Q2: Stochastic Calculus

Consider a standard Brownian motion W(t) and another stochastic process defined by X(t) being the integral from 0 to t of W(s) with respect to s. Which of the following statements about X(t) is true?


🔘 A. X(t) is a martingale.
🔘 B. X(t) has continuous paths.
🔘 C. The quadratic variation of X(t) over [0, t] is t^3/3.
🔘 D. X(t) is a standard Brownian motion.
Given the stochastic process X(t) defined as the integral from 0 to t of W(s) with respect to s, where W(t) is a standard Brownian motion:
🔘 A. X(t) has continuous paths, as it is the time-integrated version of a standard Brownian motion W(t).
🔘 B. X(t) is not a martingale. Its expected future increments, given its past, are not constant.
🔘 C.The quadratic variation of X(t) over [0, t] is not t^3/3. Even though the quadratic variation of W(t) is t, this doesn't directly apply to its integral X(t).
🔘 D. X(t) is not a standard Brownian motion. Instead, it is the integral of a Brownian motion over time, resulting in a different kind of process.
Therefore, the correct answer is:
🔘 B. X(t) has continuous paths.

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