Stochastic Calculus: Itô Integrals, Itô's Formula, and Stochastic Differential Equations

Ordinary calculus integrates against time or sufficiently regular paths. Brownian paths are continuous but too irregular for the usual pathwise rules to apply unchanged. Stochastic calculus builds an integral compatible with information arriving through time, then supplies the correct change-of-variable rule.

The central idea is Brownian squared increments accumulate to elapsed time. That fact explains the extra term in Itô’s formula, the drift of log prices, and the second derivative in the Black–Scholes PDE.

Learning goals

Define Itô processes and non-anticipating stochastic integrals.
Use the isometry to compute moments and understand convergence.
Apply Itô’s formula to time-dependent functions and general diffusions.
Distinguish a local martingale from a true martingale.
Solve geometric Brownian motion and the Ornstein–Uhlenbeck SDE.
Connect stochastic dynamics to self-financing option replication and numerical simulation.

1. Probability, information, and Brownian motion

Work on a filtered probability space (Ω,ℱ,{ℱₜ},P). Ω contains outcomes, ℱ contains measurable events, P assigns probabilities, and the increasing filtration ℱₜ represents information available by time t. Standard treatments impose appropriate completeness and right-continuity conditions on this filtration.

Let B be standard Brownian motion relative to that filtration. It starts at zero, has continuous paths, and increments Bₜ−Bₛ∼N(0,t−s) independent of ℱₛ for s<t. In particular, E[Bₜ | ℱₛ]=Bₛ and Var(Bₜ)=t.

An adapted process Xₜ is observable at time t. For stochastic integration, use suitable predictable or progressively measurable integrands with the required integrability. Adaptedness alone, without the relevant measurability and integrability, is not a complete integral-existence condition. Revealing a future B_T at earlier dates changes the information structure and can invalidate the Brownian martingale property.

2. Itô processes and the meaning of an SDE

A general scalar Itô process has the integral form:

Xₜ=X₀+∫₀ᵗbₛds+∫₀ᵗaₛdBₛ,

with appropriate measurability and, locally, ∫abs(bₛ)ds<∞ and ∫aₛ²ds<∞ almost surely. The first integral is finite-variation drift; the second is stochastic noise.

The shorthand is dXₜ=bₜdt+aₜdBₜ. It does not assert that B has an ordinary derivative dB/dt. The integral equation is the mathematical meaning of the shorthand.

A common state-dependent SDE specifies bₜ=μ(t,Xₜ) and aₜ=σ(t,Xₜ):

dXₜ=μ(t,Xₜ)dt+σ(t,Xₜ)dBₜ.

General Itô-process coefficients can depend on more of the past than Xₜ. A state-dependent Markov diffusion is an important special class, not the definition of every Itô process. Conditions on μ, σ, and the initial value determine whether a well-behaved solution exists.

3. Construct the integral from decisions made before each shock

For deterministic dates 0=t₀<…<tₙ=T, first consider a simple predictable process:

fₛ=Σₖξₖ1_(tₖ,tₖ₊₁](s),

where ξₖ is known at tₖ and has suitable second moments. Define:

∫₀ᵀfₛdBₛ=Σₖξₖ(Bₜₖ₊₁−Bₜₖ).

The coefficient is chosen before the next increment. Financially, ξₖ is a holding maintained over that interval. Allowing it to depend on the future increment would change the problem and introduce look-ahead information.

For a general square-integrable predictable f, approximate it by simple processes in the norm E[∫₀ᵀabs(fₛ)²ds]. Their integrals converge in mean square, and the limit defines the Itô integral. It is not an assertion that arbitrary Riemann sums converge pathwise for every Brownian realization and every sampling choice.

4. Itô’s isometry: the bridge from integrands to random values

If E[∫₀ᵀfₛ²ds]<∞, then:

E[∫₀ᵀfₛdBₛ]=0,

E[(∫₀ᵀfₛdBₛ)²]=E[∫₀ᵀfₛ²ds].

Because the integral has zero mean in this square-integrable setting, the second moment equals its variance. The isometry is fundamentally a second-moment identity; the variance wording relies on that zero-mean property.

For two approximations f and g:

E[abs(∫f dB−∫g dB)²]=E[∫abs(f−g)²ds].

Thus convergence of integrands implies convergence of integrals, and different approximating sequences produce the same limit in L². This is how the construction obtains a stable meaning. MIT’s stochastic-integral lecture develops these integral and diffusion ideas.

Examples and an important distributional distinction

For deterministic fₛ=3, ∫₀ᵀ3dBₛ=3B_T, with mean zero and variance 9T.

