Brownian Drawdowns

BM drawdown
Figure 1: Brownian motion and drawdown

Here, I apply the theory outlined in the previous post to fully describe the drawdown point process of a standard Brownian motion. In fact, as I will show, the drawdowns can all be constructed from independent copies of a single ‘Brownian excursion’ stochastic process. Recall that we start with a continuous stochastic process X, assumed here to be Brownian motion, and define its running maximum as {M_t=\sup_{s\le t}X_s} and drawdown process {D_t=M_t-X_t}. This is as in figure 1 above.

Next, {D^a} was defined to be the drawdown ‘excursion’ over the interval at which the maximum process is equal to the value {a \ge 0}. Precisely, if we let {\tau_a} be the first time at which X hits level {a} and {\tau_{a+}} be its right limit {\tau_{a+}=\lim_{b\downarrow a}\tau_b} then,

\displaystyle  D^a_t=D_{({\tau_a+t})\wedge\tau_{a+}}=a-X_{({\tau_a+t)}\wedge\tau_{a+}}.

Next, a random set S is defined as the collection of all nonzero drawdown excursions indexed the running maximum,

\displaystyle  S=\left\{(a,D^a)\colon D^a\not=0\right\}.

The set of drawdown excursions corresponding to the sample path from figure 1 are shown in figure 2 below.

BM drawdown excursions
Figure 2: Brownian drawdown excursions

As described in the post on semimartingale local times, the joint distribution of the drawdown and running maximum {(D,M)}, of a Brownian motion, is identical to the distribution of its absolute value and local time at zero, {(\lvert X\rvert,L^0)}. Hence, the point process consisting of the drawdown excursions indexed by the running maximum, and the absolute value of the excursions from zero indexed by the local time, both have the same distribution. So, the theory described in this post applies equally to the excursions away from zero of a Brownian motion.

Before going further, let’s recap some of the technical details. The excursions lie in the space E of continuous paths {z\colon{\mathbb R}_+\rightarrow{\mathbb R}}, on which we define a canonical process Z by sampling the path at each time t, {Z_t(z)=z_t}. This space is given the topology of uniform convergence over finite time intervals (compact open topology), which makes it into a Polish space, and whose Borel sigma-algebra {\mathcal E} is equal to the sigma-algebra generated by {\{Z_t\}_{t\ge0}}. As shown in the previous post, the counting measure {\xi(A)=\#(S\cap A)} is a random point process on {({\mathbb R}_+\times E,\mathcal B({\mathbb R}_+)\otimes\mathcal E)}. In fact, it is a Poisson point process, so its distribution is fully determined by its intensity measure {\mu={\mathbb E}\xi}.

Theorem 1 If X is a standard Brownian motion, then the drawdown point process {\xi} is Poisson with intensity measure {\mu=\lambda\otimes\nu} where,

  • {\lambda} is the standard Lebesgue measure on {{\mathbb R}_+}.
  • {\nu} is a sigma-finite measure on E given by
    \displaystyle  \nu(f) = \lim_{\epsilon\rightarrow0}\epsilon^{-1}{\mathbb E}_\epsilon[f(Z^{\sigma})] (1)

    for all bounded continuous continuous maps {f\colon E\rightarrow{\mathbb R}} which vanish on paths of length less than L (some {L > 0}). The limit is taken over {\epsilon > 0}, {{\mathbb E}_\epsilon} denotes expectation under the measure with respect to which Z is a Brownian motion started at {\epsilon}, and {\sigma} is the first time at which Z hits 0. This measure satisfies the following properties,

    • {\nu}-almost everywhere, there exists a time {T > 0} such that {Z > 0} on {(0,T)} and {Z=0} everywhere else.
    • for each {t > 0}, the distribution of {Z_t} has density
      \displaystyle  p_t(z)=z\sqrt{\frac 2{\pi t^3}}e^{-\frac{z^2}{2t}} (2)

      over the range {z > 0}.

    • over {t > 0}, {Z_t} is Markov, with transition function of a Brownian motion stopped at zero.

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