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2 years ago in Mathematics , Problem-Solving Techniques By Priyanka ghogge

If f and g are martingales, is w = f + a(g - f²) also a martingale for constant a?

I'm analyzing a stochastic model in financial mathematics where this particular transformation, w = f + a(g - f²), has emerged. I need to verify under what general conditions w remains a martingale, as the presence of the squared term f² is causing some uncertainty in my derivation.

Martingale StochasticProcess

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By Raghav V Answered 1 year ago

In general, I would say no, w is not guaranteed to be a martingale, and I've seen this trap catch many researchers. The martingale property is preserved under linear combinations, but the term f² is the issue. f² is generally not a martingale; it's a submartingale by Jensen's inequality. Therefore, its conditional expectation is not simply itself. For w to be a martingale, you'd need E[f²_{n+1} | F_n] = f²_n, which is not true unless f has zero quadratic variation (like a constant process). You must check the specific properties of f and the filtration.

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If f and g are martingales, is w = f + a(g - f²) also a martingale for constant a?

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