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If I have a Stochastic differential equation $dX_t=W_{t}^{3} dW_{t}$ and I let $Z_t=X_{t}^{2}$

Is it possible to show if or if not $Z_t$ is martingale (with respect to $(F_t)t≥0)$

Any help on this would be really appreciated. Thank you

Jacob Mitch
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1 Answers1

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Ito-formula is of great help here $$ df(X_t)= f'(X_t)dX_t + \frac12 f''(X_t)d[X]_t $$ now $d[X]_t = W_t^6dt$ hence $$ d(Z_t) = d(X_t^2) = 2X_tdX_t + d[X]_t = 2 X_t W_t^3dW_t + W_t^6dt $$ and so $$ X_t^2 = \int_0^t 2X_sW_s^3 dW_s + \int_0^t W_s^6ds. $$ $Z_t$ cannot be a martingale since its drift term is nonzero.

Mick
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  • Hi thanks for the help, is the drift term Ws^6. Is the Ito-formula, used in this way, all I would need to fully answer similar questions in the future. Thanks again. – Jacob Mitch Mar 16 '20 at 20:03