Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
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Math 489/Math 889
Stochastic Processes and
Advanced Mathematical Finance
Dunbar, Fall 2009

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Itô’s Formula

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Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

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Section Starter Question

State the Taylor expansion of a function $f (x)$ up to order $1$ . What is the relation of this expansion to the Mean Value Theorem of calculus? What is the relation of this expansion to the Fundamental Theorem of calculus?

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Key Concepts

Itô’s formula is an expansion expressing a stochastic process in terms of the deterministic differential and the Wiener process differential, that is, the stochastic differential equation for the process.
Solving stochastic differential equations follows by guessing solutions based on comparison with the form of Itô’s formula.

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Vocabulary

Itô’s formula is often also called Itô’s lemma by other authors and texts. This result is more important than a mere lemma, and so we use the alternative name of “formula”. “Formula” also emphasizes the analogy with the chain “rule” and the Taylor “expansion”.

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Mathematical Ideas

Itô’s Formula and Itô calculus

We need some operational rules that allow us to manipulate stochastic processes with stochastic calculus.

The important thing to know about traditional differential calculus is that it is the

the Fundamental Theorem of Calculus,
chain rule, and
Taylor polynomials and Taylor series

that enable us to calculate with functions. A deeper understanding of calculus recognizes that these three calculus theorems are all aspects of the same fundamental idea. Likewise we need similar rules and formulas for stochastic processes. Itô’s formula will perform that function for us. However, Itô’s formula acts in the capacity of all three of the calculus theorems, and so we have only one such theorem for stochastic calculus.

The next example will show us that we will need some new rules for stochastic calculus because the old rules from calculus will no longer make sense.

Example. Consider the process which is the square of the Wiener process:

Y (t) = W {(t)}^{2} .

We notice that this process is always non-negative, $Y (0) = 0$ , $Y$ has infinitely many zeroes on $t > 0$ and $E [Y (t)] = E [W {(t)}^{2}] = t$ . What more can we say about this process? For example, what is the stochastic differential of $Y (t)$ and what would that tell us about $Y (t)$ ?

Using naive calculus, we might conjecture using the ordinary chain rule

dY = 2 W (t) dW(t) .

If that were true then the Fundamental Theorem of Calculus would imply

Y (t) = \int_{0}^{t} dY = \int_{0}^{t} 2 W (t) dW(t)

should also be true. But consider $\int_{0}^{t} 2 W (t) dW(t)$ . It ought to correspond to a limit of a summation (for instance a Riemann-Stieltjes left sum):

\int_{0}^{t} 2 W (t) dW(t) \approx \sum_{i = 1}^{n} 2 W ((i - 1) t ∕ n) [W (i t ∕ n) - W ((i - 1) t ∕ n)]

But look at this carefully: $W ((i - 1) t ∕ n) = W ((i - 1) t ∕ n) - W (0)$ is independent of $[W (i t ∕ n) - W ((i - 1) t ∕ n)]$ by property 2 of the definition of the Wiener process. Therefore, the expected value, or mean, of the summation will be zero:

\begin{array}{l} E [Y (t)] & = E [\int_{0}^{t} 2 W (t) \underset{̣}{W (t)}] \\ = E [lim_{n \to \infty} \sum_{i = 1}^{n} 2 W ((i - 1) t ∕ n) (W (i t ∕ n) - W ((i - 1) t ∕ n))] \\ = lim_{n \to \infty} \sum_{i = 1}^{n} 2 E [[W ((i - 1) t ∕ n) - W (0)] [W (i t ∕ n) - W ((i - 1) t ∕ n)]] \\ = 0 . \end{array}

(Note the assumption that the limit and the expectation can be interchanged!)

But the mean of $Y (t) = W {(t)}^{2}$ is $t$ which is definitely not zero! The two stochastic processes don’t agree even in the mean, so something is not right! If we agree that the integral definition and limit processes should be preserved, then the rules of calculus will have to change.

We can see how the rules of calculus must change by rearranging the summation. Use the simple algebraic identity

2 b (a - b) = (a^{2} - b^{2} - {(a - b)}^{2})

to re-write

\begin{array}{l} \int_{0}^{t} 2 W (t) dW(t) & = lim_{n \to \infty} \sum_{i = 1}^{n} 2 W ((i - 1) t ∕ n) [W (i t ∕ n) - W ((i - 1) t ∕ n)] \\ = lim_{n \to \infty} \sum_{i = 1}^{n} (W {(i t ∕ n)}^{2} - W {((i - 1) t ∕ n)}^{2} - {(W (i t ∕ n) - W ((i - 1) t ∕ n))}^{2}) \\ = lim_{n \to \infty} (W {(t)}^{2} - W {(0)}^{2} - \sum_{i = 1}^{n} {(W (i t ∕ n) - W ((i - 1) t ∕ n))}^{2}) \\ = W {(t)}^{2} - lim_{n \to \infty} \sum_{i = 1}^{n} {(W (i t ∕ n) - W ((i - 1) t ∕ n))}^{2} \end{array}

We recognize the second term in the last expression as being the quadratic variation of Wiener process, which we have already evaluated, and so

\int_{0}^{t} 2 W (t) dW(t) = W {(t)}^{2} - t .

