Stieltjes integration and stochastic calculus with respect to self-affine functions

In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a [Formula: see text]-Hölder continuous function [Formula: see text] with [Formula: see text]. Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to [Formula: see text] is the extension of the Young integral [An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251–282] given by Zähle [Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998) 333–374].

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Lévy Processes and Stochastic Calculus

10.1017/cbo9780511755323 ◽

2004 ◽

Cited By ~ 439

Author(s):

David Applebaum

Keyword(s):

Lévy Processes ◽

Levy Processes ◽

Stochastic Calculus

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Introduction to Stochastic Calculus and Its Applications

SSRN Electronic Journal ◽

10.2139/ssrn.3607647 ◽

2020 ◽

Author(s):

Xinye Yang

Keyword(s):

Stochastic Calculus

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Simplified Stochastic Calculus With Applications in Economics and Finance

European Journal of Operational Research ◽

10.1016/j.ejor.2020.12.037 ◽

2020 ◽

Author(s):

Aleš Černý ◽

Johannes Ruf

Keyword(s):

Stochastic Calculus

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TRANSFORMATION OF NESTED LOOPS WITH MODULO INDEXING TO AFFINE RECURRENCES

Parallel Processing Letters ◽

10.1142/s0129626494000260 ◽

1994 ◽

Vol 04 (03) ◽

pp. 271-280 ◽

Cited By ~ 6

Author(s):

FLORIN BALASA ◽

FRANK H.M. FRANSSEN ◽

FRANCKY V.M. CATTHOOR ◽

HUGO J. DE MAN

Keyword(s):

Code Generation ◽

State Of The Art ◽

Transformation Method ◽

Control Flow ◽

Code Optimization ◽

Transformation Techniques ◽

Hermite Normal Form ◽

Nested Loops ◽

Affine Functions ◽

Systems Transformation

For multi-dimensional (M-D) signal and data processing systems, transformation of algorithmic specifications is a major instrument both in code optimization and code generation for parallelizing compilers and in control flow optimization as a preprocessor for architecture synthesis. State-of-the-art transformation techniques are limited to affine index expressions. This is however not sufficient for many important applications in image, speech and numerical processing. In this paper, a novel transformation method is introduced, oriented to the subclass of algorithm specifications that contains modulo expressions of affine functions to index M-D signals. The method employs extensively the concept of Hermite normal form. The transformation method can be carried out in polynomial time, applying only integer arithmetic.

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