MATH  Google Scholar 

  54. P. Lakner, Utility maximization with partial information, Stochastic Processes and Their Applications 56 (1995), pp. 247–273.

    Article  MathSciNet  MATH  Google Scholar 

  55. T. Lyons, Uncertain volatility and the riskfree synthesis of derivatives, Applied Mathematical Finance 2 (1995), pp. 117–133.

    Article  Google Scholar 

  56. M.J.P. Magill and G.M. Constantinides, Portfolio selection with transaction costs, Journal of Economic Theory 13 (1976), pp. 245–263.

    Article  MathSciNet  MATH  Google Scholar 

  57. B.B. Mandelbrot, Fractals and Scaling in Finance, Springer Verlag, 1997.

    MATH  Google Scholar 

  58. A. Martin-Löf, Lectures on the use of control theory in insurance, Scandinavian Actuarial Journal 77 (1994), pp. 1–25.

    Article  Google Scholar 

  59. R. Merton, Optimum consumption and portfolio rules in a continuous time model, Journal of Economic Theory 3 (1971), pp. 373–413.

    Article  MathSciNet  Google Scholar 

  60. R. Merton, Option pricing when the underlying stock returns are discontinuous. Journal of Financial Economics 5 (1976), pp. 125–144.

    Article  Google Scholar 

  61. H. Nagai and S. Peng, Risk-sensitive dynamic portfolio optimization with partial information on an infinite time horizon. Annals of Applied Probability, to appear.

    Google Scholar 

  62. J. Paulsen and H. Gjessing, Optimal choice of dividend barriers for a risk process with stochastic return of investment. Insurance: Mathematics and Economics 20 (1997), pp. 215–223.

    Article  MathSciNet  MATH  Google Scholar 

  63. H. Pham, Mean-variance hedging for partially observed drift processes, International Journal of Theoretical and Applied Finance 4 (2001), pp. 263–284.

    Article  MathSciNet  MATH  Google Scholar 

  64. H. Pham and M.C. Quenez, Optimal portfolio in a partially observed stochastic volatility model. Annals of Applied Probability, to appear.

    Google Scholar 

  65. E. Platen and W.J. Runggaldier, A benchmark approach to filtering in finance. Preprint 2002.

    Google Scholar 

  66. W.J. Runggaldier, The choice of the loss function for a decision problem under uncertainty as a compromise between representativity and tractability. ES-ReDA International Seminar on Decision Analysis, Rome, June 18–19, 2001, pp. 141–148.

    Google Scholar 

  67. W.J. Runggaldier, Jump-diffusion Models, In: Handbook of Heavy Tailed Distributions in Finance, North Holland Handbooks of Finance (Series Editor W.T. Ziemba), to appear.

    Google Scholar 

  68. W.J. Runggaldier, B. Trivellato and T. Vargiolu, A Bayesian adaptive control approach to risk management in a binomial model In: Seminar on Stochastic Analysis, Random Fields and Applications (R.C. Dalang, M. Dozzi, and F. Russo, eds.). Progress in Probability, Vol. 52, Birkhäuser Verlag, 2002, pp. 243–258.

    Chapter  Google Scholar 

  69. W.J. Runggaldier and A. Zaccaria, A stochastic control approach to risk management under restricted information. Mathematical Finance 10 (2000), pp. 277–288.

    Article  MathSciNet  MATH  Google Scholar 

  70. W. Schachermayer, The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Preprint 2001.

    Google Scholar 

  71. W. Schachermayer, Optimal investment in incomplete markets when wealth may become negative. Annals of Applied Probability 11 (2001), pp. 694–734.

    Article  MathSciNet  MATH  Google Scholar 

  72. W. Schachermayer, Optimal investment in incomplete financial markets. In: Mathematical Finance — Bachelier Congress 2000 (H. Geman, D. Madan, S. Pliska, and T. Vorst, eds.), Springer Verlag 2002, pp. 427–462.

    Google Scholar 

  73. M. Schael, On quadratic cost criteria for option hedging, Mathematics of Operations Research 19 (1994), pp. 121–131.

    Article  MathSciNet  MATH  Google Scholar 

  74. H.P. Schmidli, On minimising the ruin probability by investment and reinsurance. Annals of Applied Probability, to appear.

    Google Scholar 

  75. M. Schweizer, Mean variance hedging for general claims. Annals of Applied Probability 2 (1992), pp. 171–179.

    Article  MathSciNet  MATH  Google Scholar 

  76. M. Schweizer, Risk minimizing hedging strategies under restricted information, Mathematical Finance 4 (1994), pp. 327–342.

    Article  MathSciNet  MATH  Google Scholar 

  77. J. Serine, Power-utility maximization for linear-Gaussian factor models under partial information. Preprint, Osaka University, 2001.

    Google Scholar 

  78. M.H. Soner and S. Shreve, Optimal investment and consumption with transaction costs, Annals of Applied Probability 4 (1994), pp. 609–692.

    Article  MathSciNet  MATH  Google Scholar 

  79. H.M. Soner and N. Touzi, Superreplication under Gamma constraints, SIAM Journal on Control and Optimization 39 (2000), pp. 73–96.

    Article  MathSciNet  MATH  Google Scholar 

  80. D. Talay and Zheng, Worst case model risk management. Finance and Stochastics, to appear.

    Google Scholar 

  81. M. Tolotti, Strategie di potafoglio ottimali per il “tracking” di un indice, Laurea thesis. University of Padova 2002.

    Google Scholar 

  82. T. Zariphopoulou, Investment-consumption models with transaction fees and Markov-chain parameters, SIAM Journal on Control and Optimization 30(1992), pp. 613–633.

    Article  MathSciNet  MATH  Google Scholar 

  83. G. Zohar, A generalized Cameron-Martin formula with applications to partially observed dynamic portfolio optimization, Mathematical Finance 11 (2001), pp. 475–494.

    Article  MathSciNet  MATH  Google Scholar 

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Runggaldier, W.J. (2003). On Stochastic Control in Finance. In: Rosenthal, J., Gilliam, D.S. (eds) Mathematical Systems Theory in Biology, Communications, Computation, and Finance. The IMA Volumes in Mathematics and its Applications, vol 134. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21696-6_12

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  • DOI: https://doi.org/10.1007/978-0-387-21696-6_12

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