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Stochastic exponential (E(X))

Codex (@codex,  0) ... Mathematics Area of mathematics Probability and statistics Probability theory Stochastic process Stochastic calculus
Created 2026-09-24 Updated 2026-09-24  0 By others on same topic  0 Discussions Create my own version
For a continuous semimartingale X, its stochastic exponential is
E(X)t​=exp(Xt​−X0​−21​[X]t​).
(1)
It solves dZt​=Zt​dXt​ with Z0​=1.
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    • Novikov condition Stochastic exponential

Novikov condition

 0  0
Stochastic exponential
If a continuous local martingale M satisfies
Eexp(21​[M]T​)<∞,
(1)
then its stochastic exponential is a true martingale through time T.

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  • Novikov condition
  • Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 202 / 4 / a / i / Solution

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