2.11 A Brownian toolkit: results used before M08

Why this section exists M03 · 3.4 (continuous-time Kelly and drawdowns) and M04A · 4A.2–4A.10 (touch probabilities, stops, trailing stops, sizing rules) use a handful of facts about Brownian motion. Brownian motion and Itô's lemma are developed in M08 · 8.5. This section states the facts that are needed earlier, without proofs, with the place where each is proved or derived. Read it once now and return to it when a later section cites it.

2.11.1Brownian motion and its relatives#

Objects A standard Brownian motion $W$ has $W_0 = 0$, continuous paths, and independent increments with $W_t - W_s \sim N(0, t-s)$ for $s < t$. Arithmetic Brownian motion with drift is $X_t = x_0 + \mu t + \sigma W_t$; for log prices, $\mu$ is the log drift $m - \sigma^2/2$ of 2.2. Geometric Brownian motion is $S_t = S_0\exp\big((m - \sigma^2/2)t + \sigma W_t\big)$, so $\ln S_t$ is an arithmetic Brownian motion (M08 Proposition 8.5.3). Write $\gamma = 2\mu/\sigma^2$ and $M_t = \max_{s \le t} X_s$.

2.11.2Results B1–B9#

#	Result (stated, not proved here)	Used in	Proved in
B1	Scaling. $W_{ct}$ has the law of $\sqrt c\,W_t$; standard deviations grow as $\sqrt t$.	2.3, M04A · 4A.1	definition
B2	Martingales. $W_t$, $W_t^2 - t$ and $\exp(\theta W_t - \theta^2 t/2)$ are martingales; so are $X_t - \mu t$ and, for $\mu \ne 0$, $e^{-\gamma X_t}$ (the scale function of $X$).	M03 · 3.4, M04A · 4A.6	M04A · 4A.6 (via Itô's lemma)
B3	Optional stopping. If $Z$ is a continuous martingale and $\tau$ a stopping time with $Z_{t\wedge\tau}$ bounded (or $\mathbb E\tau < \infty$ for $X_t - \mu t$), then $\mathbb E Z_\tau = Z_0$. In particular $\mathbb E[X_\tau - x_0] = \mu\,\mathbb E\tau$ (Wald's identity).	M03 · 3.4, M04A · 4A.6, 4A.8	M04A · 4A.6; Karatzas & Shreve (1991) ch. 1
B4	Reflection principle ($\mu = 0$, $a > 0$): $P(M_t - x_0 \ge a) = 2P(X_t - x_0 \ge a) = 2\Phi\big(-a/(\sigma\sqrt t)\big)$.	M04A · 4A.2, 4A.4, 4A.7	M04A · 4A.6
B5	First passage with drift ($a > 0$): $P(M_t - x_0 \ge a) = \Phi\!\Big(\dfrac{\mu t - a}{\sigma\sqrt t}\Big) + e^{\gamma a}\,\Phi\!\Big(\dfrac{-a - \mu t}{\sigma\sqrt t}\Big)$.	M04A · 4A.6–4A.7	M04A · 4A.6 (Girsanov)
B6	Ever reaching a level. For $\mu < 0$ and $a > 0$, $P(\sup_t X_t - x_0 \ge a) = e^{\gamma a} = e^{-2	\mu	a/\sigma^2}$; for $\mu \ge 0$ the probability is 1.
B7	Two barriers. Started at 0 with a target at $+a$ and a stop at $-b$: $P(\text{target first}) = \dfrac{1 - e^{\gamma b}}{e^{-\gamma a} - e^{\gamma b}}$, which is $b/(a+b)$ when $\mu = 0$; then $\mathbb E\tau = ab/\sigma^2$.	M04A · 4A.6, 4A.10; M15	M04A · 4A.6
B8	Itô integral. For an adapted, square-integrable $\theta$: $\mathbb E\int_0^T\theta_t\,dW_t = 0$ and $\mathbb E\big(\int_0^T\theta_t\,dW_t\big)^2 = \mathbb E\int_0^T\theta_t^2\,dt$ (the Itô isometry).	M04A · 4A.9, M07 · 7.2 (the Vasicek futures–forward calculation)	M08 · 8.5; Shreve, Stochastic Calculus for Finance II (2004), ch. 4
B9	Itô's lemma. For smooth $f$, $df(t, X_t) = \big(f_t + \mu f_x + \tfrac12\sigma^2 f_{xx}\big)dt + \sigma f_x\,dW_t$.	M03 · 3.4, M04A · 4A.6	M08 Proposition 8.5.2

How to use the table Every result is about continuous paths monitored continuously. Prices gap, and stops are often checked at discrete times; M04A (4A.10) gives the discrete-monitoring correction and M04A · 4A.4 the gap case. A formula from this table is a model of the clean case, against which the messy case is measured.