Martingales and the Itô Integral

Brownian paths have infinite length (6D.1 Brownian Motion), so the integral $\int_0^tf(s)\,dB_s$ cannot be defined path by path, as a Riemann–Stieltjes integral. Kiyosi Itô's idea (1944) was to define it as a limit in $L^2(\mathbb{P})$, using the independence of Brownian increments from the past, and to evaluate the integrand at the left endpoint of each time step: you choose how much to bet before you see the outcome. The resulting integral is a martingale, a fair game, and it obeys a new chain rule, Itô's formula, with an extra second-order term coming from $(dB)^2 = dt$:

That extra term is where the Laplacian enters probability, and through it the heat equation. It is also the term that makes the Black–Scholes equation (6D.6 Black–Scholes) a heat equation rather than a transport equation.

By the end of this chapter you will be able to:

• define conditional expectation, as an $L^2$ projection and in general, and compute with it;
• define martingales and stopping times, prove optional stopping for bounded stopping times, and explain why the doubling strategy fails;
• prove Doob's $L^2$ maximal inequality;
• define the Itô integral, prove the Itô isometry, and compute $\int_0^tB\,dB = \frac12(B_t^2 - t)$;
• prove Itô's formula for $f(B_t)$, and use its multidimensional form for Itô processes;
• state Girsanov's theorem and verify it for the one-dimensional marginal.

2.1You cannot beat a fair game

2.2Conditional expectation

Let $(\Omega, \mathcal F, \mathbb{P})$ be a probability space and $\mathcal G \subseteq \mathcal F$ a sub-$\sigma$-algebra, representing partial information: an event in $\mathcal G$ is one whose occurrence we know.

Existence. For $X \in L^2$, $L^2(\Omega, \mathcal G, \mathbb{P})$ is a closed subspace of $L^2(\Omega, \mathcal F, \mathbb{P})$, and the orthogonal projection $Y$ of $X$ onto it (4A.5 Hilbert Spaces and Lax–Milgram) satisfies $\mathbb{E}[(X - Y)Z] = 0$ for every $\mathcal G$-measurable $Z \in L^2$, in particular $Z = \mathbf 1_A$. So $\mathbb{E}[X\mid\mathcal G]$ is the best prediction of $X$ in mean square given the information $\mathcal G$. For $X \geq 0$ in $L^1$, the measure $\nu(A) = \mathbb{E}[X\mathbf 1_A]$ on $\mathcal G$ is absolutely continuous with respect to $\mathbb{P}$, and $Y = d\nu/d\mathbb{P}$ (Radon–Nikodym, 3A.4 Measures, Probability and Weights); split a general $X$ into positive and negative parts. Uniqueness: if $Y, Y'$ both qualify, $\mathbb{E}[(Y - Y')\mathbf 1_A] = 0$ for $A = \{Y > Y'\} \in \mathcal G$, so $Y \leq Y'$ almost surely, and symmetrically.

2.3Martingales and stopping times

A filtration is an increasing family of $\sigma$-algebras $(\mathcal F_t)_{t\geq0}$, the information available at time $t$. For Brownian motion we use $\mathcal F_t = \sigma(B_s : s \leq t)$ (completed by the null sets). A process $X_t$ is adapted if each $X_t$ is $\mathcal F_t$-measurable: its value at time $t$ is known at time $t$.

A stopping time is a random time $\tau$ with $\{\tau \leq t\} \in \mathcal F_t$ for all $t$: whether you have stopped by time $t$ is decided by what you have seen by time $t$. Hitting times of closed sets by continuous adapted processes are stopping times; "the last time $B$ is at $0$ before time $1$" is not.

The proof is a theorem about betting: $\mathbf 1_{\{\tau > k\}}$ is a strategy (play or not) decided with the information at time $k$, and a strategy applied to a fair game gives a fair game. The same proof shows that $\sum_kH_k(X_{k+1} - X_k)$ is a martingale for every bounded adapted "betting strategy" $H$: the martingale transform, the discrete Itô integral. For continuous-time martingales with continuous paths and bounded stopping times, optional stopping follows by approximating $\tau$ from above by stopping times with values in a finite grid (statement; Karatzas and Shreve, section 1.3). For unbounded $\tau$ it can fail (Exercise 2.15); one usually applies it to $\tau\wedge n$ and lets $n \to \infty$ with a convergence theorem.

