Stochastic processes · The Lumenary

Lens

MotivationH

Random functions of time

In 1827 the botanist Robert Brown watched particles from pollen grains in water under a microscope and saw them jitter ceaselessly, in no preferred direction. In 1900 Louis Bachelier, in a thesis on the Paris stock exchange, modelled prices by a process with independent Gaussian increments. In 1905 Albert Einstein, and independently Marian Smoluchowski, explained Brown's motion by molecular collisions and predicted that the mean square displacement grows linearly in time; Jean Perrin's measurements (1908–1909) confirmed it and fixed Avogadro's number, helping to settle the reality of atoms. Meanwhile the engineer Agner Krarup Erlang, designing the Copenhagen telephone exchange (1909), and the actuary Filip Lundberg, modelling insurance claims (1903), needed a model for events arriving "at random" in continuous time.

These are stochastic processes: random functions $t \mapsto X_t$ . They raise questions the previous lesson could not ask. What does it mean to choose a function at random? Which properties of a random path hold almost surely? And since Brownian paths turn out to be nowhere differentiable, how can one write — let alone solve — a differential equation driven by them? The answer to the last question, Itô's stochastic calculus (1944), is now used from population genetics to quantitative finance.

IntuitionH

Two limits of coin tossing

Both basic processes are scaling limits of coin tossing.

Rare events. Divide $[0, t]$ into $n$ short slots and let an "arrival" occur in each slot independently with probability $\lambda t / n$ . The number of arrivals is binomial, and as $n \to \infty$ it becomes Poisson with mean $\lambda t$ (the law of rare events). Counts in disjoint intervals are independent, and the waiting time for the next arrival does not care how long you have already waited. This is the Poisson process: a random step function that jumps by $1$ at random times.

Many small steps. Take a simple random walk and speed it up: steps of size $1/\sqrt n$ every $1/n$ units of time. After time $t$ the position is a sum of $nt$ independent steps of variance $1/n$ , so by the central limit theorem it is approximately $N(0, t)$ . The limit is Brownian motion: a continuous random path with independent Gaussian increments. The scaling — space like the square root of time — is the source of everything strange about it. Over a time step $dt$ the path moves about $\sqrt{dt}$ , far more than $dt$ , so the path has no slope; but the squares of the increments add up to exactly the elapsed time. In the heuristic shorthand of stochastic calculus, $(dB)^2 = dt$ . A second-order Taylor expansion therefore keeps a term that ordinary calculus discards:

df(B_t) = f'(B_t) \, dB_t + \tfrac12 f''(B_t) \, (dB_t)^2 = f'(B_t) \, dB_t + \tfrac12 f''(B_t) \, dt.

This is Itô's formula. The rest of the lesson makes these heuristics into theorems.

DefinitionsH

Definitions

Definition— stochastic process, finite-dimensional distributions

A stochastic process on a probability space $(\Omega, \mathcal F, P)$ is a family $(X_t)_{t \in I}$ of random variables indexed by time, $I = [0, \infty)$ or $\{0, 1, 2, \dots\}$ . Its finite-dimensional distributions are the laws of the vectors $(X_{t_1}, \dots, X_{t_k})$ . For each $\omega$ , $t \mapsto X_t(\omega)$ is a sample path. A filtration $(\mathcal F_t)$ is an increasing family of sub-σ-algebras; $X$ is adapted if each $X_t$ is $\mathcal F_t$ -measurable, and an adapted integrable $X$ is a martingale if $\mathbb E[X_t \mid \mathcal F_s] = X_s$ almost surely for all $s \le t$ .

The Kolmogorov extension theorem says that any consistent family of finite-dimensional distributions (consistent under permuting and forgetting coordinates) is the family of some process. It does not control paths: the event "the path is continuous" depends on uncountably many coordinates and is not even measurable in the product σ-algebra. Path regularity has to be constructed separately — for Brownian motion by Wiener's or Lévy's construction, or by the Kolmogorov–Chentsov continuity theorem.

