Stochastic Calculus Notes 03 | Stochastic Integration

21 Oct 2026 • Ziyuan Zhang

Overview

A stochastic integral is just cumulative trading P/L: hold \(\Delta\) shares while the price moves by \(dW\), and add it up. This post builds \(\int_0^t\Delta_s\,dW_s\) in three steps — discrete time, “buy-and-hold” (simple) strategies, and general strategies by approximation — and shows the same five properties survive every step:

linear · adapted · martingale · Itô isometry \(\mathrm{Var}[I_t]=E\int_0^t\Delta_s^2ds\) · \([I,I]_t=\int_0^t\Delta_s^2ds\).

1. Discrete time: the integral as trading P/L

Let \(M=\{M_n\}\) be a martingale (think: a share price). For a trading strategy \(\Delta=\{\Delta_n\}\) define

\[I_0=0,\qquad I_n=\sum_{j=1}^n\underbrace{\Delta_j}_{\text{shares held}}\ \underbrace{dM_j}_{\text{price change}},\qquad dM_j=M_j-M_{j-1}.\]

\(\Delta_j\,dM_j\) is the P/L over \([j-1,j]\) and \(I_n\) the cumulative P/L over \([0,n]\).

Predictable strategies. We hold \(\Delta_j\) over \([j-1,j]\), and we cannot look into the future, so \(\Delta_j\) must be known at time \(j-1\): \(\Delta_j\) is \(\mathcal F_{j-1}\)-measurable. Such a \(\Delta=\{\Delta_n\}_{n\ge1}\) is called predictable (\(\Delta_0\) is not defined since \(\mathcal F_{-1}\) does not exist).

Properties of \(I=\sum\Delta\,dM\) (\(\Delta\) predictable, \(M\) a martingale)

Linear in \(\Delta\): \(I(\alpha\Delta+\beta\psi)=\alpha I(\Delta)+\beta I(\psi)\) — just split the sum.
Adapted: \(\Delta_1,\dots,\Delta_n\) are known by time \(n-1\) and \(M_0,\dots,M_n\) by time \(n\), so \(I_n\) is \(\mathcal F_n\)-measurable.
Martingale: \(I_{n+1}-I_n=\Delta_{n+1}(M_{n+1}-M_n)\) and \(\Delta_{n+1}\) is \(\mathcal F_n\)-measurable, so
\[E[I_{n+1}-I_n\mid\mathcal F_n]=\Delta_{n+1}\,E[M_{n+1}-M_n\mid\mathcal F_n]=0.\]
Quadratic variation: \([I,I]_n=\sum_j\Delta_j^2(dM_j)^2=\sum_j\Delta_j^2\,d[M,M]_j\).
Variance: since \(I\) is a martingale with \(I_0=0\), Fact 1 of Part 02 gives
\[\mathrm{Var}[I_n]=E\big[[I,I]_n\big]=E\Big[\sum_{j=1}^n\Delta_j^2\,d[M,M]_j\Big].\]

For the random walk, \(d[M,M]_j=1\), so \([I,I]_n=\sum_j\Delta_j^2\) and \(\mathrm{Var}[I_n]=E\sum_j\Delta_j^2\).

Intuition for “martingale”: you cannot beat a fair game by choosing your bet size, as long as you choose it before the coin is tossed.

2. Continuous time: why the obvious definition fails

We want \(I_t=\int_0^t\Delta_s\,dW_s\) (\(\Delta\): integrand, \(W\): integrator; same P/L reading). The natural guess is a Riemann sum,

\[\int_0^t\Delta_s\,dW_s\approx\sum_{j=1}^n\Delta_{t_{j-1}}\big(W_{t_j}-W_{t_{j-1}}\big).\]

If \(W\) were smooth, the mean value theorem would turn this into \(\int_0^t\Delta_s\dot W_s\,ds\). But \(W\) is nowhere differentiable (Part 02), so \(\dot W_s\) does not exist and path-by-path the limit does not exist in the ordinary sense. We need a new kind of limit.