2.4The Itô integral

Fix $T > 0$. Let $\mathcal L^2_{ad}$ be the space of processes $f(t, \omega)$, $0 \leq t \leq T$, that are adapted (and jointly measurable in $(t, \omega)$, more precisely progressively measurable) with $\mathbb{E}\int_0^Tf(t)^2\,dt < \infty$. A simple process has the form

with each $e_j$ an $\mathcal F_{t_j}$-measurable random variable in $L^2$. Its Itô integral is the obvious sum, with the integrand frozen at the left endpoint:

The left-endpoint rule is what makes the cross terms vanish: each bet $e_j$ is placed before the increment $\Delta_j$ it multiplies is revealed. So the Itô integral is a linear isometry from simple processes, with the norm of $L^2(\Omega\times[0, T])$, into $L^2(\Omega)$.

Extension. Simple processes are dense in $\mathcal L^2_{ad}$ (statement; Øksendal, Lemma 3.1.5. For $f$ adapted with continuous bounded paths it is elementary: $\phi_n = \sum_jf(t_j)\mathbf 1_{(t_j, t_{j+1}]}$ on finer and finer partitions converges to $f$ in $L^2(\Omega\times[0, T])$ by dominated convergence.) A linear isometry extends uniquely to the closure (4A.1 Banach Spaces and Bounded Operators), so for $f \in \mathcal L^2_{ad}$

for any simple $\phi_n \to f$ in $\mathcal L^2_{ad}$, and the isometry $\mathbb{E}\big(\int f\,dB\big)^2 = \mathbb{E}\int f^2\,dt$ and $\mathbb{E}\int f\,dB = 0$ persist.

2.5Itô's formula

The formula holds for all $f \in C^2$ (approximate $f$ by functions in $C^3_b$, and localise with stopping times to handle growth; Øksendal, section 4.1). Note the structure: $f(B_t) - \frac12\int_0^tf''(B_s)\,ds$ is a martingale. The operator $\frac12\frac{d^2}{dx^2}$ appearing here is the generator of Brownian motion, and this martingale property is the bridge from probability to PDE that 6D.3 Stochastic Differential Equations and Feynman–Kac exploits.

Itô processes and the general formula. An Itô process is

with adapted $\mu$, $\sigma$ (integrable and square-integrable in time). For several Brownian motions $B^1, \ldots, B^m$ and $X$ with values in $\mathbb{R}^n$, $dX^i = \mu^i\,dt + \sigma^i_\alpha\,dB^\alpha$. The general Itô formula (statement; Øksendal, Theorem 4.2.1) says that for $f(t, x)$ of class $C^{1,2}$,

where the products are computed by the rules $dt\cdot dt = dt\cdot dB^\alpha = 0$ and $dB^\alpha\,dB^\beta = \delta^{\alpha\beta}dt$, so $dX^i\,dX^j = (\sigma\sigma^T)^{ij}\,dt$. The multiplication table is the quadratic variation theorem in shorthand, extended to independent Brownian motions (whose cross variation vanishes). A useful special case is the product rule $d(XY) = X\,dY + Y\,dX + dX\,dY$.

2.6Changing the measure

Two more theorems complete the toolkit; we state them.

Under $\mathbb{P}$, $\tilde B$ is a Brownian motion with drift $\theta$. Girsanov's theorem says a drift can be removed by re-weighting the paths, with the exponential martingale as weight. For the one-dimensional marginal this is a completion of the square (Exercise 2.19); for the whole path it follows from Lévy's characterisation. In finance this is the passage from the real-world measure, under which stocks drift at their expected return $\mu$, to the risk-neutral measure, under which they drift at the interest rate $r$ (6D.6 Black–Scholes). Robert Cameron and William Martin found the deterministic-drift case in 1944, and Igor Girsanov the general one in 1960.

2.7History

Jean Ville introduced martingales into probability in 1939, and Joseph Doob developed their theory, with the maximal inequality and the stopping theorems, in the 1940s and in his 1953 book. Kiyosi Itô defined the stochastic integral in 1944 and proved his formula in 1951. In 2000, a sealed envelope that the young mathematician Wolfgang Doeblin had deposited with the Paris Academy of Sciences in 1940, shortly before his death as a soldier, was opened; it contained a version of the change of variables formula for diffusions, which is why some authors (Shreve among them) speak of the Itô–Doeblin formula. Ruslan Stratonovich introduced his integral in the 1960s. Cameron and Martin (1944) and Girsanov (1960) proved the change of measure theorems.

2.8Exercises

Solution

With $B_t = B_s + \Delta$: $\mathbb{E}[(B_s + \Delta)^3\mid\mathcal F_s] = B_s^3 + 3B_s^2\mathbb{E}\Delta + 3B_s\mathbb{E}\Delta^2 + \mathbb{E}\Delta^3 = B_s^3 + 3(t - s)B_s$. So $\mathbb{E}[B_t^3 - 3tB_t\mid\mathcal F_s] = B_s^3 + 3(t - s)B_s - 3tB_s = B_s^3 - 3sB_s$.