Definition— Poisson process

A Poisson process of rate $\lambda > 0$ is a process $(N_t)_{t \ge 0}$ with values in $\{0, 1, 2, \dots\}$ such that $N_0 = 0$ ; its paths are right-continuous, non-decreasing, and increase by jumps of size $1$ ; its increments over disjoint intervals are independent; and $N_t - N_s$ has the Poisson distribution with mean $\lambda (t - s)$ for $s \lt t$ :

P(N_t - N_s = k) = p_k\big(\lambda(t - s)\big), \qquad p_k(r) = e^{-r} \frac{r^k}{k!}.

Definition— continuous-time Markov chain

Let $S$ be a finite set. A generator (Q-matrix) on $S$ is a matrix $Q = (q_{ij})$ with $q_{ij} \ge 0$ for $i \ne j$ and rows summing to $0$ ; write $q_i = -q_{ii} = \sum_{j \ne i} q_{ij}$ . The continuous-time Markov chain with generator $Q$ sits in state $i$ for an exponential time with rate $q_i$ , then jumps to $j \ne i$ with probability $q_{ij}/q_i$ , independently of the past. Its transition matrix is $P(t)_{ij} = P(X_t = j \mid X_0 = i)$ .

Definition— Brownian motion

A standard Brownian motion is a real process $(B_t)_{t \ge 0}$ such that

$B_0 = 0$ ;
for $0 \le t_0 \lt t_1 \lt \dots \lt t_k$ the increments $B_{t_1} - B_{t_0}, \dots, B_{t_k} - B_{t_{k-1}}$ are independent;
for $s \lt t$ , $B_t - B_s$ is normal with mean $0$ and variance $t - s$ ;
almost every sample path $t \mapsto B_t(\omega)$ is continuous.

A Gaussian process is one whose finite-dimensional distributions are all multivariate normal; it is determined in law by its mean $\mathbb E X_t$ and covariance $\operatorname{Cov}(X_s, X_t)$ .

Definition— simple integrands and the Itô integral

Let $(B_t)$ be a Brownian motion with filtration $(\mathcal F_t)$ (for which $B_t - B_s$ is independent of $\mathcal F_s$ ). A simple process on $[0, T]$ has the form $H_t = \sum_{k=0}^{n-1} \xi_k \mathbf 1_{(t_k, t_{k+1}]}(t)$ for a partition $0 = t_0 \lt \dots \lt t_n = T$ , where each $\xi_k$ is bounded and $\mathcal F_{t_k}$ -measurable — the integrand is decided at the left endpoint. Its Itô integral is

\int_0^T H_t \, dB_t = \sum_{k=0}^{n-1} \xi_k \big(B_{t_{k+1}} - B_{t_k}\big).

It extends by an $L^2$ limit (below) to all adapted, jointly measurable $H$ with $\mathbb E \int_0^T H_t^2 \, dt \lt \infty$ .

Definition— Itô process, stochastic differential equation, quadratic variation

An Itô process is $X_t = X_0 + \int_0^t \mu_s \, ds + \int_0^t \sigma_s \, dB_s$ , written $dX_t = \mu_t \, dt + \sigma_t \, dB_t$ . A stochastic differential equation $dX_t = b(t, X_t) \, dt + \sigma(t, X_t) \, dB_t$ asks for an Itô process with $\mu_t = b(t, X_t)$ and $\sigma_t = \sigma(t, X_t)$ . The quadratic variation of a process over $[0, t]$ is the limit, if it exists, of $\sum_k (X_{t_{k+1}} - X_{t_k})^2$ as the mesh of the partition tends to $0$ .

ProofH

The Poisson process

Theorem— Construction from exponential waiting times

Let $E_1, E_2, \dots$ be independent exponential random variables with rate $\lambda$ (density $\lambda e^{-\lambda s}$ on $s > 0$ ), let $T_n = E_1 + \dots + E_n$ be the $n$ -th arrival time and $N_t = \#\{n \ge 1 : T_n \le t\}$ . Then:

$T_n$ has the Gamma density $f_n(s) = \lambda^n s^{n-1} e^{-\lambda s}/(n - 1)!$ on $s > 0$ ;
$N_t$ has the Poisson distribution with mean $\lambda t$ ;
(memorylessness) $P(E_1 > s + t \mid E_1 > s) = P(E_1 > t)$ for $s, t \ge 0$ .

