Free lesson · Stochastic calc
Itô integration and quadratic variation
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Start with the idea
- An Itô integral sums a non-anticipating exposure times the next random increment.
- Brownian paths are too irregular for ordinary pathwise Riemann calculus to apply unchanged.
- Their accumulated squared increments contribute a nonzero term.
Symbols, units & horizon
- ∫ H_t dW_t: Itô integral of exposure H against Brownian motion
- H_t: adapted square-integrable integrand
- W_T: terminal Brownian value with W₀=0
- T: elapsed time
- E: expectation
- Σ(ΔW)²: realised squared-increment sum
- Quadratic variation: limiting accumulated squared increments
- The second identity: Itô isometry, under its integrability assumptions
- Integral units: H-units multiplied by square-root-time units
- t: integration time variable
- dt: ordinary time increment
- dW_t: stochastic integrator, not an ordinary velocity
When and why to use this
Use the discrete telescoping identity to understand why ordinary chain rules miss a term, and use the isometry for variance reasoning when its assumptions apply.
- For a square-integrable adapted step process, multiply the exposure fixed at the start of an interval by the Brownian increment over that interval.
- The general Itô integral is defined by an appropriate mean-square limit.
- This explanation is a construction outline, not a full measure-theoretic proof.
- For standard Brownian motion, the sum of squared increments over refining deterministic partitions converges to elapsed time in the quadratic-variation sense.
- This produces the familiar shorthand (dW)²=dt; it is not an equality between ordinary small real numbers.
Itô integration and quadratic variation
- On a partition, expand W_next²−W_current²=2W_current ΔW+(ΔW)².
- Sum over intervals to telescope: Σ W_current ΔW=(W_T²−Σ(ΔW)²)/2. Taking the Itô limit gives (W_T²−T)/2.
- For adapted step exposures, cross terms have zero expectation by conditioning on the past, while each diagonal contribution is E[H_k²]Δt. Summing and passing to the mean-square limit gives the isometry.
For one coarse path with increments .3 and −.1, W_T=.2, squared-increment sum=.10, and the left sum is 0×.3+.3×(−.1)=−.03. The exact discrete identity gives (.04−.10)/2=−.03. A coarse realised sum need not equal T.
An analogy to remember
Choose an exposure before the next coin-like shock, then record its contribution. Choosing it after seeing that shock changes the information rule and therefore the mathematical object.
How this becomes a building block
A self-financing diffusion trading gain has stochastic-integral terms. The isometry converts a second moment of accumulated noise exposure into an ordinary expected time integral. The quadratic-variation correction is the component that becomes gamma’s Itô term in an option model.
Python implementation
Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.
def ito_left_sum(increments):
w=0.0
integral=0.0
variation=0.0
for dw in increments:
integral+=w*dw
variation+=dw*dw
w+=dw
return integral,(w*w-variation)/2,variation
def constant_exposure_variance(exposure,horizon):
return exposure**2*horizon
print(ito_left_sum([.3,-.1]),constant_exposure_variance(2,1))Continue learning
Stochastic Calculus — all lessons- Start with one random step before continuous noise
- Before Itô: random walks, Brownian motion and information
- Itô integration and quadratic variation
- SDEs, Euler–Maruyama and an Ornstein–Uhlenbeck example
- Geometric Brownian motion: the model under Black–Scholes
- Itô's lemma: why gamma exists
- Black–Scholes and delta hedging in production
- Put–call parity and a full Greek P&L budget
Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations