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GTM295 Chapter 14

In Exercises 14.1 to 14.11, \((B_t)_{t\ge0}\) is a one-dimensional Brownian motion with \(B_0=0\), and \(S_t=\sup\{B_s:0\le s\le t\}\).

Exercise 14.1 For every \(a\ge0\), set \(T_a=\inf\{t\ge0:B_t=a\}\).

  1. Prove that, for every \(0\le a<b\), the random variable \(T_b-T_a\) is independent of \(\sigma(T_c,0\le c\le a)\) and has the same distribution as \(T_{b-a}\).

  2. Prove that, for every \(a_1,\ldots,a_n\in\mathbb{R}_+\) and \(\lambda>0\), the vector \((T_{\lambda a_1},\ldots,T_{\lambda a_n})\) has the same law as \((\lambda^2T_{a_1},\ldots,\lambda^2T_{a_n})\).

  3. Let \(n\in\mathbb{N}\) and let \(T^{(1)},T^{(2)},\ldots,T^{(n)}\) be \(n\) independent random variables distributed as \(T_1\). Verify that \(T^{(1)}+\cdots+T^{(n)}\) has the same distribution as \(n^2T_1\). Comment on the relation between this result and the strong law of large numbers.

Exercise 14.2 Let \(a>0\) and \(T_a=\inf\{t\ge0:B_t=a\}\). Prove that we have almost surely

\[ T_a=\inf\{t\ge0:B_t>a\}. \]

Exercise 14.3 Prove that

\[ \left(\int_0^t e^{B_s}\,ds\right)^{1/\sqrt{t}} \xrightarrow[t\to\infty]{(d)} e^{|N|}, \]

where \(N\) is a Gaussian \(\mathcal{N}(0,1)\) random variable.

Exercise 14.4

  1. Prove that a.s.,

    \[ \limsup_{t\downarrow0}\frac{B_t}{\sqrt{t}}=+\infty, \qquad \liminf_{t\downarrow0}\frac{B_t}{\sqrt{t}}=-\infty. \]

  2. Let \(s>0\). Prove that a.s. the function \(t\mapsto B_t\) is not differentiable at \(s\).

Exercise 14.5

  1. For every integer \(n\ge1\), set

    \[ \Sigma_n = \sum_{k=1}^{2^n} \left(B_{k2^{-n}}-B_{(k-1)2^{-n}}\right)^2. \]

    Compute \(\mathbb{E}[\Sigma_n]\) and \(\operatorname{var}(\Sigma_n)\), and prove that \(\Sigma_n\) converges in \(L^2\) and a.s. to a constant as \(n\to\infty\).

  2. Prove that a.s. the function \(t\mapsto B_t(\omega)\) is not of bounded variation on the interval \([0,1]\) (see Exercise 6.1 for the definition of functions of bounded variation).

Exercise 14.6

  1. For every \(t\in[0,1]\), set \(B'_t=B_{1-t}-B_1\). Prove that the two random processes \((B_t)_{t\in[0,1]}\) and \((B'_t)_{t\in[0,1]}\) have the same law (as in the definition of the Wiener measure, this law is a probability measure on the space \(C([0,1],\mathbb{R})\)).

  2. Let \(t>0\). Prove that \(S_t-B_t\) and \(S_t\) have the same law without using Corollary 14.17.

Exercise 14.7 Let \(\tau=\inf\{t\ge0:B_t=S_1\}\).

  1. Prove that \(0<\tau<1\) a.s. (one may use the preceding exercise) and then that \(\tau\) is not a stopping time.

  2. Using question (2) of the preceding exercise, verify (without making any calculation) that, for every \(a\in(0,1)\),

    \[ \mathbb{P}(\tau>a) = \mathbb{P}(\sqrt{1-a}|N|>\sqrt{a}|N'|) \]

    where \(N\) and \(N'\) are two independent Gaussian \(\mathcal{N}(0,1)\) random variables.