Proof

(1) By induction: $f_1$ is the exponential density, and since $T_{n+1} = T_n + E_{n+1}$ with independent summands, its density is the convolution $f_{n+1}(s) = \int_0^s f_n(u) \, \lambda e^{-\lambda(s - u)} \, du = \frac{\lambda^{n+1} e^{-\lambda s}}{(n-1)!} \int_0^s u^{n-1} \, du = \frac{\lambda^{n+1} s^n e^{-\lambda s}}{n!}$ .

(2) $P(N_t = 0) = P(E_1 > t) = e^{-\lambda t}$ . For $n \ge 1$ , $N_t = n$ exactly when $T_n \le t \lt T_n + E_{n+1}$ . Since $E_{n+1}$ is independent of $T_n$ ,

P(N_t = n) = \int_0^t f_n(s) \, P(E_{n+1} > t - s) \, ds = \int_0^t \frac{\lambda^n s^{n-1} e^{-\lambda s}}{(n-1)!} \, e^{-\lambda(t - s)} \, ds = e^{-\lambda t} \frac{(\lambda t)^n}{n!}.

(3) $P(E_1 > s + t)/P(E_1 > s) = e^{-\lambda(s + t)}/e^{-\lambda s} = e^{-\lambda t}$ .

∎

Memorylessness characterises the exponential law among distributions on $(0, \infty)$ : if $G(t) = P(E > t)$ satisfies $G(s + t) = G(s) G(t)$ and $G$ is monotone with $0 \lt G(1) \lt 1$ , then $G(t) = e^{-\lambda t}$ with $\lambda = -\log G(1)$ (Cauchy's functional equation for $\log G$ ). It is also why the construction has independent, stationary increments: at any fixed time $s$ , the residual waiting time until the next arrival is again exponential with rate $\lambda$ and independent of what happened before $s$ , so $(N_{s+t} - N_s)_{t \ge 0}$ is a Poisson process independent of $(N_u)_{u \le s}$ (Norris, Markov Chains, §2.4). In particular the counts must be consistent: a Poisson( $\lambda s$ ) count followed by an independent Poisson( $\lambda t$ ) count must total a Poisson( $\lambda(s + t)$ ) count.

Theorem— Poisson counts add

For $s, t \ge 0$ and $k \ge 0$ : $\displaystyle \sum_{j=0}^k p_j(s) \, p_{k-j}(t) = p_k(s + t)$ . Equivalently, the sum of independent Poisson variables with means $s$ and $t$ is Poisson with mean $s + t$ .

Proof

$\displaystyle \sum_{j=0}^k e^{-s}\frac{s^j}{j!} \, e^{-t} \frac{t^{k-j}}{(k-j)!} = \frac{e^{-(s+t)}}{k!} \sum_{j=0}^k \binom kj s^j t^{k-j} = e^{-(s+t)} \frac{(s+t)^k}{k!}$ , by the binomial theorem.

∎

Theorem— Law of rare events

If $n p_n \to r$ , then $\binom nk p_n^k (1 - p_n)^{n-k} \to e^{-r} r^k/k!$ for every fixed $k$ .

Proof

Write $\binom nk p_n^k = \frac{n (n-1) \cdots (n - k + 1)}{n^k} \cdot \frac{(n p_n)^k}{k!} \to \frac{r^k}{k!}$ . Also $p_n \to 0$ , so $(1 - p_n)^{-k} \to 1$ , and $n \log(1 - p_n) = -n p_n (1 + O(p_n)) \to -r$ gives $(1 - p_n)^n \to e^{-r}$ .

∎

Two further operations keep us inside the family (Kingman, Poisson Processes): the superposition of independent Poisson processes of rates $\lambda$ and $\mu$ is a Poisson process of rate $\lambda + \mu$ , and thinning — keeping each arrival independently with probability $p$ — turns a Poisson process of rate $\lambda$ into one of rate $p\lambda$ , independent of the process of discarded arrivals.

