Integral de Itô y fórmula de Itô. Modelos en finanzas y algunas extensiones
Integral de Itô y fórmula de Itô. Modelos en finanzas y algunas extensiones
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Moreno Trujillo, J.F. 2016. Integral de Itô y fórmula de Itô. Modelos en finanzas y algunas extensiones. ODEON. 10 (oct. 2016), 7–27. DOI:https://doi.org/10.18601/17941113.n10.02.
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Se presenta una definición formal de la integral estocástica y una demostración completa de la fórmula de Itô, junto con algunas extensiones de estos resultados en el contexto de la modelación financiera.
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Referencias (VER)
Bachelier, L. (1900). Th´eorie de la sp´eculation. Gauthier-Villars.
Bjork, T. (2004). Arbitrage theory in continuous time. Oxford University Press.
Black, F., y Scholes, M. (1973). The pricing of options and corporate liabilities. The journal of political economy, 637-654.
Davie, M. H. (1997). Option pricing in incomplete markets.
Durrett, R. (1996). Stochastic calculus: a practical introduction (Vol. 6). CRC press.
Einstein, A. (1956). Investigations on the theory of the brownian movement. Courier Corporation.
Itô, K. (1944). 109. stochastic integral. Proceedings of the Imperial Academy, 20(8), 519-524.
Itô, K. (1975). Stochastic differentials. Applied Mathematics & Optimization, 1(4), 374-381.
Itô, K., y Watanabe, S. (1965). Transformation of markov processes by multiplicative functionals. En Annales de l’institut fourier (Vol. 15, p. 13-30).
Karatzas, I., y Shreve, S. (2012). Brownian motion and stochastic calculus (Vol. 113). Springer Science & Business Media.
Klebaner, F. C., y cols. (2005). Introduction to stochastic calculus with applications (Vol. 57). World Scientific.
Kunita, H., Watanabe, S., y cols. (1967). On square integrable martingales. Nagoya Mathematical Journal, 30, 209-245.
McCauley, J. L. (2013). Stochastic calculus and differential equations for physics and finance. Cambridge University Press.
Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of
economics and management science, 141-183.
Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. The Journal of finance, 29(2), 449-470.
Meyer, P. (1967). On square integrable martingales. Nagoya Mathematical
Journal, 30, 209.
Meyer, P.-A. (1976). Un cours sur les int´egrales stochastiques (expos´es 1 `a 6). Séminaire de probabilit´es de Strasbourg, 10, 245-400.
Oksendal, B. (2013). Stochastic differential equations: an introduction with applications. Springer Science & Business Media.
Shreve, S. E. (2004). Stochastic calculus for finance ii: Continuous-time models (Vol. 11). Springer Science & Business Media.
Sottinen, T., y Viitasaari, L. (2014). Pathwise integrals and ito–tanaka formula for gaussian processes. Journal of Theoretical Probability, 1-27.
Tanaka, H. (1963). Note on continuous additive functional of the 1-dimensional brownian path. Probability Theory and Related Fields, 1(3), 251-257.
Trujillo, J. F. M. (2011). Estimación de parámetros en ecuaciones diferenciales estocásticas aplicadas a finanzas (parameter estimation in stochastic differential equations (sdes): Applications to financial modeling). ODEON-Observatorio de Economía y Operaciones Numéricas, 6.
Trujillo, J. F. M. (2016). Transformaciones integrales y sus aplicaciones en
finanzas. ODEON(9), 257-265.
Uhlenbeck, G. E., y Ornstein, L. S. (1930). On the theory of the brownian
motion. Physical review, 36(5), 823.
Wiener, N. (1966). Nonlinear problems in random theory. Nonlinear Problems in Random Theory, by Norbert Wiener, pp. 142. ISBN 0-262-73012-X. Cambridge, Massachusetts, USA: The MIT Press, August 1966.(Paper), 1.
Yamada, T., Watanabe, S., y cols. (1971). On the uniqueness of solutions of stochastic differential equations. Journal of Mathematics of Kyoto University, 11(1), 155-167.
Bjork, T. (2004). Arbitrage theory in continuous time. Oxford University Press.
Black, F., y Scholes, M. (1973). The pricing of options and corporate liabilities. The journal of political economy, 637-654.
Davie, M. H. (1997). Option pricing in incomplete markets.
Durrett, R. (1996). Stochastic calculus: a practical introduction (Vol. 6). CRC press.
Einstein, A. (1956). Investigations on the theory of the brownian movement. Courier Corporation.
Itô, K. (1944). 109. stochastic integral. Proceedings of the Imperial Academy, 20(8), 519-524.
Itô, K. (1975). Stochastic differentials. Applied Mathematics & Optimization, 1(4), 374-381.
Itô, K., y Watanabe, S. (1965). Transformation of markov processes by multiplicative functionals. En Annales de l’institut fourier (Vol. 15, p. 13-30).
Karatzas, I., y Shreve, S. (2012). Brownian motion and stochastic calculus (Vol. 113). Springer Science & Business Media.
Klebaner, F. C., y cols. (2005). Introduction to stochastic calculus with applications (Vol. 57). World Scientific.
Kunita, H., Watanabe, S., y cols. (1967). On square integrable martingales. Nagoya Mathematical Journal, 30, 209-245.
McCauley, J. L. (2013). Stochastic calculus and differential equations for physics and finance. Cambridge University Press.
Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of
economics and management science, 141-183.
Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. The Journal of finance, 29(2), 449-470.
Meyer, P. (1967). On square integrable martingales. Nagoya Mathematical
Journal, 30, 209.
Meyer, P.-A. (1976). Un cours sur les int´egrales stochastiques (expos´es 1 `a 6). Séminaire de probabilit´es de Strasbourg, 10, 245-400.
Oksendal, B. (2013). Stochastic differential equations: an introduction with applications. Springer Science & Business Media.
Shreve, S. E. (2004). Stochastic calculus for finance ii: Continuous-time models (Vol. 11). Springer Science & Business Media.
Sottinen, T., y Viitasaari, L. (2014). Pathwise integrals and ito–tanaka formula for gaussian processes. Journal of Theoretical Probability, 1-27.
Tanaka, H. (1963). Note on continuous additive functional of the 1-dimensional brownian path. Probability Theory and Related Fields, 1(3), 251-257.
Trujillo, J. F. M. (2011). Estimación de parámetros en ecuaciones diferenciales estocásticas aplicadas a finanzas (parameter estimation in stochastic differential equations (sdes): Applications to financial modeling). ODEON-Observatorio de Economía y Operaciones Numéricas, 6.
Trujillo, J. F. M. (2016). Transformaciones integrales y sus aplicaciones en
finanzas. ODEON(9), 257-265.
Uhlenbeck, G. E., y Ornstein, L. S. (1930). On the theory of the brownian
motion. Physical review, 36(5), 823.
Wiener, N. (1966). Nonlinear problems in random theory. Nonlinear Problems in Random Theory, by Norbert Wiener, pp. 142. ISBN 0-262-73012-X. Cambridge, Massachusetts, USA: The MIT Press, August 1966.(Paper), 1.
Yamada, T., Watanabe, S., y cols. (1971). On the uniqueness of solutions of stochastic differential equations. Journal of Mathematics of Kyoto University, 11(1), 155-167.
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