hard · Quantitative Finance stochastic
For a standard Brownian motion W_t, define the exponential local martingale Z_t=exp!big(λ W_t-tfrac12λ^2 tbig) and the related Y_t=exp!big(λ W_tbig).
Which statement correctly relates their Itô dynamics and martingale status?
- Both Z_t and Y_t are true martingales here under this measure, since each one is merely a smooth exponential function of the single martingale W_t
- Y_t satisfies dY_t=tfrac12λ^2 Y_t,dt+λ Y_t,dW_t, so Y_t is a submartingale, while Z_t satisfies dZ_t=λ Z_t,dW_t and is a (true) martingale
- Y_t is the actual martingale here and Z_t is only a submartingale, because the extra -tfrac12λ^2 t term subtracted in the exponent adds positive drift into Z_t instead
- dY_t=λ Y_t,dW_t with a zero drift term, exactly identical to the dynamics of Z_t, since the exponent λ W_t is linear in W_t and carries absolutely no curvature
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