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Consider the following sequence of SDEs:

$dX^n_t = \sin(nX^n_t)dt + dW_t, X^n_0 = 0\,\,\,$

Show that the solutions $X^n$ converge in finite dimensional distribution to Brownian Motion.

I have been working on this for a long time and can't get anywhere. Any help would be appreciated. Thank you.

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  • $\begingroup$ What is $W_t$ ? $\endgroup$ – gmath Oct 27 '14 at 19:15
  • $\begingroup$ @gmath $(W_t)_{t \geq 0}$ is a (one-dimensional) Brownian motion. Seamus: Is this an exercise from a book? $\endgroup$ – saz Oct 27 '14 at 19:25
  • $\begingroup$ No. This is not from any book that I know of. It was asked on an old probability exam. $\endgroup$ – user30201 Oct 27 '14 at 23:15
  • $\begingroup$ I would start by solving the SDE using Ito's formula for some good f(x). $\endgroup$ – TKM Oct 29 '14 at 17:53
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    $\begingroup$ @TKM So what's a "good $f(x)$" in this particular case? $\endgroup$ – saz Oct 30 '14 at 6:19
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Surely not an exam proof, what is going to come.

We will use Theorem 5.4.1 from Ethier and Kurtz, Markov Processes, Wiley, 1986 (EK86); the important part is quoted below. Let us first write down the generator $A_n$ of the process $X^{(n)}$ \begin{equation} A_n f(x) = \sin (nx) f'(x) + \frac{1}{2} f''(x) , x \in \mathbb{R}. \end{equation} The expected limit process, Brownian motion, has generator $A f(x) = \frac{1}{2}f''(x)$, $x \in \mathbb{R}$. Both definitions are understood for functions $f \in C_c^\infty(\mathbb{R})$, the compactly supported, infinitely often differentiable functions on $\mathbb{R}$.

Lemma 1: The sequence $\{X^{(n)}_t)_{t\geq 0} : n \in \mathbb{N}\}$ is tight in the Skorohod space $D([0,\infty), \mathbb{R})$.

Lemma 2: For any $f \in C_c^\infty(\mathbb{R})$ we find a function $f_n$ such that \begin{equation} \| f_n - f\|_\infty \to 0, \qquad \| A_n f_n - A f \|_\infty \to 0 \end{equation} as $n \to \infty$.

Proof of your statement : Let $Y=(Y_t)_{t\geq 0}$ be a limit point of the sequence $\{X^{(n)}_t)_{t\geq 0} : n \in \mathbb{N}\}$ which exists by Lemma 1. Then Lemma 2 (combined with Theorem 5.4.1 in EK86) tells us that $Y$ solves the martingale problem related to the generator $A$. That means that $Y$ is a Brownian motion. Since this holds for any limit point $Y$, we know that $\{X^{(n)}_t)_{t\geq 0} : n \in \mathbb{N}\}$ converges weakly in Skorohod space to Brownian motion. This implies converges in finite dimensional distributions by Theorem 3.7.5 in EK86.

Proof of Lemma 2: Fix $f \in C_c^2(\mathbb{R})$ and let $a := \inf \text{supp } f$, $A:= \sup \text{supp } f$ such that $f(x) = 0$ for all $x \not \in [a,A]$. \begin{equation} f_n (x) := f(x) - 2\int_a^x \exp (2n^{-1} \cos (ny) ) \left(-C + \int_a^y \exp(-2n^{-1}\cos(nz)) \sin(nz) f'(z) d z \right) . \end{equation} Here $C = C(n) = \int_a^A \exp(-2n^{-1}\cos(nz)) \sin(nz)f'(z) d z$.

