Drift

In the stochastic differential equation $dX_t = \mu_t\,dt + \sigma_t\,dW_t$, the coefficient $\mu_t$ of the $\mu_t\,dt$ term is the drift. If the drift is a constant $\mu$ and $\sigma$ is also constant, then $X_t = \mu t + \sigma W_t$, which the MIT notes call Brownian motion with drift $\mu$. Its expected value is $E[X_t] = \mu t$; the $W_t$ term does not affect the mean¹².

For example, with $\mu = 2$ (per year) and $X_0 = 100$, the expected value after 3 years is $100 + 2 \times 3 = 106$. The actual value will be above or below this by the movement of $W_t$.

The drift need not be constant and can depend on the current value. In the Hull-White model, the drift of the short rate is $\theta(t) - a r_t$, which expresses mean reversion pulling the rate toward the level $\theta(t)/a$³. The drift of the same process also changes when the probability measure is changed. In the LIBOR market model, under the forward measure that takes the bond maturing at $T_{M+1}$ as the reference, the drift of the last forward rate is 0, while the drifts of the other forward rates are not 0⁴.

Notes

1. Massachusetts Institute of Technology, "Lecture 18: Itō Calculus", Example 1.3 (i) (page 2) and Section 3, "Change of measure" (pages 5 to 7). The text replaces the source's $B_t$ with $W_t$. ↩

2. Massachusetts Institute of Technology, "Lecture 17: Stochastic Processes II", Example 2.6 (page 5; the expected value of Brownian motion with drift $\mu$ is $E[X(t)] = \mu t$). The example is calculated from this formula. ↩

3. "Interest Rate Modelling and Derivative Pricing" (Humboldt-Universität zu Berlin, Sebastian Schlenkrich, winter semester 2020/21), Part IV, slide p. 254 (the Hull-White equation $dr = (\theta(t) - a r)\,dt + \sigma\,dW$). That $\theta(t)/a$ is the level follows from the equation. ↩

4. Martin Haugh, "Market Models" (Columbia University, Term Structure Models: IEOR E4710, 2010), page 6, equation (14), and page 8, equations (24) and (25) and Remark 3. ↩