Question

Let Z be a standard normal random variable and define a stochastic process X = (X_t)_t≥0 by X_t=√t*Z. The stochastic process X has continuous paths and, at each fixed time t, X_t is normally distributed.

Answer 1

The stochastic process X defined by X_t = √t * Z has continuous paths and at each fixed time t, X_t is normally distributed with a mean of 0 and standard deviation of √t.

Step-by-step explanation:

In this question, we are given a standard normal random variable Z and a stochastic process X defined as X_t = √t * Z. The stochastic process X has continuous paths and, at each fixed time t, X_t is normally distributed.

A standard normal distribution is a continuous random variable (RV) with a mean of 0 and a standard deviation of 1. It is often denoted as Z ~ N(0, 1).

In the given stochastic process, X_t is the product of √t and Z. Since Z follows a standard normal distribution, X_t will also be normally distributed with mean 0 and standard deviation √t.