Brownian Motion and Stochastic Processes

Let \(\{X_n:n\geq 1\}\) be a stochastic process adapted to a filtration \(\{\mathcal {F}_n:n\geq 1\}\) and satisfying \(\mathbb {E}[|X_n|]<\infty \) for all \(n\geq 1\). Prove that \(\{X_n:n\geq 1\}\) is a martingale if and only if \(\mathbb {E}[X_T]=\mathbb {E}[X_1]\) for all bounded stopping times \(T\).

(a) Find the distribution of \(\int _0^1 B_t\,dt\), where \((B_t)\) is Brownian motion.

(b) Given an i.i.d. sequence \(X_1,X_2,\ldots \) with mean 0 and variance 1, what does Donsker’s invariance principle tell us about this sequence in connection with the distribution of \(\int _0^1 B_t\,dt\)?

Let \(\{W(t)\}_{t\geq 0}\) be a standard Brownian motion, and define \[ U(t):=e^{-t/2}W(e^t)\qquad \text {for all }t\geq 0. \]

(a) Prove that \(\{U(t)\}_{t\geq 0}\) is Gaussian process; compute its mean and covariance functions.

(b) Prove that \[ \limsup _{t\to \infty }\frac {U(t)}{\sqrt {2\ln t}} = -\liminf _{t\to \infty }\frac {U(t)}{\sqrt {2\ln t}} =1 \qquad \text {a.s.} \]

Let \(P\) be the transition matrix for a finite irreducible aperiodic Markov chain, and assume that \(P\) is doubly stochastic (row and column sums are 1). Find the limit \(\lim _{n\to \infty }P_{ij}^n\) for all \(i\) and \(j\), and provide justification.

Let \(\{B(t),t\geq 0\}\) be a standard Brownian motion. Prove directly, using \[ V_n=\sum _{i=1}^{2^n}\left |B(i/2^n)-B((i-1)/2^n)\right |; \] that \(B(\cdot ,\omega )\) is of unbounded variation on \([0,1]\) for a.e. \(\omega \).

Let \(\{B(t),\, t\ge 0\}\) be a standard Brownian motion. Prove directly, using \[ V_n=\sum _{i=1}^{2^n}\left |B(i/2^n)-B((i-1)/2^n)\right |, \] that \(B(\cdot ,\omega )\) is of unbounded variation on \([0,1]\) for a.e. \(\omega \).

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 \(\int _0^1 B(t)\,dt\).

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 \(\int _0^1 B(t)dt\).