ProofH

Continuous-time Markov chains

Theorem— Kolmogorov's equations (finite state space)

Let $Q$ be a generator on a finite set $S$ and $P(t) = e^{tQ} = \sum_{m \ge 0} t^m Q^m/m!$ . Then $P(0) = I$ , $P(s + t) = P(s) P(t)$ , each $P(t)$ is a stochastic matrix (non-negative entries, rows summing to $1$ ), and

P'(t) = Q \, P(t) \quad \text{(backward equation)}, \qquad P'(t) = P(t) \, Q \quad \text{(forward equation)}.

Proof

The exponential series converges absolutely; $e^{(s+t)Q} = e^{sQ} e^{tQ}$ because $sQ$ and $tQ$ commute, and differentiating the series termwise gives $P' = QP = PQ$ . Rows: $Q \mathbf 1 = 0$ (rows of $Q$ sum to zero), so $P(t) \mathbf 1 = \mathbf 1 + \sum_{m \ge 1} t^m Q^m \mathbf 1/m! = \mathbf 1$ . Non-negativity: with $c = \max_i q_i$ , the matrix $Q + cI$ has non-negative entries, so $e^{t(Q + cI)}$ does too, and $e^{tQ} = e^{-ct} e^{t(Q + cI)}$ (as $cI$ commutes with $Q$ ).

∎

One shows that the jump-and-hold chain of the definition has exactly these transition probabilities (Norris, §2.8): the backward equation comes from conditioning on the first jump, the forward equation from conditioning on the last. A distribution $\pi$ is stationary if $\pi P(t) = \pi$ for all $t$ , equivalently $\pi Q = 0$ ; for an irreducible finite chain it is unique and $P(t)_{ij} \to \pi_j$ as $t \to \infty$ (no periodicity can arise in continuous time).

Example— The two-state chain

On $S = \{0, 1\}$ with $q_{01} = a$ , $q_{10} = b$ and $a + b > 0$ , the forward equation for $p(t) = P(t)_{00}$ reads $p' = -a p + b (1 - p)$ : probability leaves $0$ at rate $a$ and returns at rate $b$ . With $p(0) = 1$ its solution is

p(t) = \frac{b}{a + b} + \frac{a}{a + b} e^{-(a+b) t} \xrightarrow[t \to \infty]{} \frac{b}{a+b} = \pi_0.

The relaxation rate $a + b$ is the non-zero eigenvalue of $-Q$ .

The Poisson process is the chain on $\{0, 1, 2, \dots\}$ with $q_{i, i+1} = \lambda$ . On infinite state spaces a new phenomenon appears: a pure birth chain with rates $\lambda_n$ makes infinitely many jumps in finite time — it explodes — with probability one if $\sum_n 1/\lambda_n \lt \infty$ (the expected total holding time is finite), and with probability zero if $\sum_n 1/\lambda_n = \infty$ .

ProofH

Brownian motion

Brownian motion exists. Wiener (1923) constructed its law on the space of continuous functions; Lévy gave a construction by successive dyadic interpolation with independent Gaussian corrections; alternatively, the finite-dimensional distributions exist by Kolmogorov's extension theorem, and since $\mathbb E|B_t - B_s|^{2p} = C_p |t - s|^p$ , the Kolmogorov–Chentsov theorem provides a modification whose paths are locally Hölder continuous of every order $\gamma \lt \frac{p-1}{2p}$ , hence of every order $\gamma \lt \tfrac12$ (Mörters–Peres, §1.1; Le Gall, Ch. 2).

Theorem— Brownian motion as a Gaussian process

A process $(B_t)$ with $B_0 = 0$ and continuous paths is a standard Brownian motion if and only if it is a Gaussian process with $\mathbb E B_t = 0$ and $\operatorname{Cov}(B_s, B_t) = \min(s, t)$ .

Proof

( $\Rightarrow$ ) Any vector $(B_{t_1}, \dots, B_{t_k})$ with $t_1 \lt \dots \lt t_k$ is a linear image of the independent Gaussian increments, hence Gaussian with mean $0$ . For $s \le t$ , independence of $B_s$ and $B_t - B_s$ gives $\operatorname{Cov}(B_s, B_t) = \operatorname{Var} B_s + \operatorname{Cov}(B_s, B_t - B_s) = s$ .

