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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1994 • Question 2

Question

Math 6010 • Linear Models 1994 • Question 2

Let \((\Omega ,\mathcal A,P)\) be a probability space and \(\{B(t),0\le t\le 1\}\) be a Brownian motion on it. Since almost all sample paths of \(B\) are continuous, \(\int _0^t B(t)dt\) can be defined as a usual Rieman integral. Compute the distribution of \(\int _0^1 B(t)dt\).

Raw LaTeX

Let \((\Omega,\mathcal A,P)\) be a probability space and
\(\{B(t),0\le t\le 1\}\) be a Brownian motion on it. Since almost all sample paths
of \(B\) are continuous, \(\int_0^t B(t)dt\) can be defined as a usual Rieman
integral. Compute the distribution of \(\int_0^1 B(t)dt\).

Topic Tags

Brownian Motion and Stochastic Processes Measure-Theoretic Probability Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits