Mathematics.

stochastic calculus

Stochastic Integration

Stochastic Processes75 minDifficulty8 out of 10

You should know: brownian motion, martingales

Overview

Stochastic integration extends the classical Riemann-Stieltjes integral to integrators with rough paths, particularly Brownian motion. Because Brownian paths have infinite variation, classical integration theory fails. The Ito integral is defined as a limit of Riemann sums evaluated at the left endpoint, producing a martingale. The Stratonovich integral uses midpoint evaluations and obeys classical chain rules. Stochastic integration is the foundation of Ito's formula and stochastic differential equations.

Intuition

Classical integration int_0^T H_s dB_s treats the Brownian increments dB_s as infinitesimal 'noise' multiplied by a process H. Because B has unbounded variation, the integral is not a path-by-path Stieltjes integral -- it requires a probabilistic definition. The key insight: using left-endpoint evaluations (non-anticipating) makes the integral a martingale and gives the Ito isometry. Midpoint evaluations give the Stratonovich integral, which satisfies the classical chain rule but is not a martingale.

Formal Definition

Definition

Let {B_t} be a standard Brownian motion and {H_t} be an adapted process with E[int_0^T H_s^2 ds] < infinity. The Ito integral I_T = int_0^T H_s dB_s is defined as the L^2 limit of left-Riemann sums on a partition 0 = t_0 < t_1 < ... < t_n = T as the mesh tends to 0.

∫0THs dBs=lim⁡∥Π∥→0∑k=0n−1Htk(Btk+1−Btk)\int_0^T H_s\,dB_s = \lim_{\|\Pi\| \to 0} \sum_{k=0}^{n-1} H_{t_k}(B_{t_{k+1}} - B_{t_k})
Ito integral (left Riemann sum)
E ⁣[∫0THs dBs]=0\mathbb{E}\!\left[\int_0^T H_s\,dB_s\right] = 0
Zero mean (martingale property)
E ⁣[(∫0THs dBs)2]=E ⁣[∫0THs2 ds]\mathbb{E}\!\left[\left(\int_0^T H_s\,dB_s\right)^2\right] = \mathbb{E}\!\left[\int_0^T H_s^2\,ds\right]
Ito isometry
Mt=∫0tHs dBs is a (local) martingaleM_t = \int_0^t H_s\,dB_s \text{ is a (local) martingale}
Martingale property
[M]t=∫0tHs2 ds(quadratic variation)[M]_t = \int_0^t H_s^2\,ds \quad \text{(quadratic variation)}
Quadratic variation of Ito integral

Notation

NotationMeaning
∫0THs dBs\int_0^T H_s\,dB_sIto stochastic integral of H with respect to Brownian motion B
[M]t[M]_tQuadratic variation of the process M up to time t
L2(W)L^2(W)Space of square-integrable adapted processes: E[int_0^T H_s^2 ds] < infinity

Theorems

Theorem 1: Theorem 1 (Ito Isometry)
For H∈L2(W),E ⁣[(∫0THs dBs)2]=E ⁣[∫0THs2 ds].\text{For } H \in L^2(W), \quad \mathbb{E}\!\left[\left(\int_0^T H_s\,dB_s\right)^2\right] = \mathbb{E}\!\left[\int_0^T H_s^2\,ds\right].
Theorem 2: Theorem 2 (Martingale Property)
If H∈L2(W), then Mt=∫0tHs dBs is a square-integrable martingale with M0=0 and ⟨M⟩t=∫0tHs2 ds.\text{If } H \in L^2(W), \text{ then } M_t = \int_0^t H_s\,dB_s \text{ is a square-integrable martingale with } M_0 = 0 \text{ and } \langle M \rangle_t = \int_0^t H_s^2\,ds.
Theorem 3: Theorem 3 (Ito vs Stratonovich)
The Stratonovich integral ∫0THs∘dBs=∫0THs dBs+12∫0T∂Hs∂Bs ds. Stratonovich satisfies the classical chain rule; Ito does not.\text{The Stratonovich integral } \int_0^T H_s \circ dB_s = \int_0^T H_s\,dB_s + \frac{1}{2}\int_0^T \frac{\partial H_s}{\partial B_s}\,ds. \text{ Stratonovich satisfies the classical chain rule; Ito does not.}

Worked Examples

  1. 1

    This is the classic computation. Use the Ito sum approximation: sum B_{t_k}(B_{t_{k+1}} - B_{t_k}) and expand.

    ∑kBtk(Btk+1−Btk)=12∑k(Btk+12−Btk2)−12∑k(Btk+1−Btk)2\sum_k B_{t_k}(B_{t_{k+1}} - B_{t_k}) = \frac{1}{2}\sum_k (B_{t_{k+1}}^2 - B_{t_k}^2) - \frac{1}{2}\sum_k (B_{t_{k+1}} - B_{t_k})^2
  2. 2

    The first sum telescopes to (B_T^2 - B_0^2)/2 = B_T^2/2. The second sum converges to [B]_T = T (quadratic variation of Brownian motion).

    ∫0TBs dBs=BT22−T2\int_0^T B_s\,dB_s = \frac{B_T^2}{2} - \frac{T}{2}

✓ Answer

int_0^T B_s dB_s = B_T^2/2 - T/2. This differs from the classical result B_T^2/2 by the Ito correction -T/2.

Practice Problems

Mediumfree response

Show that int_0^T t dB_t is a Gaussian random variable and find its variance.

Hardfree response

Let M_t = int_0^t B_s dB_s. Compute E[M_t] and E[M_t^2].

Common Mistakes

Common Mistake

Applying the classical chain rule d(f(B_t)) = f'(B_t) dB_t

Ito's formula adds a second-order correction: d(f(B_t)) = f'(B_t) dB_t + (1/2) f''(B_t) dt. Ignoring the (1/2)f'' dt term is the most common error in stochastic calculus.

Common Mistake

Treating dB_s like a Riemann integrator

Brownian motion has unbounded variation, so path-by-path Riemann-Stieltjes integration fails. The Ito integral must be defined as an L^2 limit and is fundamentally probabilistic.

Quiz

The Ito isometry states that for an adapted square-integrable process H:
The classical computation int_0^T B_s dB_s equals:
The Ito integral int_0^t H_s dB_s (for H adapted and square-integrable) is:

Summary

  • The Ito integral int H dB is defined as an L^2 limit of left-endpoint Riemann sums over adapted integrands.
  • Ito isometry: E[(int_0^T H_s dB_s)^2] = E[int_0^T H_s^2 ds].
  • The Ito integral is a martingale with zero mean.
  • Key result: int_0^T B_s dB_s = B_T^2/2 - T/2 (the -T/2 is the Ito correction).
  • The Stratonovich integral satisfies the classical chain rule but is not a martingale.

References

  1. BookKaratzas, I. and Shreve, S. -- Brownian Motion and Stochastic Calculus, Chapter 3
  2. BookOksendal, B. -- Stochastic Differential Equations, Chapter 3
  3. WebsiteWikipedia -- Ito calculus