Let \( W_t \) be a standard Wiener process (Brownian motion), and define

\[ X_t = \int_0^t W_\tau\,d\tau. \]

We answer two questions:

1. Distribution of \( X_t \)

The process \( X_t \) is a stochastic integral of Brownian motion with respect to Lebesgue measure (not Itô). It can be evaluated in distribution as follows:

  • \( X_t \) is a Gaussian random variable (being a linear transformation of a Gaussian process).
  • We compute its mean and variance.

Mean:

\[ \mathbb{E}[X_t] = \mathbb{E}\left[\int_0^t W_\tau\,d\tau\right] = \int_0^t \mathbb{E}[W_\tau]\,d\tau = \int_0^t 0\,d\tau = 0. \]

Variance:

Using the covariance of Brownian motion:

\[ \text{Var}(X_t) = \mathbb{E}[X_t^2] = \mathbb{E}\left[ \left( \int_0^t W_\tau\,d\tau \right)^2 \right] = \int_0^t\int_0^t \mathbb{E}[W_s W_u]\,ds\,du. \]

Since \( \mathbb{E}[W_s W_u] = \min(s, u) \), we compute:

\[ \text{Var}(X_t) = \int_0^t \int_0^t \min(s, u)\,ds\,du = \frac{t^3}{3}. \]

Thus,

\[ X_t \sim \mathcal{N}\left(0, \frac{t^3}{3} \right). \]

2. Is \( {X_t}_{t \ge 0} \) a martingale?

We check whether \( X_t \) satisfies the martingale property with respect to the natural filtration \( \mathcal{F}_t = \sigma(W_s: s \le t) \):

We evaluate \( \mathbb{E}[X_t \mid \mathcal{F}_s] \) for \( s < t \). Note that:

\[ X_t = \int_0^t W_\tau\,d\tau = \int_0^s W_\tau\,d\tau + \int_s^t W_\tau\,d\tau = X_s + \int_s^t W_\tau\,d\tau. \]

Then,

\[ \mathbb{E}[X_t \mid \mathcal{F}s] = X_s + \mathbb{E}\left[ \int_s^t W\tau\,d\tau \mid \mathcal{F}_s \right]. \]

However, for \( \tau > s \), \( W_\tau \) is not \( \mathcal{F}_s \)-measurable. In fact,

\[ \mathbb{E}[W_\tau \mid \mathcal{F}_s] = W_s, \]

so:

\[ \mathbb{E}\left[ \int_s^t W_\tau\,d\tau \mid \mathcal{F}s \right] = \int_s^t \mathbb{E}[W\tau \mid \mathcal{F}_s]\,d\tau = \int_s^t W_s\,d\tau = (t - s)W_s. \]

Therefore,

\[ \mathbb{E}[X_t \mid \mathcal{F}_s] = X_s + (t - s)W_s \ne X_s. \]

So \( X_t \) is not a martingale.

Conclusion

  • \( X_t = \int_0^t W_\tau\,d\tau \sim \mathcal{N}(0, t^3/3) \)
  • \( X_t \) is not a martingale with respect to the natural filtration of \( W_t \)

Reference