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# Discrete Stochastic Calculus

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## Abstract

The theory of stochastic processes deals with random functions of time such as asset prices, interest rates, and trading strategies. As is also the case for Mathematical Finance, it can be developed in both discrete and continuous time.

## References

- 18.N. Bäuerle, U. Rieder,
*Markov Decision Processes with Applications to Finance*(Springer, Heidelberg, 2011)CrossRefGoogle Scholar - 27.J.-M. Bismut, Growth and optimal intertemporal allocation of risks. J. Econom. Theory
**10**(2), 239–257 (1975)MathSciNetCrossRefGoogle Scholar - 28.J.-M. Bismut, An introductory approach to duality in optimal stochastic control. SIAM Rev.
**20**(1), 62–78 (1978)MathSciNetCrossRefGoogle Scholar - 65.J. Cvitanić, I. Karatzas, Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance
**6**(2), 133–165 (1996)MathSciNetCrossRefGoogle Scholar - 96.N. El Karoui, Les aspects probabilistes du contrôle stochastique, in
*Ninth Saint Flour Probability Summer School—1979 (Saint Flour, 1979)*, volume 876 of*Lecture Notes in Math.*(Springer, Berlin, 1981), pp. 73–238Google Scholar - 102.T. Ferguson, Who solved the secretary problem? Stat. Sci.
**4**(3), 282–296 (1989)MathSciNetCrossRefGoogle Scholar - 114.H. Föllmer, Yu. Kabanov, Optional decomposition and Lagrange multipliers. Finance Stochast.
**2**(1), 69–81 (1998)MathSciNetzbMATHGoogle Scholar - 117.H. Föllmer, A. Schied,
*Stochastic Finance: An Introduction in Discrete Time*, 3rd edn. (De Gruyter, Berlin, 2011)CrossRefGoogle Scholar - 125.T. Goll, J. Kallsen, Optimal portfolios for logarithmic utility. Stoch. Process. Appl.
**89**(1), 31–48 (2000)MathSciNetCrossRefGoogle Scholar - 126.T. Goll, J. Kallsen, A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab.
**13**(2), 774–799 (2003)MathSciNetCrossRefGoogle Scholar - 135.M. Haugh, L. Kogan, Pricing American options: a duality approach. Oper. Res.
**52**(2), 258–270 (2004)MathSciNetCrossRefGoogle Scholar - 144.J.-B. Hiriart-Urruty, C. Lemaréchal,
*Convex Analysis and Minimization Algorithms I*(Springer, Berlin, 1993)CrossRefGoogle Scholar - 152.J. Jacod,
*Calcul Stochastique et Problèmes de Martingales*, volume 714 of*Lecture Notes in Math*(Springer, Berlin, 1979)CrossRefGoogle Scholar - 153.J. Jacod, P. Protter,
*Probability Essentials*, 2nd edn. (Springer, Berlin, 2004)CrossRefGoogle Scholar - 154.J. Jacod, A. Shiryaev,
*Limit Theorems for Stochastic Processes*, 2nd edn. (Springer, Berlin, 2003)CrossRefGoogle Scholar - 161.E. Jouini, H. Kallal, Martingales and arbitrage in securities markets with transaction costs. J. Econom. Theory
**66**(1), 178–197 (1995)MathSciNetCrossRefGoogle Scholar - 188.I. Karatzas, G. Žitković, Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab.
**31**(4), 1821–1858 (2003)MathSciNetCrossRefGoogle Scholar - 197.D. Kramkov, W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab.
**9**(3), 904–950 (1999)MathSciNetCrossRefGoogle Scholar - 203.D. Lamberton, Optimal stopping and American options. Lecture Notes (2009)Google Scholar
- 222.J. Mossin, Optimal multiperiod portfolio policies. J. Bus.
**41**(2), 215–229 (1968)CrossRefGoogle Scholar - 226.B. Øksendal,
*Stochastic Differential Equations*, 6th edn. (Springer, Berlin, 2003)CrossRefGoogle Scholar - 237.S. Pliska,
*Introduction to Mathematical Finance*(Blackwell, Malden, 1997)Google Scholar - 238.P. Protter,
*Stochastic Integration and Differential Equations*, 2nd edn. (Springer, Berlin, 2004)zbMATHGoogle Scholar - 249.C. Rogers, Monte Carlo valuation of American options. Math. Finance
**12**(3), 271–286 (2002)MathSciNetCrossRefGoogle Scholar - 258.P. Samuelson, Lifetime portfolio selection by dynamic stochastic programming. Rev. Econ. Stat.
**51**, 239–246 (1969)CrossRefGoogle Scholar - 270.M. Schweizer, A guided tour through quadratic hedging approaches.
*Option Pricing, Interest Rates and Risk Management*(Cambridge Univ. Press, Cambridge, 2001), pp. 538–574Google Scholar - 271.A. Seierstad,
*Stochastic Control in Discrete and Continuous Time*(Springer, New York, 2009)CrossRefGoogle Scholar - 275.A. Shiryaev,
*Probability*, 2nd edn. (Springer, New York, 1995)zbMATHGoogle Scholar - 277.A. Shiryaev, Yu. Kabanov, D. Kramkov, A. Mel’nikov, Toward a theory of pricing options of European and American types. I. Discrete time. Theory Probab. Appl.
**39**(1), 14–60 (1995)MathSciNetCrossRefGoogle Scholar - 278.S. Shreve,
*Stochastic Calculus for Finance I: The Binomial Asset Pricing Model*(Springer, New York, 2004)CrossRefGoogle Scholar

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