In stochastic calculus, the Doléans-Dade exponential, Doléans exponential, or stochastic exponential, of a semimartingale X is defined to be the solution to the stochastic differential equation dYt = Yt dXt with initial condition Y0 = 1. The concept is named after Catherine Doléans-Dade. It is sometimes denoted by Ɛ(X). In the case where X is differentiable, then Y is given by the differential equation dY/dt = Y dX/dt to which the solution is Y = exp(X − X0). Alternatively, if Xt = σBt + μt for a Brownian motion B, then the Doléans-Dade exponential is a geometric Brownian motion. For any continuous semimartingale X, applying Itō's lemma with ƒ (Y) = log(Y) gives

Exponentiating gives the solution

This differs from what might be expected by comparison with the case where X is differentiable due to the existence of the quadratic variation term [X] in the solution.
The Doléans-Dade exponential is useful in the case when X is a local martingale. Then, Ɛ(X) will also be a local martingale whereas the normal exponential exp(X) is not. This is used in the Girsanov theorem. Criteria for a continuous local martingale X to ensure that its stochastic exponential Ɛ(X) is actually a martingale are given by Kazamaki's condition, Novikov's condition and Beneš' condition.
It is possible to apply Itō's lemma for non-continuous semimartingales in a similar way to show that the Doléans-Dade exponential of any semimartingale X is

where the product extents over the (countable many) jumps of X up to time t.

