10
$\begingroup$

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 beginner 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?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$
    – HMPanzo
    Commented Sep 27, 2014 at 23:04

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .