8.8 Stochastic Exponential

The stochastic exponential (also known as the semimartingale, or Dol ansDade exponential) is a stochastic analogue of the exponential function. Recall that if f( t) is a smooth function then g( t) = e ^{f( t)} is the solution to the differential equation dg( t) = g( t) df( t). The stochastic exponential is defined as a solution to a similar stochastic equation. The stochastic exponential of It processes was introduced in Section 5.2. For a semimartingale X, its stochastic exponential ?( X)( t) = U( t) is defined as the unique solution to the equation

As an application of It 's formula and the rules of stochastic calculus we prove

Theorem 8.12

Let X be a continuous semimartingale. Then its stochastic exponential is given by

PROOF: Write U( t) = e ^{V( t)}, with V( t) = X( t) ? X(0) ? [ X,X]( t). Then

Using the fact that [ X, X] ( t) is a continuous process of finite variation, we obtain [ X, [ X,X]]( t) = 0, and [ V, V]( t) = [ X,X]( t). Using this, we obtain

or dU( t) = U( t) dX( t). Thus U( t) defined by (8.38) satisfies (8.37).