A process $X_t$ is a local martingale if for each increasing sequence of stopping times $\{\tau_k,k=1,2,...\}$ the stopped process is a martingale. All true martingales are local martingales, but the inverse is not true. A strict local martingale is a local martingale which is not a true martigale. In fact, a positive strict local martingale is a supermartingale -- i.e. the expectation decreases with horizon.
In quant finance strictly local martingales have appeared as models which exhibit volatility induced stationarity or models that describe financial bubbles.
All Ito processes desrcibed by a 'driftless SDE' are in fact local martingales, but not martingales (which is surprising to many). For example the familiar Geometric Brownian motion $$dX_t = X_t\ dW_t$$ is a local martingale and a true martingale, while the CEV model with exponent greater than one $$dY_t = Y_t^\alpha\ dW_t \text{ (for given }\alpha > 1\text{)}$$ is a local martingale but not a true martingale. In fact, starting from $Y_0$ the expectation $$E[Y_t|\mathcal{F}_0] < Y_0 \text{ (for all }t>0\text{)}$$
My question is on the intution behind their dynamics and paths:
- What are the qualitative features that break the martingality for such a process as $\alpha$ crosses 1?
- How do paths of strict local martingales look like (against true martingales)? They are not explosive, they are not hitting zero, and I cannot see anything particularly 'strange' or 'unusual' when I plot them.
- How come that even when I simulate this process (with Euler) I get a negative drift, even though I am adding up a finite number of Gaussians with zero mean (although strictly local martingales are only a continuous time phenomenon)?
This blog post discusses these points, but I am looking for something more high level and (if possible) intuitive.
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