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Stochastic Calculus / Stochastic Integration and Itô's Formula

Extended Itô Formulas: Products, Quotients, and Time-Dependent Functions

The basic Itô formula handles f(Wt)f(W_t) — a fixed function of Brownian motion at the current time. In practice, options pricing requires f(t,St)f(t, S_t) (where ff carries explicit time-dependence — the discount factor, the remaining time to expiry), and portfolio analysis requires d(XtYt)d(X_t Y_t) (the product of a position size and a price). These demand three extensions:
  1. Formula II (time-dependent ff): adds ∂tf\partial_t f to the drift.
  2. Formula III (Itô process): replaces dWtdW_t by a general dXt=Rtdt+AtdWtdX_t = R_t dt + A_t dW_t.
  3. Product rule: d(XtYt)=Xt dYt+Yt dXt+d[X,Y]td(X_t Y_t) = X_t \, dY_t + Y_t \, dX_t + d[X,Y]_t — the analogue of integration by parts.

The product rule is the most frequently used extension: it underlies the derivation of the Black-Scholes PDE, the self-financing condition for a portfolio, and the Itô exponential (Doléans-Dade exponential) used in Girsanov's theorem.

Learning objectives. By the end of this note you should be able to:
  • State Itô's Formula II (f(t,Wt)f(t, W_t)) and apply it to time-dependent examples.
  • State Itô's Formula III (f(t,Xt)f(t, X_t) for general Itô processes) and compute df(t,Xt)df(t, X_t) in explicit problems.
  • Apply the stochastic product rule d(XY)=X dY+Y dX+d[X,Y]d(XY) = X \, dY + Y \, dX + d[X, Y].
  • Derive the quotient rule d(X/Y)d(X/Y) from the product rule plus Itô's formula.
  • Solve the exponential SDE dXt=AtXt dWtdX_t = A_t X_t \, dW_t and identify the Itô exponential.

In this lesson

  1. Formula II: Time-Dependent Functions
  2. Formula III: Itô Processes
  3. The Product Rule
  4. The Quotient Rule
  5. The Itô (Doléans-Dade) Exponential

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