Why Positivity Matters in Martingale Representation

Summary

The document examines a proof claiming that a positive martingale can be expressed as an exponential driven by a Brownian stochastic integral. The proof applies the martingale representation theorem to write the martingale differential in terms of Brownian motion, then divides by the martingale to obtain a proportional-integral form.

The response identifies the step where positivity matters: division by the martingale requires it to be nonzero. This is a narrow mathematical clarification relevant to stochastic modeling, but the note does not discuss the conditions needed for the representation theorem, integrability, or whether strict positivity is maintained over time. It therefore highlights a key algebraic requirement without presenting a complete proof or a full account of the theorem's assumptions.

Key ideas

• The proof uses positivity when it divides the martingale differential by the martingale itself.
• Division requires the martingale to be nonzero at the times considered.
• The note does not specify the full assumptions needed for martingale representation or the exponential form.

Tags

# Why can any positive martingale be written as the exponential of an Ito integral w.r.t. Brownian motion?


# Why can any positive martingale be written as the exponential of an Ito integral w.r.t. Brownian motion?












Why is any positive martingale the exponential of an Ito integral w.r.t. Brownian motion?

Here is a little proof.

> For any positive P-martingale M, $dM_t = Mt ·1/M_tdM_t$. By Martingale Representation Theorem, $dMt = \Gamma_tdW(t)$ for some adapted process $\Gamma_t$. So $dM_t = M_t(\Gamma_t/M_t)dW$, i.e. any positive martingale must be the exponential of an integral w.r.t. Brownian motion.

Where in this proof are we using the fact that $M$ is positive?

## Answer by M. Jeunesse (score 1)

https://quant.stackexchange.com/a/29994

You are using the fact that $M$ is positive because you divide by $M$.

Dividing by zero is the root of evil.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.