Predictability of a Process Defined from Lagged Price Movements

Summary

The document asks whether a process used in a two-asset market model is predictable when its values depend on the direction of earlier price movements. The concern is whether each value can be known one period in advance, as required by predictability in a discrete-time filtration.

The answer notes that the process definition is incomplete in the text: an earlier part of the question may define its initial value. If that initial value is specified, the process is predictable because each subsequent value is determined using information already available. Without it, the process is not properly defined, so later values cannot be assessed. This is a conditional clarification rather than a general test for all stochastic processes; the omitted definition prevents a definitive conclusion about the particular process as presented.

Key ideas

• Predictability depends on whether each process value is determined by information available at the prior time.
• A process based on lagged price movements may be predictable if its initial value is specified.
• If the initial value is missing, the process definition is incomplete.
• The answer is conditional because the document omits part of the process definition.

Tags

# Is this process predictable or not?


# Is this process predictable or not?












Consider a market model with two assets which are modeled as usual by the stochastic process $S^0$ and $S^1$, that is adapted to the filtration.

Can anyone tell if this process is predictable or not:

What I think is that it is not predictable as we know only $\xi_1$ and the other values of $\xi_t$ are given only when $S_{t-1}$ is greater than $S_{t-2}$, which contradicts predictability which says that we should be able to calculate the process when we are one period behind.

## Answer by SRKX (score 1)

https://quant.stackexchange.com/a/15867

First, I don't see $S^0$ appear anywhere, so I assume it is just used somewhere else.

Second, there is probably a point (i) in your question, because you included the point (ii). I'd expect that $\xi_0$ is actually defined there.

If that's the case, then $\xi_t$ is predictable.

Otherwise, if $\xi_0$ is not defined anywhere then the process $\xi_t$ is simply not defined properly as there is no way to know $\xi_2$. I'd guess this is a mistake.

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