For deterministic fₛ=s on [0,2], the integral is Gaussian with mean zero and variance ∫₀²s²ds=8/3. Deterministic square-integrable functions of time produce Gaussian Brownian integrals.

For random fₛ=Bₛ, the variance is ∫₀ᵀE[Bₛ²]ds=T²/2. The integral will turn out to be (B_T²−T)/2, which is not Gaussian. The isometry does not say that a stochastic integral with a random integrand is generally normal.

5. Why ∫B dB is not just B²/2

For each interval:

Bₜₖ₊₁²−Bₜₖ²=2Bₜₖ(Bₜₖ₊₁−Bₜₖ)+(Bₜₖ₊₁−Bₜₖ)².

Sum over the partition and rearrange:

ΣₖBₜₖΔBₖ=½B_T²−½Σₖ(ΔBₖ)².

Along suitable refining deterministic partitions, the left side converges to the Itô integral and the quadratic sum converges to T in the relevant sense. Therefore:

∫₀ᵀBₛdBₛ=½B_T²−½T.

Taking expectations gives zero because E[B_T²]=T. Classical substitution would miss the correction and incorrectly give expected value T/2. At T=2, the integral has mean zero and variance 2, consistent with both the isometry and the variance of the squared Gaussian expression.

Integration conventions matter

The Itô integral uses non-anticipating left-endpoint values in its elementary construction. A Stratonovich integral uses a different limiting convention and, here, gives ∫B∘dB=B_T²/2. These are different integrals, not contradictory answers. For a scalar smooth state-dependent diffusion, converting a Stratonovich SDE to Itô form adds the appropriate ½σσ_x drift correction. Always state the convention.

6. Itô’s formula: the stochastic chain rule

For f∈C² and standard Brownian motion:

df(Bₜ)=f′(Bₜ)dBₜ+½f″(Bₜ)dt.

For f(t,x)∈C^(1,2), with one time derivative and two spatial derivatives:

df(t,Bₜ)=f_t+½f_xxdt+f_x(t,Bₜ)dBₜ.

For dXₜ=μ(t,Xₜ)dt+σ(t,Xₜ)dBₜ, the general scalar version is:

df(t,Xₜ)=f_t+μf_x+½σ²f_xxdt+σ(t,Xₜ)f_x(t,Xₜ)dBₜ.

The derivatives and coefficients are evaluated at the current time and state. Required regularity and any integrability or localization assumptions should be matched to the application. Nonsmooth payoffs at expiry can require limiting arguments or generalized formulas, rather than direct substitution into a classical C² theorem at the kink.

The differential multiplication table is shorthand

The mnemonic (dB)²=dt, dt dB=0, (dt)²=0 records which accumulated Taylor terms survive in the limiting stochastic calculation. It is not a literal equality for each random increment. For X, quadratic variation is [X]ₜ=∫₀ᵗσ(s,Xₛ)²ds; the drift contributes zero.

7. Product rules and multidimensional shocks

For continuous Itô processes X and Y:

d(XY)=X dY+Y dX+d[X,Y].

If dX contains a dB term and dY contains c dB, their instantaneous cross-variation is d[X,Y]=ac dt. With correlated Brownian drivers B¹,B² satisfying d[B¹,B²]ₜ=ρdt, the corresponding cross term incorporates ρ.

In several state dimensions, Itô’s formula uses the gradient and Hessian: the second-order drift includes ½ tr(aaᵀ Hessian(f)), where a is the diffusion loading matrix on independent Brownian shocks. Ignoring covariance terms would miss risk interactions in multi-asset or stochastic-volatility models.

For a simple check, f(B)=B² gives d(B²)=2B dB+dt, recovering the integral result above.

8. Zero drift: local martingale first, true martingale with conditions

If f_t+½f_xx=0, Itô’s formula removes the finite-variation drift from f(t,Bₜ). Under the standard local hypotheses this produces a local martingale. To assert a true martingale and its conditional-expectation identity, additional integrability is needed.

A useful sufficient condition over each finite horizon is E[∫₀ᵀf_x(s,Bₛ)²ds]<∞, together with an integrable initial value. Then:

E[f(t,Bₜ) | ℱₛ]=f(s,Bₛ) for s≤t.

Zero drift alone should not be used to claim every local martingale has constant expectation. The same distinction appears in financial arguments about discounted prices and admissible trading strategies.

Two explicit martingales

For f(t,x)=x²−t, f_t=−1 and f_xx=2, so their drift contribution cancels. Hence Bₜ²−t is a true martingale on finite horizons; its integral representation has the required moments.

