Problem bank

Problem 322 of 333EasyStochastic CalculusP322

Which of these are Brownian motions?

  1. Let WW be a standard Brownian motion. Which of the following processes, defined for t≥0t \ge 0, are also standard Brownian motions?

    • Xt=2Wt/4X_t = 2W_{t/4}
    • Yt=14W4tY_t = \tfrac14 W_{4t}
    • Zt=W1+t−W1Z_t = W_{1+t} - W_1
    • Vt=t ξV_t = \sqrt t\,\xi, where ξ∼N(0,1)\xi \sim N(0,1) is a single draw fixed for the whole path

    For each one that fails, say which property breaks and give a number that shows it.