Let X be a semimartingale. Then the stochastic equation

has a unique solution, called the stochastic exponential of X, and this solution is given by

Formula (8.62) can be written by using quadratic variation as follows

PROOF: Let and . Note that although the product is taken for all s ? t, there are at most countably many points at which ? X( s) ? 0, (by the regularity property of the process), hence there are at most countably many elements different from 1 in the product. We show that the product converges. Since by (8.51) ? _{s? t}( ? X( s)) ² < ?, there are only finitely many points s at which ? X( s) > 0.5, which give a finite non-zero contribution to the product. Taking the product with over s at which ? X( s) ? 1/2, and taking logarithm, it is enough to show that ? _{s ? t}In(1 + ? X( s)) ? ? X( s) converges. But this follows from the inequality In(1 + ? X( s)) ? ? X( s) ? ( ? X( s)) ² by (8.51). To see that U