Journal
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
Volume 120, Issue 5, Pages 622-652Publisher
ELSEVIER
DOI: 10.1016/j.spa.2010.01.013
Keywords
Ito's formula; Stochastic differential equation; Jump-diffusion; Black-Scholes equation; Merton's equation
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We trace Ito's early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Ito's formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Ito's formula in mathematical finance in the 1970s. Throughout the paper, we treat Ito's jump SDEs driven by Brownian motions and Poisson random measures, as well as the well-known continuous SDEs driven by Brownian motions. (C) 2010 Elsevier B.V. All rights reserved.
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