It is easy to see that $A_n f_n(x) - Af(x) = 0$ for any $x \in \mathbb{R}$. So we are left with the verification of $\sup_{x} | f_n(x) -f(x)| \to 0$ as $n\to \infty$. Note that since $1-(2/n) \leq \exp(-n^{-1} \xi) \leq 1+(4/n)$ for any $\xi \in [-2,2]$, $n \geq 4$ we have \begin{align} \int_a^y & \exp(-2n^{-1}\cos(nz)) 2\sin(nz) f'(z) d z \\ & = - \int_a^y \exp(-2n^{-1}\cos(nz)) f''(z) dz + f'(y) \exp(-2n^{-1}\cos(ny)) - f'(a) \exp(-2n^{-1}) \\ & = -\int_a^y f''(z) dz + f'(y) - f'(a) + \mathcal{O}\left(n^{-1} \|f''\|_\infty + n^{-1} \|f'\|_\infty \right) \\ & = \mathcal{O}\left(n^{-1} \|f''\|_\infty + n^{-1} \|f'\|_\infty \right) \end{align} for any $y \in \mathbb{R}$. This allows to say that for $x \in [a,A]$: \begin{align} f(x) - f_n(x) & = 2\int_a^x \exp(-n^{-1}\cos (ny)) dy \ \cdot \mathcal{O}\left(n^{-1} \|f''\|_\infty + n^{-1} \|f'\|_\infty \right) \\ & \leq 4 |A-a| \cdot \mathcal{O}\left(n^{-1} \|f''\|_\infty + n^{-1} \|f'\|_\infty \right) \to 0 \end{align} as $n \to \infty$. If we are not in the interval of support, i.e. $x \not \in [a,A]$, then we have for $x \geq A$: \begin{align} f(x) -f_n(x) &= -f_n(x) = 2\int_a^x \exp (2n^{-1} \cos (ny) ) \left(-C + \int_a^y \exp(-2n^{-1}\cos(nz)) \sin(nz) f'(z) d z \right) \\ & = |A-a| \cdot \mathcal{O}\left(n^{-1} \|f''\|_\infty + n^{-1} \|f'\|_\infty \right) \\ & \qquad + 2\int_A^x \exp (2n^{-1} \cos (ny) ) \left( -C + \int_a^A \cdots dz + \int_A^y \cdots dz \right) \, . \end{align} The first term vanishes as before uniformly in $x$, which leaves us with the terms of the last line. By definition of $C$ we are only left with the last term which is zero anyway, since we are not in the support of $f$ anymore. A similar, even easier reasoning applies for $x <a$. The previous lines include saying that $f_n$ is a bounded function.

Proof of Lemma 1: Use Remark 5.4.2 in EK86 which requires standard conditions on $A$ which are fulfilled here and the compact containment condition for $\{X^{(n)}_t)_{t\geq 0} : n \in \mathbb{N}\}$. For the compact containment condition hold define two processes $Z^{i}$, $i = -1, 1$ solving $dZ_t^i = i\ dt + dB_t$. Then Theorem 5.2.18 in Karatzas and Shreve, Brownian Motion and Stochastic Calculus, 1991 allows to say we can construct probability spaces $(\Omega^n,\mathcal{A}^n,P^n)$ such that $Z_t^{-1} \leq X_t^n \leq Z_t^1$ for all $t \geq 0$ $P^n$ almost surely for all $n \in \mathbb{N}$.

Theorem 5.4.1 in EK86

Let $A \subset C_b(\mathbb{R}) \times C_b(\mathbb{R})$ and $A_n \subset \mathbb{B}_b(\mathbb{R}) \times \mathbb{B}_b(\mathbb{R})$, the bounded measurable functions, $n \in \mathbb{N}$. Suppose that for each $(f,g) \in A$ there is an $(f_n,g_n) \in A_n$ such that \begin{equation} \| f_n -f \| \to 0, \quad \| g_n -g\| \to 0. \end{equation} If for each $n\in \mathbb{N}$, $X^{(n)}$ is a solution of the $A_n$ martingale problem with paths in the Skorohod space and $X_n \Rightarrow X$, then $X$ is a solution to the $A$-martingale problem.

Note: The authors of EK86 prefer to denote a generator $A: D(A) \to \mathbb{B}_b(E)$ by its graph $\{(f,Af): f \in D(A) \} \subset (\mathbb{B}_b(E),\mathbb{B}_b(E))$.

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  • $\begingroup$ wow, this is much harder than I thought it would be. Thanks! $\endgroup$ – user30201 Nov 10 '14 at 2:22

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