For a fixed real λ, Mₜ=exp(λBₜ−½λ²t) is also a true martingale. Gaussian exponential moments give E[Mₜ]=1, and independent increments give E[Mₜ | ℱₛ]=Mₛ. This is the continuous counterpart of normalizing an exponential random walk by its MGF.

The drift-cancellation equation is a backward heat equation in this space-time formulation. A PDE connection is powerful, but its solution must satisfy the boundary, terminal, and integrability conditions relevant to the problem.

9. Geometric Brownian motion and volatility drag

Let dPₜ=μPₜdt+σPₜdBₜ, with P₀>0 and constant μ,σ. Apply f(p)=log p on positive states:

f′(p)=1/p; f″(p)=−1/p²,

giving:

d log Pₜ=(μ−½σ²)dt+σdBₜ.

Integrating and exponentiating yields:

Pₜ=P₀exp[(μ−½σ²)t+σBₜ].

The process remains positive at finite times in this model. Its expected level is E[Pₜ]=P₀exp(μt), while E[log(Pₜ/P₀)]=(μ−½σ²)t. The median is P₀exp[(μ−½σ²)t]. These are distinct summaries.

Worked growth example

Take μ=8% per year, σ=20% per square-root year, and P₀=100. The log-growth rate is 0.08−0.5×0.2²=6%. After one year:

Expected price is 100exp(0.08)≈108.33.
Median price is 100exp(0.06)≈106.18.
Expected log return is 6%, not the same as a 6% simple return.

“Volatility drag” describes the −½σ² adjustment between arithmetic drift and expected log growth in this model. It is not a transaction fee, and holding μ fixed means it does not lower E[Pₜ] below P₀exp(μt). Compounded outcomes and expected wealth are different objectives.

10. Existence and uniqueness: what the theorem guarantees

Consider dXₜ=μ(t,Xₜ)dt+σ(t,Xₜ)dBₜ with an ℱ₀-measurable initial value satisfying E[abs(X₀)²]<∞. A standard sufficient theorem assumes measurable coefficients with bounds, on each finite horizon, of the form:

abs(μ(t,x)−μ(t,y))+abs(σ(t,x)−σ(t,y))≤L abs(x−y),

and abs(μ(t,x))²+abs(σ(t,x))²≤C(1+abs(x)²).

These global spatial Lipschitz and linear-growth conditions give a continuous adapted strong solution, with finite second-moment bounds on finite intervals, and pathwise uniqueness. Strong means the solution uses the specified Brownian motion and initial information; pathwise uniqueness means two solutions driven by the same data agree as processes, up to indistinguishability.

The theorem does not mean moments must be bounded uniformly over an infinite horizon, or that every SDE can be solved in closed form. Its conditions are sufficient, not necessary. Some useful square-root diffusions require more specialized existence and boundary arguments. Local conditions can yield a solution only up to an explosion time unless nonexplosion is established. University of Chicago’s SDE notes discuss strong solutions and uniqueness.

11. Solving by coefficient matching—and its limits

For a proposed representation Xₜ=f(t,Bₜ), Itô’s formula gives drift f_t+½f_xx and diffusion f_x. Matching these to the desired μ(t,f) and σ(t,f) can identify a solution when such a representation exists.

For geometric Brownian motion, try f(t,b)=P₀exp(at+cb). Its diffusion is cf and drift is (a+½c²)f. Matching gives c=σ, a=μ−½σ², reproducing the solution above.

Not every SDE solution is a function of just today’s Bₜ and time. The OU solution below contains a weighted integral of the past Brownian path; generally it cannot be recovered from Bₜ alone. Coefficient matching is a useful method, not a universal closed-form ansatz.

12. Ornstein–Uhlenbeck: an integrating-factor solution

For α>0, consider:

dXₜ=−α(Xₜ−m)dt+σdBₜ.

Here m is the mean-reversion level and α is the speed. Multiply Xₜ−m by exp(αt). Since the integrating factor has finite variation, the product rule gives:

d[exp(αt)(Xₜ−m)]=σexp(αt)dBₜ.

Therefore:

Xₜ=m+(X₀−m)exp(−αt)+σ∫₀ᵗexp[−α(t−s)]dBₛ.

For deterministic X₀=x, the integral has a deterministic kernel and is Gaussian. Its moments are:

E[Xₜ]=m+(x−m)exp(−αt),

Var(Xₜ)=σ²[1−exp(−2αt)]/(2α).

