000			02332 a2200229 4500
008			230127b |||||||| |||| 00| 0 eng d
020			_a9780387287201
082			_a519.23 _bKUO
100			_aKuo, Hui-Hsiung
245			_aIntroduction to stochastic integration
260			_bSpringer Science + Business media, _c2006. _aNew York:
300			_axiii, 278p.; _bpbk; _c23cm.
440			_aUniversitext
504			_aInclude Index and Reference
520			_aIn the Leibniz–Newton calculus, one learns the di?erentiation and integration of deterministic functions. A basic theorem in di?erentiation is the chain rule, which gives the derivative of a composite of two di?erentiable functions. The chain rule, when written in an inde?nite integral form, yields the method of substitution. In advanced calculus, the Riemann–Stieltjes integral is de?ned through the same procedure of “partition-evaluation-summation-limit” as in the Riemann integral. In dealing with random functions such as functions of a Brownian motion, the chain rule for the Leibniz–Newton calculus breaks down. A Brownian motionmovessorapidlyandirregularlythatalmostallofitssamplepathsare nowhere di?erentiable. Thus we cannot di?erentiate functions of a Brownian motion in the same way as in the Leibniz–Newton calculus. In 1944 Kiyosi Itˆ o published the celebrated paper “Stochastic Integral” in the Proceedings of the Imperial Academy (Tokyo). It was the beginning of the Itˆ o calculus, the counterpart of the Leibniz–Newton calculus for random functions. In this six-page paper, Itˆ o introduced the stochastic integral and a formula, known since then as Itˆ o’s formula. The Itˆ o formula is the chain rule for the Itˆocalculus.Butitcannotbe expressed as in the Leibniz–Newton calculus in terms of derivatives, since a Brownian motion path is nowhere di?erentiable. The Itˆ o formula can be interpreted only in the integral form. Moreover, there is an additional term in the formula, called the Itˆ o correction term, resulting from the nonzero quadratic variation of a Brownian motion. https://link.springer.com/book/10.1007/0-387-31057-6
650			_aMartingale
650			_aGaussian measure
650			_aBrownian motion
650			_aProbability theory
650			_aDiffusion process
942			_2ddc _cTD
999			_c58532 _d58532