{"product_id":"introduction-to-stochastic-integration-universitext-0387287205","title":"Introduction to Stochastic Integration (Universitext)","description":"\u003cp\u003e\u003cstrong\u003eISBN:\u003c\/strong\u003e 0387287205\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eAuthor:\u003c\/strong\u003e Kuo, Hui-Hsiung\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eCondition:\u003c\/strong\u003e New\u003c\/p\u003e\u003cp\u003eIn 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 It o published the celebrated paper Stochastic Integral in the Proceedings of the Imperial Academy (Tokyo). It was the beginning of the It o calculus, the counterpart of the Leibniz-Newton calculus for random functions. In this six-page paper, It o introduced the stochastic integral and a formula, known since then as It os formula. The It o formula is the chain rule for the Itocalculus.Butitcannotbe expressed as in the Leibniz-Newton calculus in terms of derivatives, since a Brownian motion path is nowhere di?erentiable. The It o formula can be interpreted only in the integral form. Moreover, there is an additional term in the formula, called the It o correction term, resulting from the nonzero quadratic variation of a Brownian motion.\u003c\/p\u003e","brand":"Mia Karts","offers":[{"title":"Default Title","offer_id":51823864381728,"sku":"NEW0387287205","price":74.21,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0980\/7426\/3840\/files\/51MAXF_XnOL.jpg?v=1781204169","url":"https:\/\/miakarts.com\/products\/introduction-to-stochastic-integration-universitext-0387287205","provider":"Miakarts Books","version":"1.0","type":"link"}