Functional Itô calculus was introduced in order to expand a functional F(t,X.+t,Xt) depending on time t, past and present values of the process X. Another possibility to expand F(t,X.+t,Xt) consists in considering the path X.+t = Xx+t, x ∈ [-T, 0] as an element of the Banach space of continuous functions on C([-T, 0]) and to use Banach space stochastic calculus. The aim of this paper is threefold. (1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which match more naturally the notion of horizontal derivative which is one of the tools of that calculus. (2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. (3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional Itô calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an Itô stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.
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机译：引入函数演算是为了扩展函数F（t，X。+ t，Xt），具体取决于时间t，过程X的过去和现在值。扩展F（t，X。+ t，Xt的另一种可能性）包括考虑路径X. + t = Xx + t，x∈[-T，0]作为C（[-T，0]）上连续函数的Banach空间的元素，并使用Banach空间随机演算。本文的目的是三个方面。 （1）重新构造功能性Itô演算，将时间和过去分开，利用更自然地匹配作为该演算工具之一的水平导数概念的正则化过程。 （2）利用这种重新表述，以讨论功能性方法与Banach空间方法之间的（不明显的）关系。 （3）研究在函数Itô微积分研究中自然产生的与路径相关的偏微分方程的光滑解的存在性和唯一性。更确切地说，我们研究Kolmogorov类型的路径相关方程，该方程与具有路径相关系数的Itô随机微分方程解的窗口处理有关。我们还研究了该方程的半线性形式。
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