( $\Leftarrow$ ) The increments over $t_0 \lt \dots \lt t_k$ are jointly Gaussian. For $s \le t \le u \le v$ , $\operatorname{Cov}(B_t - B_s, B_v - B_u) = t - s - t + s = 0$ , and jointly Gaussian uncorrelated variables are independent. Finally $\operatorname{Var}(B_t - B_s) = t - 2s + s = t - s$ .

∎

Theorem— Invariances

If $B$ is a standard Brownian motion, so are: $-B$ (symmetry); $c^{-1/2} B_{ct}$ for each $c > 0$ (scaling); $B_{s+t} - B_s$ for each fixed $s \ge 0$ , and it is independent of $(B_u)_{u \le s}$ (Markov property); and $t B_{1/t}$ (with value $0$ at $t = 0$ ) (time inversion).

Proof

Each new process is a centred Gaussian process; by the previous theorem it suffices to compute covariances and check continuity. For scaling, $\operatorname{Cov}(c^{-1/2} B_{cs}, c^{-1/2} B_{ct}) = c^{-1} \min(cs, ct) = \min(s, t)$ . For time inversion with $0 \lt s \le t$ , $\operatorname{Cov}(s B_{1/s}, t B_{1/t}) = st \min(1/s, 1/t) = st/t = s$ . Continuity of $t B_{1/t}$ at $t = 0$ amounts to $B_u/u \to 0$ as $u \to \infty$ , the Brownian strong law; it follows, for instance, from Doob's maximal inequality applied on dyadic blocks (Mörters–Peres, Ch. 1). The Markov property: the increments of $B_{s + \cdot} - B_s$ are increments of $B$ over $[s, \infty)$ , independent of those over $[0, s]$ .

∎

Theorem— Brownian martingales

With respect to the natural filtration, $B_t$ , $B_t^2 - t$ and $\exp(\theta B_t - \theta^2 t/2)$ (for each $\theta \in \R$ ) are martingales.

Proof

For $s \lt t$ write $B_t = B_s + D$ with $D = B_t - B_s \sim N(0, t - s)$ independent of $\mathcal F_s$ . Then $\mathbb E[B_t \mid \mathcal F_s] = B_s$ ; $\mathbb E[B_t^2 \mid \mathcal F_s] = B_s^2 + 2 B_s \mathbb E D + \mathbb E D^2 = B_s^2 + t - s$ ; and $\mathbb E[e^{\theta B_t} \mid \mathcal F_s] = e^{\theta B_s} \mathbb E e^{\theta D} = e^{\theta B_s + \theta^2 (t - s)/2}$ , using the Gaussian moment generating function.

∎

Theorem— Quadratic variation of Brownian motion

Let $\Pi_n = \{0 = t_0^n \lt \dots \lt t_{m_n}^n = t\}$ be partitions of $[0, t]$ with mesh $|\Pi_n| = \max_k (t^n_{k+1} - t^n_k) \to 0$ . Then $Q_n = \sum_k \big(B_{t^n_{k+1}} - B_{t^n_k}\big)^2 \to t$ in $L^2$ .

Proof

Write $\Delta_k = B_{t_{k+1}} - B_{t_k}$ and $\delta_k = t_{k+1} - t_k$ (dropping $n$ ). Then $\mathbb E \Delta_k^2 = \delta_k$ , so $\mathbb E Q_n = t$ . The $\Delta_k^2$ are independent with $\operatorname{Var} \Delta_k^2 = \mathbb E \Delta_k^4 - \delta_k^2 = 3\delta_k^2 - \delta_k^2 = 2\delta_k^2$ . Hence

\mathbb E (Q_n - t)^2 = \operatorname{Var} Q_n = 2 \sum_k \delta_k^2 \le 2 \, |\Pi_n| \sum_k \delta_k = 2 t \, |\Pi_n| \to 0.

∎

If $\sum_n |\Pi_n| \lt \infty$ (for instance dyadic partitions), Chebyshev and Borel–Cantelli upgrade this to almost-sure convergence.

