## [PDS] Ito Calculus: Main Results

**Ito Calculus: Main Results**

**1. Definition of the Ito Integral**: Given the Brownian Motion , the Ito Integral of a function is defined by the limit:

It is important that the limit be over the ‘lower sum’, i.e. the increment of the Brownian Motion is considered *after* the function evaluation at . This is a crucial point. In finance, the hedgers and speculators are worried about the change in their portfolio \textit{after} the trade has been made. One can think of as the trading strategy, and as a ‘shock’ to the portfolio (the change in the prices of underlying securities).

**2. Ito’s Lemma**: Given a function of the Brownian Motion , Ito’s Lemma is a relationship between the ‘value’ of the function of the Brownian Motion in terms of its derivatives:

Note that integral is an Ito integral and has to be interpreted in the Ito sense (as above).

**3. The SDE (‘short-hand’) Version of Ito’s Lemma**: Since the Brownian Motion is not differentiable anywhere, technically one can’t differentiate the integral version of Ito’s Lemma on both sides to write the corresponding SDE. That said, since the SDE version of Ito’s Lemma ‘looks like’ the ever-so-familiar Taylor series, it is sometimes easier to remember Ito’s Lemma as an SDE, as long as one keeps in mind that the correct way is the integral way.

**4. Ito’s Lemma for **: If there is an explicit dependence of on time , Ito’s Lemma changes predictably as:

**5. Ito’s Lemma for **: Given the stock price process :

Ito’s Lemma for a function of the stock price and time, is written as:

Ignoring all terms , and setting in the mean-square sense:

In general, Ito’s Lemma for a function of any linear SDE of the form:

takes the form:

**6. Ito’s Lemma for **: Given two processes and :

Ito’s Lemma for a function is given as:

Later we use this result to write the Chain Rule in Ito Calculus.

**7. Ito integrals are Martingales**: Given any process , under mild integrability conditions (see Bjork or Shreve for details), the following holds:

We can generalize to write that, modulo an integrability condition, is a martingale. That is, every Ito integral is a martingale.

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