Doléans-Dade exponential

A stochastic exponential is a stochastic process, which is an analogue of the exponential function of habitual Analysis in the mathematical subfield of stochastic analysis. After the French mathematician Catherine Doléans -Dade it is also called Doléans -Dade exponential or short as Doléans exponential.

The exponential function can be characterized in that it coincides with its derivative thereby. If you want to achieve a similar behavior for the exponential function of a stochastic process, it must be taken into account because of the lemma of Itō its quadratic variation, if this does not go away, such as the Wiener process.

Stochastic exponentials play among other things, an important role in the explicit solution of stochastic differential equations and, at the rate of Girsanow that describes the behavior of stochastic processes with a change of measure. An important question in this context is the conditions under which a stochastic exponential is a martingale. Many models of financial mathematics include processes that are stochastic exponentials, such as the geometric Brownian motion in the Black-Scholes model.

Introduction

The exponential function is uniquely determined by the two conditions and. More generally follows the chain rule that the unique solution of the linear ordinary differential equation with the initial condition is.

These relationships are valid for stochastic differential equations in this simple form no more, since here the chain rule must be replaced by the lemma of Itō, which takes into account the quadratic variation of processes. Example, is a standard Wiener process, we obtain for the differential of the process due to the Ito 's lemma

The additional term in the stochastic differential equation can be avoided if, instead of the exponential function of the "corrected " approach is used: Then arises, analogous to the case of ordinary differential equations. In addition, now is the process such as the Wiener process is a martingale.

Definition

It is a semimartingale. Then called the (uniquely determined ) Semimartigal, the solution of the stochastic differential equation

Is the initial condition, the exponential of stochastic and is referred to, i.e..

That the process solution of said initial value problem is not explicitly mean that he the Itō integral equation

Met.

Explicit representation and calculation rules

Is a steady Semimartigal, the stochastic exponential has the explicit representation

The quadratic variation of designated.

In the general case the additional discontinuities to be considered of. Here the resulting

With the jump process.

Instead of the functional equation of the exponential function applies to the stochastic exponential of semi Marti Galen and the calculation rule

Is continuous with, the following applies

Martingaleigenschaften

Below is a continuous semimartingale and without restriction applies, ie. By definition, the stochastic exponential is always a semimartingale. Is a local martingale, it shows the representation as Itō integral, that is also a local martingale. However, it must, even if a martingale, the stochastic exponential be a real martingale; as a non-negative local martingale By then it is a supermartingale.

For many applications it is important to simply have nachzuprüfende criteria which guarantee that the stochastic exponential of a local martingale is a (true ) martingale. The best-known sufficient condition is the Novikov condition ( after the Russian mathematician Alexander Novikov ): Let be a continuous local martingale. Applies to all, then a martingale on.

Applications

Linear stochastic differential equations

With the help of the stochastic Exponentials can explicitly specify the solutions of linear stochastic differential equations. A linear stochastic differential equation has the form

With continuous functions or continuous adapted stochastic processes. The associated homogeneous equation

Has the solution with and without restriction. The general solution is thus explicitly

With

And

A particular solution of the inhomogeneous equation can it be interpreted by variation of constants, see, so by the approach.

Set of Girsanow

There are a Wiener process on the interval with respect to the probability measure and a process. If the stochastic exponential is a martingale, then apply and can be interpreted as Radon Nikodým - density of a probability measure in terms of:

Regarding the measure thus defined is the drift process

A standard Wiener process.