# Joint law of the time integral of Brownian motion and its maximum

Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively:

$$M_t=\max_{0\leq s\leq t}\,W_s$$
$$I_t=\int\limits_0^tW_s\,\mathrm{d}s$$

The laws of $M_t$ and $I_t$ can be easily derived by any beginnner studying stochastic processes. However, I haven't seen anything in the literature about their joint law. Is the joint law of $M_t$ and $I_t$ known?

## 1 Answer

You can find explicit formula of the joint density of these two variables in (2.29)---(2.31) of the following articles. The authors cited the formula from other books.

Gerber, Hans U., Elias SW Shiu, and Hailiang Yang. "Valuing equity-linked death benefits and other contingent options: a discounted density approach." Insurance: Mathematics and Economics 51.1 (2012): 73-92.

• Unfortunately, the formula appearing in that paper is actually for the joint density of $M_t$ and $X_t$ where $M_t$ is the running maximum of $X_t$ and $X_t$ is a Brownian motion with constant diffusion and drift coefficients $\sigma >0$ and $\mu$, respectively. – HMPanzo Sep 27 '14 at 23:04