Stochastic Processes – Problem 49: Prove that is also a Gaussian process, and find ,

Given that $\{B_{t}:t\geqslant0\}$ is a Gaussian process, define $X_{t}=B_{t}-t B_{1}$, where $0<t<1$. Prove that $X_{t}$ is also a Gaussian process, and find $cor\left(X_{t},X_{s}\right)$, $0<t,s<1$.

Step 1. Prove that $X_t$ is a Gaussian process. Since $\{B_t\}$ is a Gaussian process, for any time points $t_1, \dots, t_n$ and $t=1$, the random vector $(B_{t_1}, \dots, B_{t_n}, B_1)$ follows a multivariate normal distribution. The random variable $X_t = B_t - t B_1$ is a linear combination of $B_t$ and $B_1$. For any $t_1, \dots, t_n \in (0, 1)$, $(X_{t_1}, \dots, X_{t_n})$ is a linear transformation of the multivariate normal vector $(B_{t_1}, \dots, B_{t_n}, B_1)$. Since multivariate normal distributions are preserved under linear transformations, $(X_{t_1}, \dots, X_{t_n})$ is also multivariate normal. This proves $\{X_t\}$ is a Gaussian process.

Compute the mean: $\mathbb{E}[X_t] = \mathbb{E}[B_t] - t \mathbb{E}[B_1] = 0$. Compute the covariance: $\text{Cov}(X_t, X_s) = \mathbb{E}[X_t X_s] = \min(t, s) - st$. Compute the variance: $\text{Var}(X_t) = t(1-t)$. The correlation coefficient is: $\text{cor}(X_t, X_s) = \frac{\min(t, s) - st}{\sqrt{t(1-t)s(1-s)}}$.

QED. $\frac{\min(t, s) - st}{\sqrt{t(1-t)s(1-s)}}$