medium · Quantitative Finance stochastic
Let W_t be a standard Brownian motion and define X_t = W_t^3 - 3tW_t.
Which statement about X_t is correct?
- X_t is a martingale because applying Itô's formula to f(t,w)=w^3-3tw yields zero drift, leaving dX_t=(3W_t^2-3t),dW_t
- X_t is a strict submartingale since W_t^3 dominates the -3tW_t correction for large t
- X_t is a martingale only after the additional compensator -3!int_0^t W_s,ds is subtracted
- X_t has drift 3W_t^2,dt because the -3tW_t term contributes -3W_t,dt that fails to cancel the tfrac12 f_ww term
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