3. Simple (buy-and-hold) integrands

First bring things back to almost discrete time. Fix \(0=t_0<t_1<\dots<t_n\) and \(\mathcal F_{t_j}\)-measurable r.v.’s \(\Delta_j\), and set

\[\Delta_t=\Delta_0\mathbf 1_{\{t=0\}}+\sum_{j=1}^n\Delta_{j-1}\mathbf 1_{\{t\in(t_{j-1},t_j]\}}:\]

hold \(\Delta_{j-1}\) shares over \((t_{j-1},t_j]\). For such a simple integrand the integral is explicit:

\[I_t=\sum_{j=1}^n\underbrace{\Delta_{j-1}}_{\text{position}}\ \underbrace{\big(W_{t\wedge t_j}-W_{t\wedge t_{j-1}}\big)}_{\text{price change}},\qquad t\wedge s=\min\{t,s\}.\]

Reading it piece by piece: for \(t_{j-1}\le t\le t_j\),

\[I_t=\underbrace{\sum_{i=1}^{j-1}\Delta_{i-1}\big(W_{t_i}-W_{t_{i-1}}\big)}_{\text{P/L over }[0,t_{j-1}]}+\underbrace{\Delta_{j-1}\big(W_t-W_{t_{j-1}}\big)}_{\text{P/L over }[t_{j-1},t]},\]

and for \(t\ge t_n\) nothing changes any more — we stopped trading.

Properties for simple \(\Delta\)

Linear and adapted: as in discrete time.
Continuous paths (new): each term \(t\mapsto\Delta_{j-1}(W_{t\wedge t_j}-W_{t\wedge t_{j-1}})\) is continuous because \(W\) is, and a finite sum of continuous processes is continuous.
Martingale. For \(t_{j-1}<s<t_j<t\) (adjacent buckets),
\[\begin{aligned} E[I_t-I_s\mid\mathcal F_s]&=E[\Delta_{j-1}(W_{t_j}-W_s)\mid\mathcal F_s]+E\big[E[\Delta_j(W_t-W_{t_j})\mid\mathcal F_{t_j}]\mid\mathcal F_s\big]\\ &=\Delta_{j-1}\cdot0+E[\Delta_j\cdot0\mid\mathcal F_s]=0. \end{aligned}\]
It is crucial that the position is chosen at the beginning of each period.
Itô isometry: \(\mathrm{Var}[I_t]=E\Big[\int_0^t\Delta_s^2\,ds\Big]\).
Quadratic variation: \([I,I]_t=\int_0^t\Delta_s^2\,ds\).

Proof: Itô isometry for simple integrands

\(I_0=0\) and \(I\) is a martingale, so \(\mathrm{Var}[I_t]=E[I_t^2]\). Relabel so that \(t=t_j\) and \(I_t=\sum_{i=1}^j\Delta_{t_{i-1}}(W_{t_i}-W_{t_{i-1}})\). Cross terms vanish exactly as in Part 02 (condition on the later left endpoint), and each diagonal term is

\[E\big[\Delta_{t_{i-1}}^2(W_{t_i}-W_{t_{i-1}})^2\big]=E\big[\Delta_{t_{i-1}}^2\,E[(W_{t_i}-W_{t_{i-1}})^2\mid\mathcal F_{t_{i-1}}]\big]=E\big[\Delta_{t_{i-1}}^2(t_i-t_{i-1})\big].\]

Since \(\Delta\) is constant on each \((t_{i-1},t_i]\),

\[\mathrm{Var}[I_t]=E\Big[\sum_{i=1}^j\Delta_{t_{i-1}}^2(t_i-t_{i-1})\Big]=E\Big[\int_0^t\Delta_s^2\,ds\Big].\]

Proof: quadratic variation for simple integrands

Partition one bucket \([t_{i-1},t_i]\) finely by \(s_0<s_1<\dots<s_n\). Inside the bucket the position is constant, so \(I_{s_k}-I_{s_{k-1}}=\Delta_{t_{i-1}}(W_{s_k}-W_{s_{k-1}})\) and

\[[I,I]^\Pi_{[t_{i-1},t_i]}=\Delta_{t_{i-1}}^2\,[W,W]^\Pi_{[t_{i-1},t_i]}\ \longrightarrow\ \Delta_{t_{i-1}}^2(t_i-t_{i-1}).\]

Summing over buckets, \([I,I]_t=\sum_i\Delta_{t_{i-1}}^2(t_i-t_{i-1})=\int_0^t\Delta_s^2\,ds\).

