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I am trying to solve the following:

Define the characteristic functional of a random process $X_t, T=\mathbb{R}$ by $$L(\varphi)=E\left[\exp \left(i\int_{T}\varphi(t)X_{t}\, dt \right) \right],$$ where $\varphi(t)$ is continuous differentiable with compact support. Determine the characteristic functional for Brownian motion.

I want to compute $L(\varphi)=E[\exp(i\int_{T}\varphi(t)W_{t} \, dt)]$, where $W_t$ is a Brownian motion with continuous trajectories and independent gaussian distributed increments. How can I apply these facts to compute $L(\varphi)$? Am I supposed to estimate $\int_{T}\varphi(t)W_{t} \, dt$ using a Riemann sum?

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  • $\begingroup$ Are you integrating from $0$ to a certain time $T$? $\endgroup$ – BGM Apr 20 '17 at 4:20
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Choose $a<b$ such that the support of $\varphi$ is contained in $[a,b]$. Then $$\int_{\mathbb{R}} \varphi(t) W_t \, dt = \int_a^b \varphi(t) W_t \, dt.$$ Approximating the integral by Riemann sums we find

$$\int_{a}^b \varphi(t) W_t \, dt = \lim_{n \to \infty} \frac{b-a}{n} \sum_{k=0}^n \varphi(t_k^{(n)}) W(t_k^{(n)})$$

for $t_k^{(n)} := a+ (b-a) \frac{k}{n}$, $k=0,\ldots,n$. Therefore it follows from the dominated convergence theorem that

$$L(\varphi) = \lim_{n \to \infty} \mathbb{E} \exp \left(i \frac{b-a}{n} \sum_{k=0}^n \varphi(t_k^{(n)}) W(t_k^{(n)}) \right).$$

Since the vector $(W(t_k^{(n)}))_{k=0,\ldots,n}$ is Gaussian with mean $0$ and covariance matrix $C_n := (\min\{t_k^{(n)},t_j^{(n)}\})_{k,j=0,\ldots,n}$, the right-hand side can be calculated explicitly:

$$\begin{align*} L(\varphi) &= \lim_{n \to \infty} \exp \left(- \frac{1}{2} \frac{(b-a)^2}{n^2} \sum_{j=0}^n \sum_{k=0}^n \min\{t_k^{(n)},t_j^{(n)}\} \varphi(t_k^{(n)}) \varphi(t_j^{(n)}) \right) \\ &= \exp \left(- \frac{1}{2} \int_a^b \int_a^b \min\{s,t\} \varphi(s) \varphi(t) \, ds \, dt \right). \end{align*}$$

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