Solution

(a) The bet at time $k$ is $H_k = 2^k$ if no win yet (decided at time $k$), so $X$ is a martingale transform of a fair game, hence a martingale. At the first win at toss $\tau$, the net is $2^{\tau-1} - (1 + 2 + \cdots + 2^{\tau-2}) = 1$. $\tau$ is finite but unbounded. (b) $X_{\tau\wedge N} = 1$ with probability $1 - 2^{-N}$, and $-(2^N - 1)$ with probability $2^{-N}$ (all $N$ tosses lost). So $\mathbb{E}X_{\tau\wedge N} = 1 - 2^{-N} - (1 - 2^{-N}) = 0$, as optional stopping says.

Solution

$\mathbb{E}\exp(\theta B_{\tau_a\wedge n} - \lambda(\tau_a\wedge n)) = 1$. Up to $\tau_a$, $B \leq a$, so the integrand is at most $e^{\theta a}$; as $n \to \infty$ it tends to $e^{\theta a - \lambda\tau_a}$ on $\{\tau_a < \infty\}$ (probability $1$, 6D.1 Brownian Motion). By dominated convergence, $e^{\theta a}\mathbb{E}e^{-\lambda\tau_a} = 1$.

Solution

The sum is $\sum B_{t_i}\Delta_i + \sum(B_{m_i} - B_{t_i})\Delta_i$. The first part tends to $\frac12(B_t^2 - t)$. Write $\Delta_i = (B_{m_i} - B_{t_i}) + (B_{t_{i+1}} - B_{m_i})$; the second part is $\sum(B_{m_i} - B_{t_i})^2 + \sum(B_{m_i} - B_{t_i})(B_{t_{i+1}} - B_{m_i})$. The first of these is a quadratic variation over half-intervals, tending to $\frac t2$; the second has mean zero and variance $\sum\frac{\delta_i}{2}\cdot\frac{\delta_i}{2} \to 0$. Total: $\frac12B_t^2$.

Solution

$d(B^4) = 4B^3\,dB + 6B^2\,dt$; the stochastic integral has mean zero, so $\mathbb{E}B_t^4 = 6\int_0^t\mathbb{E}B_s^2\,ds = 6\int_0^ts\,ds = 3t^2$. Isometry: $\mathbb{E}\int_0^tB_s^2\,ds = \frac{t^2}{2}$. Directly: $\operatorname{Var}\big(\frac12(B_t^2 - t)\big) = \frac14\operatorname{Var}(B_t^2) = \frac14(3t^2 - t^2) = \frac{t^2}{2}$.

Solution

$\mathbb{E}_{\mathbb{P}}[Z_Tg(B_T + \theta T)] = \int g(x + \theta T)e^{-\theta x - \theta^2T/2}\frac{e^{-x^2/(2T)}}{\sqrt{2\pi T}}dx$, and $-\theta x - \frac{\theta^2T}{2} - \frac{x^2}{2T} = -\frac{(x + \theta T)^2}{2T}$. Substitute $y = x + \theta T$: $\int g(y)\frac{e^{-y^2/(2T)}}{\sqrt{2\pi T}}dy$. With $g = 1$, $\mathbb{E}Z_T = 1$.

Solution

$d|B|^2 = \sum_i(2B^i\,dB^i + dt) = 2B\cdot dB + n\,dt$, so $|B_t|^2 - nt - |x|^2$ is a stochastic integral, a martingale. Optional stopping at the bounded time $\tau\wedge k$: $\mathbb{E}_x|B_{\tau\wedge k}|^2 - n\mathbb{E}_x(\tau\wedge k) = |x|^2$. As $k \to \infty$, $\mathbb{E}(\tau\wedge k) \to \mathbb{E}\tau$ (monotone convergence) and $|B_{\tau\wedge k}|^2 \to R^2$ boundedly ($\tau < \infty$ and $|B| \leq R$ before $\tau$). So $R^2 - n\mathbb{E}_x\tau = |x|^2$.

Solution

(a) $df = \partial_if\,dX^i + \frac12\partial_{ij}f\,dX^idX^j$, with $dX^idX^j = \sigma^i_\alpha\sigma^j_\alpha\,dt = g^{ij}\,dt$. So $df = \partial_if\sigma^i_\alpha\,dB^\alpha - \frac12\partial_ifg^{jk}\Gamma^i_{jk}\,dt + \frac12g^{ij}\partial_{ij}f\,dt$, and the $dt$ terms are $\frac12g^{ij}(\partial_{ij}f - \Gamma^k_{ij}\partial_kf)$ after renaming indices. (b) Without it the generator would be $\frac12g^{ij}\partial_{ij}$, which is not coordinate-invariant. The drift makes the generator the invariant operator $\frac12\Delta_g$; it is $\frac12\Delta_g$ of the coordinate functions, the tension field of the coordinate chart.