Note the difference between the last two: \([I,I]_t\) is a random quantity path by path; the isometry is its expectation.

4. General integrands: approximate, then take an \(L^2\) limit

Fix \(T>0\). Let \(\mathcal L\) be the set of integrands \(\Delta\) that are

adapted, and
square-integrable: \(E\Big[\int_0^T\Delta_s^2\,ds\Big]<\infty\).

For \(\Delta\in\mathcal L\) and partitions \(\Pi^n\) of \([0,T]\) with mesh \(\to0\), freeze \(\Delta\) at the left endpoints:

\[\Delta^n_t=\Delta_0\mathbf 1_{\{t=0\}}+\sum_{j=1}^n\Delta_{t_{j-1}}\mathbf 1_{\{t\in(t_{j-1},t_j]\}},\qquad I^n_t=I(\Delta^n)_t .\]

\(\Delta^n\) is a simple “rebalance only at grid times” version of \(\Delta\), and \(I^n\) is perfectly well defined. For a fixed \(\omega\), \(\lim I^n_t(\omega)\) need not exist; instead we take the limit in mean square.

Step 1 — the approximation is good. A key result of the course is that for \(\Delta\in\mathcal L\),

\[\lim_{n\to\infty}E\Big[\int_0^T(\Delta^n_s-\Delta_s)^2\,ds\Big]=0,\]

written \(\Delta^n\to\Delta\) in \(L^2([0,T]\times\Omega)\). (Intuitively: as the rebalancing grid gets finer, the frozen strategy tracks the real one, on average over both time and scenarios.)

Step 2 — so the approximations are Cauchy. From \((a+b)^2\le2a^2+2b^2\),

\[(\Delta^n_u-\Delta^m_u)^2\le2(\Delta^n_u-\Delta_u)^2+2(\Delta_u-\Delta^m_u)^2\ \Longrightarrow\ \lim_{n,m\to\infty}E\int_0^T(\Delta^n_s-\Delta^m_s)^2ds=0.\]

Step 3 — the isometry transfers this to the integrals. By linearity, \(I^n_t-I^m_t=I(\Delta^n-\Delta^m)_t\), and \(\Delta^n-\Delta^m\) is simple, so

\[\mathrm{Var}\big[I^n_t-I^m_t\big]=E\Big[\int_0^t(\Delta^n_u-\Delta^m_u)^2du\Big]\le E\Big[\int_0^T(\Delta^n_u-\Delta^m_u)^2du\Big]\xrightarrow{n,m\to\infty}0.\]

Step 4 — define the integral as the limit. The integrals \(I^n\) form a Cauchy sequence in mean square, so they converge (as processes) to a limit \(I\), and we define

\[\Big(\int_0^t\Delta_s\,dW_s\Big)(\omega):=I_t(\omega),\qquad t\le T.\]

Why the isometry is the hero. It says the map “strategy \(\mapsto\) P/L” preserves distance: two strategies that are close in \(E\int(\cdot)^2\) produce P/Ls that are close in variance. So “approximate the strategy” automatically gives “approximate the integral” — even though no single path converges in the classical sense.

5. Properties for general integrands

Everything carries over to \(\Delta\in\mathcal L\):

linear: \(I(\alpha\Delta+\beta\psi)=\alpha I(\Delta)+\beta I(\psi)\);
continuous sample paths, \(I_0=0\);
adapted;
martingale;
Itô isometry: \(\mathrm{Var}[I_t]=E\Big[\int_0^t\Delta_s^2\,ds\Big]\);
quadratic variation: \([I,I]_t=\int_0^t\Delta_s^2\,ds\) path by path.

In differential shorthand, \(dI_t=\Delta_t\,dW_t\) and \(d[I,I]_t=\Delta_t^2\,dt=(\Delta_t\,dW_t)^2\) — consistent with \((dW_t)^2=dt\).

Series contents

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This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.