Theorem— Brownian paths have infinite variation

Almost surely, the total variation $V = \sup_\Pi \sum_k |B_{t_{k+1}} - B_{t_k}|$ of $B$ on $[0, t]$ (with $t > 0$ ) is infinite.

Proof

Take the dyadic partitions $\Pi_n$ of $[0, t]$ , along which $Q_n \to t$ almost surely. For every $n$ , $Q_n \le \max_k |\Delta_k| \cdot \sum_k |\Delta_k| \le \max_k |\Delta_k| \cdot V$ . Each path is uniformly continuous on $[0, t]$ , so $\max_k |\Delta_k| \to 0$ . If $V(\omega) \lt \infty$ then $Q_n(\omega) \to 0$ , which happens only on a null set because $Q_n \to t > 0$ almost surely.

∎

So Brownian paths are not of bounded variation, and $\int H \, dB$ cannot be defined path by path as a Riemann–Stieltjes integral. In fact much more is true.

Theorem— Nowhere differentiability (Paley–Wiener–Zygmund, 1933)

Almost surely, $t \mapsto B_t$ is differentiable at no point of $[0, 1)$ .

Proof

This proof is due to Dvoretzky, Erdős and Kakutani (1961). Suppose $B$ is differentiable at some $s \in [0, 1)$ . Then there are $M \in \N$ and $\delta > 0$ with $|B_u - B_s| \le M |u - s|$ for $|u - s| \le \delta$ . Let $n > 4/\delta$ and $k = \lceil ns \rceil$ , so $0 \le k/n - s \lt 1/n$ . For $j = 1, 2, 3$ both $(k + j)/n$ and $(k + j - 1)/n$ lie within $4/n$ of $s$ , and

\big|B_{(k+j)/n} - B_{(k+j-1)/n}\big| \le M \frac{j + 1}{n} + M \frac{j}{n} \le \frac{7M}{n}.

So $\omega$ lies in $A_{n, M} = \bigcup_{k=0}^n \bigcap_{j=1}^3 \big\{|B_{(k+j)/n} - B_{(k+j-1)/n}| \le 7M/n\big\}$ for all large $n$ . The three increments are independent $N(0, 1/n)$ , and a $N(0, 1/n)$ density is at most $\sqrt{n/(2\pi)} \le \sqrt n/2$ , so each event in the intersection has probability at most $2 \cdot \frac{7M}{n} \cdot \frac{\sqrt n}{2} = 7M/\sqrt n$ . Hence

P(A_{n, M}) \le (n + 1) \Big(\frac{7M}{\sqrt n}\Big)^3 \xrightarrow[n \to \infty]{} 0.

The set of paths differentiable somewhere in $[0, 1)$ is therefore contained in $\bigcup_{M} \liminf_n A_{n, M}$ , and $P(\liminf_n A_{n, M}) \le \liminf_n P(A_{n, M}) = 0$ for each $M$ .

∎

Finally, the reflection principle: for $a > 0$ , $P\big(\max_{s \le t} B_s \ge a\big) = 2 P(B_t \ge a)$ . Reflect the path after the first time it hits $a$ ; by the strong Markov property the reflected path is again a Brownian motion (Mörters–Peres, §2.2).

ProofH

Itô's stochastic calculus

Theorem— Itô isometry (simple integrands)

For a simple process $H$ on $[0, T]$ : $\mathbb E \int_0^T H_t \, dB_t = 0$ and $\displaystyle \mathbb E \Big(\int_0^T H_t \, dB_t\Big)^2 = \mathbb E \int_0^T H_t^2 \, dt$ .

Proof

Let $\Delta_k = B_{t_{k+1}} - B_{t_k}$ , independent of $\mathcal F_{t_k}$ , with mean $0$ and variance $\delta_k = t_{k+1} - t_k$ . Since $\xi_k$ is $\mathcal F_{t_k}$ -measurable, $\mathbb E[\xi_k \Delta_k] = \mathbb E[\xi_k] \, \mathbb E[\Delta_k] = 0$ . Expanding the square, for $j \lt k$ the variable $\xi_j \Delta_j \xi_k$ is $\mathcal F_{t_k}$ -measurable, so $\mathbb E[\xi_j \Delta_j \xi_k \Delta_k] = \mathbb E[\xi_j \Delta_j \xi_k] \, \mathbb E \Delta_k = 0$ . The diagonal terms give $\mathbb E[\xi_k^2 \Delta_k^2] = \mathbb E[\xi_k^2] \, \delta_k$ , and $\sum_k \mathbb E[\xi_k^2] \delta_k = \mathbb E \int_0^T H_t^2 \, dt$ .