  3. Conclude that the law of \(\tau\) is the arcsine distribution of Exercise 9.3.

Exercise 14.8 Show the local maxima of \(B\) are almost surely distinct. In other words, a.s. for any rationals \(0\le a<b<c<d\), we have

\[ \sup_{a\le t\le b}B_t\ne\sup_{c\le t\le d}B_t. \]

Exercise 14.9 Let \(H=\{t\in[0,1]:B_t=0\}\). Using Corollary 14.10 and the strong Markov property, prove that \(H\) is a.s. a compact subset of \([0,1]\) with no isolated points and zero Lebesgue measure.

Exercise 14.10

  1. For every \(a>0\), set \(\sigma_a=\inf\{t\ge0:|B_t|\ge a\}\). Show that there is a constant \(\gamma\in(0,1)\) depending on \(a\) such that, for every integer \(N\ge1\),

    \[ \mathbb{P}(\sigma_a>N)\le\gamma^N. \]

  2. For every \(n\ge1\), define a sequence \(T_0^n,T_1^n,\ldots\) by induction by setting

    \[ T_0^n=0,\qquad T_1^n=\sigma_{2^{-n}},\qquad T_{k+1}^n=\inf\{t>T_k^n:|B_t-B_{T_k^n}|=2^{-n}\}. \]

    Verify that the random times \(T_k^n\) are almost surely finite and are stopping times. Prove that the random variables \(T_k^n-T_{k-1}^n\), \(k=1,2,\ldots\), are independent and identically distributed, and similarly the random variables \(B_{T_k^n}-B_{T_{k-1}^n}\), \(k=1,2,\ldots\), are independent and identically distributed.

  3. We set \(X_k^n=2^nB_{T_k^n}\) for every \(k\in\mathbb{Z}_+\). Verify that \((X_k^n)_{k\in\mathbb{Z}_+}\) is a simple random walk on \(\mathbb{Z}\).

  4. Show that there exists a constant \(c>0\) such that, for every \(t\ge0\),

    \[ \lim_{n\to\infty}T_{\lfloor 2^{2n}t\rfloor}^n=ct, \qquad \text{a.s.} \]

  5. Infer that, for every \(t>0\),

    \[ \lim_{n\to\infty} \sup_{0\le s\le t} \left| \frac1{2^n}X_{\lfloor 2^{2n}s\rfloor}^n-B_{cs} \right| =0, \qquad \text{a.s.} \]

    and finally that \(c=1\).

Exercise 14.11 For \(\alpha\in(0,1]\), a continuous function \(f:[0,1]\to\mathbb{R}\) is said to be \(\alpha\)-Holder if there exists a constant \(C\) such that \(|f(s)-f(t)|\le C|s-t|^\alpha\) for every \(s,t\in[0,1]\).

  1. Prove that the function \([0,1]\ni t\mapsto B_t(\omega)\) is a.s. not \(\frac12\)-Holder.

  2. Let \(\delta\in(0,\frac12)\). Prove that a.s. there exists an integer \(n_0(\omega)\) such that the bound

    \[ |B_{k2^{-n}}-B_{(k-1)2^{-n}}|\le2^{-n\delta} \]

    holds for every \(n\ge n_0(\omega)\) and every \(k\in\{1,\ldots,2^n\}\).

  3. Prove that the function \([0,1]\ni t\mapsto B_t(\omega)\) is a.s. \(\delta\)-Holder.

Exercise 14.12 Let \(d\ge3\) and let \(B\) be a \(d\)-dimensional Brownian motion started from \(0\). Fix \(A>1\) and \(\delta\in(0,1)\), and set

\[ \mathcal{R}_{\delta,A}=\{x\in\mathbb{R}^d:\delta\le|x|\le A\}. \]

For every \(n\ge1\) and every \(k_1,\ldots,k_d\in\mathbb{Z}\), define the cube

\[ C_{k_1,\ldots,k_d}^{(n)} = [k_12^{-n},(k_1+1)2^{-n}] \times[k_22^{-n},(k_2+1)2^{-n}] \times\cdots\times [k_d2^{-n},(k_d+1)2^{-n}]. \]