The mean deviation decays exponentially while new shocks keep arriving. As t→∞, the distribution converges to N(m,σ²/(2α)). Individual paths do not settle at m or stop fluctuating. Initializing from the invariant distribution independently of future Brownian increments gives a stationary process; an arbitrary fixed initial value generally gives a transient process approaching that law.

In the stationary model, Cov(Xₛ,Xₜ)=[σ²/(2α)]exp(−α abs(t−s)). Standard OU dynamics are ergodic under these assumptions, connecting long-run time averages with stationary expectations.

Worked OU example

Use m=0.03, α=0.5 per year, σ=0.01 per square-root year, and X₀=0.05. At t=2 years:

E[X₂]=0.03+0.02exp(−1)≈0.037358, or 3.7358%.

Var(X₂)=0.0001[1−exp(−2)]≈0.000086466,

with standard deviation about 0.9299 percentage points. The long-run standard deviation is 1 percentage point, and the expected-deviation half-life is log 2/0.5≈1.3863 years.

Connection to AR(1) and rates

Sampling at interval Δ gives the exact transition:

Xₜ₊Δ=m+(Xₜ−m)exp(−αΔ)+σ√{[1−exp(−2αΔ)]/(2α)} Z,

with a fresh standard normal Z independent of current information. Thus a sampled OU is an AR(1) with coefficient exp(−αΔ) and a precisely specified innovation variance.

The Vasicek short-rate model uses this OU form. It allows negative rates because its Gaussian state is unbounded below; whether that is acceptable depends on the purpose and period. Physical-measure parameters used for forecasting and risk-neutral parameters used for bond pricing need not coincide. OU models are also useful for stylized spread or physical-velocity dynamics, but mean reversion must be supported rather than assumed from a convenient formula.

13. From Itô calculus to the Black–Scholes PDE

Let a non-dividend-paying stock obey dP=μPdt+σPdB and a derivative have value G(t,P). Itô’s formula gives:

dG=[G_t+μPG_P+½σ²P²G_PP]dt+σPG_PdB.

A self-financing replicating strategy holds a=G_P shares and cash C=G−PG_P. Its gains satisfy dV=a dP+rCdt; rebalancing the stock position is financed through cash rather than external deposits.

Matching the stochastic terms and drifts, with V=G, cancels the physical drift μ and gives:

G_t+rPG_P+½σ²P²G_PP−rG=0.

Terminal payoff and suitable boundary conditions complete the pricing problem. The derivative formula alone does not make every market complete or every actual hedge riskless. Continuous trading, the model’s volatility, financing assumptions, and feasible replication are doing important work. Our Black–Scholes lecture develops the binomial and risk-neutral expectation connections.

14. Simulation: discretization is another approximation

Euler–Maruyama approximates a general SDE by:

Xₖ₊₁=Xₖ+μ(tₖ,Xₖ)Δt+σ(tₖ,Xₖ)√Δt Zₖ,

with independent standard normals Zₖ under the basic simulation setup. The √Δt scaling is essential. Numerical convergence needs assumptions, and boundaries or nonlinear coefficients can make a naive scheme behave badly.

For geometric Brownian motion and OU, the exact finite-step transitions above are preferable when their assumptions match the desired model. They avoid Euler discretization error at the sampled dates, though discrete observation can still miss continuous barrier crossings. A simulated path is not a proof of a theorem or of a market model’s empirical validity.

15. Practice questions

If fₛ=2 on [0,3], what are the mean and variance of ∫f dB?
What is ∫₀ᵀBₛdBₛ, and why is it not generally Gaussian?
For f(t,x)=x²−t, what is f_t+½f_xx?
Does cancellation of the Itô drift automatically establish a true martingale with constant expectation?
With GBM drift μ=10% and volatility σ=30%, what is the expected log-growth rate?
For an OU process with α=2 and σ=3, what is its invariant variance?
If an OU process starts from a fixed value away from its mean, is it immediately stationary? Do its paths converge to the mean and remain there?
What extra term appears in d(XY) compared with the ordinary product rule?

Answers

Mean zero, variance 4×3=12 under the stated standard-Brownian convention.
(B_T²−T)/2. It is a shifted and scaled squared Gaussian, not a Gaussian variable.
−1+½×2=0.
No. Local martingale behavior still needs suitable integrability to imply the true martingale identity.
0.10−½×0.3²=0.055, or 5.5% per year in expected log-return units.
σ²/(2α)=9/4=2.25.
No to both. Its distribution approaches the invariant law, while paths keep fluctuating; invariant initialization gives stationarity from the start.
d[X,Y], the quadratic covariation term.