Abstract
Itô’s formula is establish for real-valued and \({\mathbb{R}}^d\)-valued continuous and arbitrary semimartingales and its use is illustrated by numerous examples. The relationship between the Itô and Stratonovich integrals is examined prior to presenting a broad range of applications of Itô’s formula. The applications include stochastic differential equations with Gaussian and non-Gaussian white noise, Tanaka’s formula, local solutions for a class of partial differential equations, and improved Monte Carlo estimates based on Girsanov’s theorem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Boresi AP, Chong KP (2000) Elasticity in engineering mechanics. Wiley, New York
Brabenec RL (1990) Introduction to real analysis. PWS-KENT Publishing Company, Boston
Chung KL, Williams RJ (1990) Introduction to stochastic integration. Birkhäuser, Boston
Durrett R (1996) Stochastic calculus: A practical introduction. CRC Press, New York
Grigoriu M (1997) Local solutions of Laplace, heat, and other equations by Itô processes. J Eng Mech ASCE 123(8):823–829
Grigoriu M (1997) Solution of some elasticity problems by the random walk method. ACTA Mechanica 125:197–209
Grigoriu M (2002) Stochastic calculus. Applications in science and engineering. Birkhäuser, Boston
Grigoriu M, Papoulia KD (2005) Effective conductivity by a probability-based local method. J Appl Phys 98:033706 (1–10)
Keskin RSO, Grigoriu M (2010) A probability-based method for calculating effective diffusion coefficients of composite media. Probab Eng Mech 25(2):249–254
Kloeden PE, Platen E (1992) Numerical solutions of stochastic differential equations. Springer, New York
Krishnan V (1984) Nonlinear filtering and smoothing: An introduction to martingales, stochastic integrals and estimation. Wiley, New York
Kuo H-H (2005) Introduction to stochastic integration. Springer, New York
Mikosch T (1998) Elementary stochastic calculus. World Scientific, New Jersey
Øksendal B (1998) Stochastic differential equations. An introduction with applications. Springer, New York
Protter P (1990) Stochastic integration and differential equations. Springer, New York
Schuss Z (1980) Theory and applications of stochastic differential equations. Wiley, New York
Snyder DL (1975) Random point processes. Wiley, New York
Steele JM (2001) Stochastic calculus and financial applications. Springer, New York
Wong E, Zakai M (1965) On the convergence of ordinary integrals to stochastic integrals. Ann Math Stat 36:1560–1564
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag London Limited
About this chapter
Cite this chapter
Grigoriu, M. (2012). Itô’s Formula and Applications. In: Stochastic Systems. Springer Series in Reliability Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-2327-9_5
Download citation
DOI: https://doi.org/10.1007/978-1-4471-2327-9_5
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-4471-2326-2
Online ISBN: 978-1-4471-2327-9
eBook Packages: EngineeringEngineering (R0)