Abstract
Most of the book pursues a probabilistic approach to diffusion processes as originated in the work by Itô. However, we demonstrate in the book that when computing explicit formulas for functionals of diffusion processes, it can also be useful to resort to an analytic approach to diffusions, as developed by Kolmogorov and Feller. We recall the approach in a concise manner and demonstrate how it can be used to compute functionals of diffusion processes. This approach also lends itself very well to the study of the boundary behavior of diffusion processes, as the approach allows us to compute the relevant functionals explicitly. As we demonstrate in the next chapter, these results allow us to determine easily if equivalent martingale measures exist or not.
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
Feller, W.: Zur Theorie der stochastischen Prozesse (Existenz- und Eindeutigkeitssätze). Math. Ann. 113(1), 113–160 (1936)
Hulley, H.: Strict local martingales in continuous financial market models. PhD thesis, UTS, Sydney (2009)
Hulley, H., Platen, E.: Laplace transform identities for diffusions, with applications to rebates and barrier options. In: Stettner, L. (ed.) Advances in Mathematical Finance. Banach Center Publications, vol. 83, pp. 139–157 (2008)
Itô, K.: Stochastic integral. Proc. Imp. Acad. (Tokyo) 20(8), 519–524 (1944)
Itô, K., McKean Jr., H.P.: Diffusion Processes and their Sample Paths. Springer, Berlin (1996)
Kolmogorov, A.: Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104(1), 415–458 (1932)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Baldeaux, J., Platen, E. (2013). Time-Homogeneous Scalar Diffusions. In: Functionals of Multidimensional Diffusions with Applications to Finance. Bocconi & Springer Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-00747-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-00747-2_16
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00746-5
Online ISBN: 978-3-319-00747-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)