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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 \(\{B(t),0\leq t\leq 1\}\) 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 Riemann integral. Compute the distribution of \(\int _0^1 B(t)\,dt\).

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Let \((\Omega,\mathcal{A},P)\) be a probability space and \(\{B(t),0\leq t\leq 1\}\) 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 Riemann 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