This is an excerpt from the article Doleans-Dade exponential from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.
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Doléans-Dade exponential - Wikipedia, the free encyclopedia
In stochastic calculus, the Doléans-Dade exponential, Doléans exponential, or stochastic exponential, of a semimartingale X is defined to be the solution to the ...
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arXiv:1112.0430v2 [math.PR] 23 Jan 2014
known as Doleans-Dade exponential. It is well known that z is a nonnegative local martingale and, therefore, it is a supermartingale with Ezt ≤ 1 for any t ≥ 0. If.
arxiv.org/pdf/1112.0430
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Stochastic calculus for Lévy processes.
Z = E(X) is called the Doléans-Dade exponential (or stochastic exponential) ... If X is a Lévy process and a martingale, then its stochastic exponential. Z = E(X) is ...
perso.univ-lemans.fr/~apopier/enseignement/ENSTA/slides_calcul_sto_levy_proc.pdf
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Stochastic Calculus for Jump Processes - Stony Brook University
Stochastic (Doléans-Dade) exponential. Theorem (Doléans-Dade exponential). Xt be a Levy process with triplet (σ. 2,v,γ).There exists unique cadlag process Zt ...
www.ams.sunysb.edu/~xizhou/resources/Stochastic%20Calculus%20for%20Jump%20Processes.pdf
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Doleans-Dade exponential formula - Math StackExchange
Dec 11, 2010 ... How do I apply the Doleans-Dade exponential formula for the following levy stochastic differential equation: ...
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Is there a discrete-time analogue of Doléans-Dade exponential?
Nov 22, 2011 ... For a continuous martingale $X$, we have the Doléans-Dade exponential: $$\ epsilon(X)_t=\exp\left(X_t-\frac{1}{2}[X]_t\right)$$. What is the ...
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Serge Cohen
Motivations. Doléans Dade exponential in stochastic calculus. If Zt is a semimartingale with jumps. dXt. = Xt− dZt. Xt. = exp{Zt −. 1. 2. [Z]c t } ∏. 0≤s≤t. ( 1 + ∆Zs)e.
www.math.ku.dk/english/research/conferences/levy2007/presentations/SergeCohen.pdf
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The Cumulant Process and Esscher's Change of Measure
criteria for uniform integrability of exponential martingales. Keywords: cumulant ... the stochastic or Doléans-Dade exponential of a semimartingale. Definition 2.1 ...
www.stochastik.uni-freiburg.de/~kallsen/esscher4.pdf
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AN EXPONENTIAL MARTINGALE EQUATION 1 Introduction
Abstract. We prove an existence of a unique solution of an exponential martingale equation in the class ... Here E(X) is the Doleans-Dade exponential of X.
bing: approx. 2
Doléans-Dade exponential - Wikipedia, the free encyclopedia
In stochastic calculus, the Doléans-Dade exponential, Doléans exponential, or stochastic exponential, of a semimartingale X is defined to be the solution to the ...
Exponential martingales and changes of measure for counting ...
May 10, 2012 ... Abstract: We give sufficient criteria for the Dol\'eans-Dade exponential of a stochastic integral with respect to a counting process local ...
[PDF]Serge Cohen
University of Copenhagen Department of Mathematics. Copenhague August the 14th, 2007 ... Motivations. Doléans Dade exponential in stochastic calculus.
[PDF]The full report (pdf) - KTH
Using Equation (4.4) we see that the Doléans-Dade exponential for the process in. Equation (4.6) is. Zt = z exp. {. Xt −. 1. 2 σ2t. }∏. 0≤s≤t. (1 + Uj) exp{−Uj}.
[PDF]AN EXPONENTIAL MARTINGALE EQUATION 1 Introduction
Georgian–American University, Business School, 3, Alleyway II, Chavchavadze Ave. 17a,. Tbilisi ... Here E(X) is the Doleans-Dade exponential of X. It is evident ...
Is there a discrete-time analogue of Doléans-Dade exponential?
Nov 22, 2011 ... For a continuous martingale $X$, we have the Doléans-Dade exponential: $$\ epsilon(X)_t=\exp\left(X_t-\frac{1}{2}[X]_t\right)$$. What is the ...
[PDF]Stochastic Calculus for Jump Processes - Stony Brook University
Xiaoping Zhou. Stony Brook University ..... exponential-Lévy models and stochastic exponentials. (Xiaoping ... Stochastic (Doléans-Dade) exponential. Theorem ...
[PDF]How to explain a corporate credit spread - ResearchGate
Dol eans Dade exponential, equivalent martingale measure, Girsanov's ... of Operations Research and Engineering, 235 Rhodes Hall, Cornell University, Ithaca, ...
[PDF]A great probabilist: Catherine Doléans-Dade - EPFL
Jun 16, 2010 ... Catherine Doleans-Dade. Catherine ... of P. A. Meyer at the University of Strasburg in the late 1960's. ... Dolean-Dade exponential [4]. Let X be ...
Books on the term Doleans-Dade exponential
Semimartingale Theory and Stochastic Calculus
Sheng-wu He, Jia-gang Wang, Jia-An Yan, 1992
Ito Formula and Doleans-Dade exponential Formula In this paragraph we will establish the change of variables formula for semimartingales, that is, the famous Ito formula. It is the most powerful tool in stochastic calculus. As its applications ...
Stochastic Calculus for Finance II: Continuous-Time Models (Springer Finance)
2008
"A wonderful display of the use of mathematical probability to derive a large set of results from a small set of assumptions. In summary, this is a well-written text that treats the key classical models of finance through an applied probability approach....It should serve as an excellent introduction for anyone studying the mathematics of the ...
Introduction to Stochastic Calculus with Applications
Fima C. Klebaner, 2005
8.8 Stochastic Exponential The stochastic exponential (also known as the semimartingale, or Doleans- Dade exponential) is a stochastic analogue of the exponential function. Recall that if f(t) is a smooth function then g(t) = e^W is the solution ...
Statistical Models Based on Counting Processes (Springer Series in Statistics)
Richard D. Gill, 1997
Modern survival analysis and more general event history analysis may be effectively handled within the mathematical framework of counting processes. This book presents this theory, which has been the subject of intense research activity over the past 15 years. The exposition of the theory is integrated with careful presentation of many practical ex...
Approximation, Optimization and Mathematical Economics: With ...
Marc Lassonde, 2001
This corollary makes the link between the change of probability and the Doleans-Dade exponential (L(u, F, f) is the ... Although the Doleans-Dade exponential does not appear in the first results we give in this section, it plays a key role in their ...
Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning (Cambridge Series...
2012
Derived from extensive teaching experience in Paris, this second edition now includes over 100 exercises in probability. New exercises have been added to reflect important areas of current research in probability theory, including infinite divisibility of stochastic processes, past-future martingales and fluctuation theory. For each exercise the au...
Risk-Neutral Valuation: Pricing and Hedging of Financial ...
Nicholas H. Bingham, Rüdiger Kiesel, 2004
... need the stochastic (or Doleans, or Doleans-Dade) exponential (see §5.10 below), giving Y = £(X) = exp{X - \ (X)) (with X a continuous semi-martingale) as the unique solution to the stochastic differential equation dY{t) = Y(t-)dX(t), Y(0) = 1.
Financial Modelling with Jump Processes (Chapman & Hall/CRC Financial Mathematics Series)
Rama Cont, 2003
WINNER of a Riskbook.com Best of 2004 Book Award!During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematical tools...
Weak Convergence of the Variations, Iterated Integrals, and ...
F. Avram, NORTH CAROLINA UNIV AT CHAPEL HILL CENTER FOR STOCHASTIC PROCESSES., 1986
This document investigates the weak convergence of variations, iterated integrals and Doleans Dade exponentials.
Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives, 2nd Ed.
2004
This second edition - completely up to date with new exercises - provides a comprehensive and self-contained treatment of the probabilistic theory behind the risk-neutral valuation principle and its application to the pricing and hedging of financial derivatives. On the probabilistic side, both discrete- and continuous-time stochastic processes are...

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