∎

The isometry says that $H \mapsto \int H \, dB$ is a linear isometry from simple processes, with the norm $\big(\mathbb E \int_0^T H^2 \, dt\big)^{1/2}$ , into $L^2(\Omega)$ . Simple processes are dense in the space of adapted, jointly measurable $H$ with $\mathbb E \int_0^T H^2 \, dt \lt \infty$ , so the integral extends uniquely, still isometric. The resulting process $t \mapsto \int_0^t H \, dB$ has a continuous modification which is a martingale (Karatzas–Shreve, §3.2). The choice of the left endpoint is what makes the integral a martingale: the integrand cannot anticipate the increment it multiplies.

Theorem— Integral of B dB

$\displaystyle \int_0^t B_s \, dB_s = \frac12 B_t^2 - \frac12 t$ .

Proof

Fix partitions $\Pi_n$ of $[0, t]$ with mesh $\to 0$ and let $H^n = \sum_k B_{t_k} \mathbf 1_{(t_k, t_{k+1}]}$ . (These are not bounded, but they are square-integrable, which suffices for the isometry.) Then

\mathbb E \int_0^t (B_s - H^n_s)^2 \, ds = \sum_k \int_{t_k}^{t_{k+1}} (s - t_k) \, ds = \frac12 \sum_k \delta_k^2 \le \frac t2 |\Pi_n| \to 0,

so by the isometry $\int H^n \, dB \to \int B \, dB$ in $L^2$ . On the other hand, the algebraic identity $a(b - a) = \frac12 (b^2 - a^2) - \frac12 (b - a)^2$ , summed over the partition (the discrete Itô formula, formalised in Lean below), gives

\int_0^t H^n \, dB = \sum_k B_{t_k} (B_{t_{k+1}} - B_{t_k}) = \frac12 B_t^2 - \frac12 \sum_k (B_{t_{k+1}} - B_{t_k})^2 \xrightarrow{L^2} \frac12 B_t^2 - \frac12 t

by the quadratic variation theorem.

∎

Ordinary calculus would predict $\frac12 B_t^2$ . The extra $-\frac12 t$ is forced: the left side is a martingale with mean $0$ , and $\mathbb E[\frac12 B_t^2] = \frac t2 \ne 0$ . With midpoints $\frac12(B_{t_k} + B_{t_{k+1}})$ instead of left endpoints the sum telescopes to exactly $\frac12 B_t^2$ : this is the Stratonovich integral $\int B \circ dB$ , which obeys the ordinary chain rule but is not a martingale.

Theorem— Itô's formula

Let $f(t, x)$ be continuously differentiable once in $t$ and twice in $x$ , and let $dX_t = \mu_t \, dt + \sigma_t \, dB_t$ be an Itô process. Then, almost surely for all $t$ ,

f(t, X_t) = f(0, X_0) + \int_0^t \Big(\partial_t f + \mu_s \, \partial_x f + \tfrac12 \sigma_s^2 \, \partial_{xx} f\Big)(s, X_s) \, ds + \int_0^t \sigma_s \, \partial_x f(s, X_s) \, dB_s.

In particular $df(B_t) = f'(B_t) \, dB_t + \frac12 f''(B_t) \, dt$ .

About the proof. Expand $f(t_{k+1}, X_{t_{k+1}}) - f(t_k, X_{t_k})$ by Taylor's theorem to first order in time and second order in space and sum over a partition. The first-order space terms converge to the stochastic integral, the second-order terms $\frac12 \partial_{xx} f \, (\Delta X_k)^2$ converge to $\frac12 \int \partial_{xx} f \, \sigma^2 \, ds$ by the same variance computation as for quadratic variation, and the remainders vanish because $\sum_k |\Delta X_k|^3 \to 0$ . A localisation argument removes boundedness assumptions (Karatzas–Shreve, §3.3; Le Gall, Ch. 5). The formula with $f(x) = x^2$ and $X = B$ is exactly the previous theorem.