Set

\[ N_n = \operatorname{card} \left\{ (k_1,\ldots,k_d)\in\mathbb{Z}^d: C_{k_1,\ldots,k_d}^{(n)}\cap\mathcal{R}_{\delta,A}\cap\{B_t,t\ge0\}\ne\varnothing \right\}. \]

  1. Prove that there is a constant \(K\) depending on \(\delta\) and \(A\) such that, for every \(n\ge1\),

    \[ \mathbb{E}[N_n]\le K2^{2n}. \]

  2. Recall from Exercise 3.4 the definition of the Hausdorff dimension \(\dim(A)\) of a subset \(A\) of \(\mathbb{R}^d\). Prove that \(\dim(\{B_t,t\ge0\})\le2\) a.s.

Exercise 14.13 Let \(D\) be a bounded domain in \(\mathbb{R}^d\), \(d\ge2\), and let \(g\) be a continuous function on \(\partial D\). Suppose that \(h\) solves the Dirichlet problem with boundary condition \(g\). Show that, for every \(x\in D\),

\[ h(x)=\mathbb{E}_x[g(B_T)], \]

with the notation of Theorem 14.23 (in particular \(T=\inf\{t\ge0:B_t\notin D\}\)).

Exercise 14.14 Let \(d\ge2\), and let \(D=\{x\in\mathbb{R}^d:0<|x|<1\}\) be the punctured open unit ball. Define \(g:\partial D\to\mathbb{R}\) by setting \(g(x)=1\) if \(|x|=1\) and \(g(0)=0\). Prove that the Dirichlet problem in \(D\) with boundary condition \(g\) has no solution. (Hint: Use the result of the preceding exercise.)

Exercise 14.15 Let \(d\ge3\). Let \(K\) be a compact subset of the closed unit ball, and \(D=\mathbb{R}^d\setminus K\). We assume that \(D\) is connected and satisfies the exterior cone condition. Let \(g:\partial D\to\mathbb{R}\) be a continuous function. We consider a function \(u\) that satisfies the Dirichlet problem in \(D\) with boundary condition \(g\), and assume that \(u\) is bounded.

We use the canonical representation of Brownian motion in \(\mathbb{R}^d\), and set \(T_K=\inf\{t\ge0:B_t\in K\}\in[0,\infty]\).

  1. Let \((R_n)_{n\in\mathbb{N}}\) be a sequence of real numbers in \((1,\infty)\) such that \(R_n\uparrow\infty\) as \(n\to\infty\). For every \(n\), set \(T_{(n)}=\inf\{t\ge0:|B_t|\ge R_n\}\). Prove that, for every \(n\ge1\) and every \(x\in D\) such that \(|x|<R_n\),

    \[ u(x) = \mathbb{E}_x[g(B_{T_K})\mathbf{1}_{\{T_K<T_{(n)}\}}] + \mathbb{E}_x[u(B_{T_{(n)}})\mathbf{1}_{\{T_{(n)}<T_K\}}]. \]

  2. Prove that, up to replacing the sequence \((R_n)_{n\in\mathbb{N}}\) by a subsequence, we can assume that there exists a constant \(\alpha\in\mathbb{R}\) such that, for every \(x\in\mathbb{R}\),

    \[ \lim_{n\to\infty}\mathbb{E}_x[u(B_{T_{(n)}})]=\alpha. \]

  3. Deduce from questions (1) and (2) that

    \[ \lim_{|x|\to\infty}u(x)=\alpha \]

    and then prove that, for every \(x\in D\),

    \[ u(x) = \mathbb{E}_x[g(B_{T_K})\mathbf{1}_{\{T_K<\infty\}}] + \alpha\,\mathbb{P}_x(T_K=\infty). \]

  4. Conversely, verify that, for any \(\alpha\in\mathbb{R}\), the right-hand side of the last display gives a solution of the Dirichlet problem in \(D\) with boundary condition \(g\).


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