hard · Quantitative Finance
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 martingales, since each is a smooth exponential of the 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 a martingale and Z_t is a submartingale, because the -tfrac12λ^2 t term adds positive drift
- dY_t=λ Y_t,dW_t with zero drift, identical to Z_t, since the exponent is linear in W_t
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