scholarly journals Constructive Martingale Representation in Functional Itô Calculus: A Local Martingale Extension

2018 ◽  
pp. 165-172
Author(s):  
Kristoffer Lindensjö
Keyword(s):  
Local Martingale ◽  
Martingale Representation ◽  
Itô Calculus

scholarly journals Local martingale and pathwise solutions for an abstract fluids model

2011 ◽  
Vol 240 (14-15) ◽  
pp. 1123-1144 ◽  
Author(s):  
Arnaud Debussche ◽  
Nathan Glatt-Holtz ◽  
Roger Temam
Keyword(s):  
Local Martingale ◽  
Pathwise Solutions

scholarly journals A weak version of path-dependent functional Itô calculus

2018 ◽  
Vol 46 (6) ◽  
pp. 3399-3441 ◽  
Author(s):  
Dorival Leão ◽  
Alberto Ohashi ◽  
Alexandre B. Simas
Keyword(s):  
Weak Version ◽  
Itô Calculus ◽  
Path Dependent

Stochastic Integration by Parts and Functional Itô Calculus

2016 ◽  
Author(s):  
Vlad Bally ◽  
Lucia Caramellino ◽  
Rama Cont
Keyword(s):  
Stochastic Integration ◽  
Integration By Parts ◽  
Itô Calculus

The Black–Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral

2021 ◽  
Vol 15 (01) ◽  
pp. 45-59
Author(s):  
E. M. Bonotto ◽  
M. Federson ◽  
P. Muldowney
Keyword(s):  
Extension Theorem ◽  
Continuous Functions ◽  
Sample Space ◽  
Asset Valuation ◽  
Itô Calculus ◽  
Black Scholes ◽  
Pricing Theory ◽  
Simple Sets ◽  
Random Times ◽  
Riemann Integration

The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon–Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue’s by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black–Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman–Ka[Formula: see text] formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.

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