Let (left(B_{t}, mathscr{F}_{t}ight)_{t geqslant 0}) be a one-dimensional Brownian motion. Which of the following processes are martingales?

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Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a one-dimensional Brownian motion. Which of the following processes are martingales?

a) \(U_{t}=e^{c B_{t}}, c \in \mathbb{R}\);

b) \(V_{t}=t B_{t}-\int_{0}^{t} B_{s} d s\);

c) \(W_{t}=B_{t}^{3}-t B_{t}\);

d) \(X_{t}=B_{t}^{3}-3 \int_{0}^{t} B_{s} d s\);

e) \(Y_{t}=\frac{1}{3} B_{t}^{3}-t B_{t}\)

f) \(Z_{t}=e^{B_{t}-c t}, c \in \mathbb{R}\).

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