A weak version of path-dependent functional Itô calculus

In this paper, we extend the first-order asymptotics analysis of Fouque et al. to general path-dependent financial derivatives using Dupire’s functional Itô calculus. The main conclusion is that the market group parameters calibrated to vanilla options can be used to price to the same order exotic, path-dependent derivatives as well. Under general conditions, the first-order condition is represented by a conditional expectation that could be numerically evaluated. Moreover, if the path-dependence is not too severe, we are able to find path-dependent closed-form solutions equivalent to the first-order approximation of path-independent options derived in Fouque et al. Additionally, we exemplify the results with Asian options and options on quadratic variation.

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Classical Solutions of Path-Dependent PDEs and Functional Forward-Backward Stochastic Systems

Mathematical Problems in Engineering ◽

10.1155/2013/423101 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Shaolin Ji ◽

Shuzhen Yang

Keyword(s):

Stochastic System ◽

Stochastic Systems ◽

Classical Solutions ◽

Itô Calculus ◽

Path Dependent ◽

The Relationship

In this paper we study the relationship between functional forward-backward stochastic systems and path-dependent PDEs. In the framework of functional Itô calculus, we introduce a path-dependent PDE and prove that its solution is uniquely determined by a functional forward-backward stochastic system.

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Functional Itô versus Banach space stochastic calculus and strict solutions of semilinear path-dependent equations

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025716500247 ◽

2016 ◽

Vol 19 (04) ◽

pp. 1650024 ◽

Cited By ~ 9

Author(s):

Andrea Cosso ◽

Francesco Russo

Keyword(s):

Banach Space ◽

Stochastic Calculus ◽

Continuous Functions ◽

Smooth Solutions ◽

Space Of Continuous Functions ◽

Kolmogorov Type ◽

Itô Calculus ◽

Strict Solutions ◽

Path Dependent ◽

Obvious Relation

Functional Itô calculus was introduced in order to expand a functional [Formula: see text] depending on time [Formula: see text], past and present values of the process [Formula: see text]. Another possibility to expand [Formula: see text] consists in considering the path [Formula: see text] as an element of the Banach space of continuous functions on [Formula: see text] 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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Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients

Stochastic Processes and their Applications ◽

10.1016/j.spa.2015.09.018 ◽

2016 ◽

Vol 126 (3) ◽

pp. 735-766 ◽

Cited By ~ 4

Author(s):

Christian Keller ◽

Jianfeng Zhang

Keyword(s):

Rough Paths ◽

Itô Calculus ◽

Path Dependent

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Moral reasoning is the process of asking moral questions and answering them

Behavioral and Brain Sciences ◽

10.1017/s0140525x18002534 ◽

2019 ◽

Vol 42 ◽

Author(s):

Mark Alfano

Keyword(s):

Moral Reasoning ◽

Dependent Process ◽

Asking Questions ◽

Path Dependent ◽

Path Dependent Process ◽

Moral Questions

Abstract Reasoning is the iterative, path-dependent process of asking questions and answering them. Moral reasoning is a species of such reasoning, so it is a matter of asking and answering moral questions, which requires both creativity and curiosity. As such, interventions and practices that help people ask more and better moral questions promise to improve moral reasoning.

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