Example— Geometric Brownian motion

Consider $dS_t = \mu S_t \, dt + \sigma S_t \, dB_t$ with $S_0 > 0$ , the Black–Scholes model of a stock price. Guess $S_t = f(t, B_t)$ with $f(t, x) = S_0 \exp\big((\mu - \frac{\sigma^2}{2}) t + \sigma x\big)$ . Then $\partial_x f = \sigma f$ , $\partial_{xx} f = \sigma^2 f$ , $\partial_t f = (\mu - \frac{\sigma^2}{2}) f$ , and Itô's formula gives

dS_t = \Big(\mu - \frac{\sigma^2}{2} + \frac{\sigma^2}{2}\Big) S_t \, dt + \sigma S_t \, dB_t = \mu S_t \, dt + \sigma S_t \, dB_t.

Since the coefficients are Lipschitz, this is the unique solution (Øksendal, Thm 5.2.1). Its behaviour is instructive. $\mathbb E S_t = S_0 e^{\mu t}$ (the mean of a log-normal), but the median is $S_0 e^{(\mu - \sigma^2/2)t}$ , and since $B_t/t \to 0$ almost surely, $\frac1t \log S_t \to \mu - \frac{\sigma^2}{2}$ . If $0 \lt \mu \lt \sigma^2/2$ , the expectation grows exponentially while almost every path tends to $0$ : the mean is carried by ever rarer, ever larger paths.

Example— The Ornstein–Uhlenbeck process

The Langevin equation $dX_t = -\theta X_t \, dt + \sigma \, dB_t$ ( $\theta > 0$ ) models the velocity of a particle with friction. Itô's formula for $e^{\theta t} X_t$ (here $\partial_{xx} = 0$ ) gives $d(e^{\theta t} X_t) = \sigma e^{\theta t} \, dB_t$ , so

X_t = e^{-\theta t} X_0 + \sigma \int_0^t e^{-\theta(t - s)} \, dB_s.

The integral is Gaussian with mean $0$ and, by the isometry, variance $\sigma^2 \int_0^t e^{-2\theta(t-s)} \, ds = \frac{\sigma^2}{2\theta}(1 - e^{-2\theta t})$ . So $X_t$ converges in distribution to $N(0, \sigma^2/2\theta)$ , the stationary law — in physics, the Maxwell distribution of velocities, with $\sigma^2/2\theta$ proportional to temperature.

Formal mathematicsF

Stochastic processes in Lean and Mathlib

Mathlib has no Itô integral yet, but the algebra that produces Itô's correction term can be formalised for any real sequence. The discrete Itô formula — left-point sums equal half the change of the square minus half the quadratic variation — is proved by induction, one line of ring per step:

discrete_itoLean 4Proved here from basic factsFormally verified

/-- **The discrete Itô formula.** For any real sequence `S₀, S₁, …` the left-point sums satisfy
`∑_{k<n} Sₖ (Sₖ₊₁ − Sₖ) = (Sₙ² − S₀²)/2 − ½ ∑_{k<n} (Sₖ₊₁ − Sₖ)²`.
The last sum is the quadratic variation. For a smooth path it tends to `0` under refinement and we
recover `∫ S dS = (Sₙ² − S₀²)/2`; for Brownian motion it tends to `t`, which gives
`∫₀ᵗ B dB = (B_t² − t)/2`. Proof: induction on `n`; each step is the identity
`a (b − a) = (b² − a²)/2 − (b − a)²/2`. -/
theorem discrete_ito (S : ℕ → ℝ) (n : ℕ) :
    ∑ k ∈ Finset.range n, S k * (S (k + 1) - S k) =
      (S n ^ 2 - S 0 ^ 2) / 2 - (∑ k ∈ Finset.range n, (S (k + 1) - S k) ^ 2) / 2 := by
  induction n with
  | zero => simp
  | succ n ih =>
    rw [Finset.sum_range_succ, Finset.sum_range_succ, ih]
    ring