Theorem 1 (Itô’s formula for functions of scaled Wiener process with drift). If $Y (t)$ is scaled Wiener process with drift, satisfying $dY = r dt + σ dW$ and $f$ is a twice continuously differentiable function, then $Z (t) = f (Y (t))$ is also a stochastic process satisfying the stochastic differential equation

dZ = (r f^{'} (Y) + (σ^{2} ∕ 2) f^{''} (Y)) dt + (σ f^{'} (Y)) dW .

In words, Itô’s formula in this form tells us how to expand (in analogy with the chain rule or Taylor’s formula) the differential of a process which is defined as an elementary function of scaled Brownian motion with drift.

Example. Consider $Z (t) = W {(t)}^{2}$ . Here the stochastic process is standard Brownian Motion, so $r = 0$ and $σ = 1$ so $dY = dW$ . The twice continuously differentiable function $f$ is the squaring function, $f (x) = x^{2}$ , $f^{'} (x) = 2 x$ and $f^{''} (x) = 2$ . Then according to Itô’s formula:

d(W^{2}) = (0 \cdot (2 W (t)) + (1 ∕ 2) (2)) dt + (1 \cdot 2 W (t)) dW = dt + 2 W (t) dW

Notice the additional $dt$ term! Note also that if we repeated the integration steps above in the example, we would obtain $W {(t)}^{2}$ as expected!

The case where $dY = dW$ , that is the base process is Standard Brownian Motion so $Z = f (W)$ , occurs commonly enough that we record Itô’s formula for this special case:

Corollary 1 (Itô’s Formula applied to functions of Standard Brownian Motion). If $f$ is a twice continuously differentiable function, then $Z (t) = f (W (t))$ is also a stochastic process satisfying the stochastic differential equation

dZ = d(W) = (1 ∕ 2) f^{''} (W) dt + f^{'} (W) dW .

Example. Consider Geometric Brownian Motion

exp (r t + σ W (t)) .

What SDE does Geometric Brownian Motion follow? Take $Y (t) = r t + σ W (t)$ , so that $dY = r dt + σ dW$ . Then Geometric Brownian Motion can be written as $Z (t) = exp (Y (t))$ , so $f$ is the exponential function. Itô’s formula is

dZ = (r f^{'} (Y (t)) + (1 ∕ 2) σ^{2} f^{''} Y (t)) dt + σ f^{'} (Y) dW

Computing the derivative of the exponential function and evaluating, $f^{'} (Y (t)) = exp (Y (t)) = Z (t)$ and likewise for the second derivative. Hence

dZ = (r + (1 ∕ 2) σ^{2}) Z (t) dt + σ Z (t) dW

The corollary above simplifies Itô’s Formula to functions of a single variable evaluated along a Standard Brownian Motion. This will be sufficient for the purposes of this text. Consult the references, e.g. Allen [1], for more general versions of Itô’s Formula which consider functions of two variables applied to a process which is the solution of a stochastic differential equation.

Guessing Processes from SDEs with Itô’s Formula

One of the key needs we will have is to go in the opposite direction and convert SDEs to processes, in other words to solve SDEs. We take guidance from ordinary differential equations, where finding solutions to differential equations comes from judicious guessing based on a through understanding and familiarity with the chain rule. For SDEs the solution depends on inspired guesses based on a thorough understanding of the formulas of stochastic calculus. Following the guess we require a proof that the proposed solution is an actual solution, again using the formulas of stochastic calculus.

A few rare examples of SDEs can be solved with explicit familiar functions. This is just like ODEs in that the solutions of many simple differential equations cannot be solved in terms of elementary functions. The solutions of the differential equations define new functions which are useful in applications. Likewise, the solution of an SDE gives us a way of defining new processes which are useful.

Example. Suppose we are asked to solve the SDE

dZ(t) = σ Z (t) dW .

We need an inspired guess, so we try

exp (r t + σ W (t))

where $r$ is a constant to be determined while the $σ$ term is given in the SDE. Itô’s formula for the guess is

dZ = (r + (1 ∕ 2) σ^{2}) Z (t) dt + σ Z (t) dW .

We notice that the stochastic term (or Wiener process differential term) is the same as the SDE. We need to choose the constant $r$ appropriately in the deterministic or drift differential term. If we choose $r$ to be $- (1 ∕ 2) σ^{2}$ then the drift term in the differential equation would match the SDE we have to solve as well. We therefore guess

Y (t) = exp (σ W (t) - (1 ∕ 2) σ^{2} t) .

We should double check by applying Itô’s formula.

Solvable SDEs are scarce, and this one is special enough to give a name. It is the Dolèan’s exponential of Brownian motion.

Sources

This discussion is adapted from Financial Calculus: An introduction to derivative pricing by M Baxter, and A. Rennie, Cambridge University Press, 1996, pages 52–62, “An Algorithmic Introduction to the Numerical Simulation of Stochastic Differential Equations”, by Desmond J. Higham, in SIAM Review, Vol. 43, No. 3, pages 525–546, 2001, and An Introduction to Stochastic Processes with Applications to Biology, by L. J. S. Allen, Pearson Prentice-Hall, 2003, pages 342-343.

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Problems to Work

Problems to Work for Understanding

Find the stochastic differential equation governing the process $Y (t) = sin (W (t))$ , where $W (t)$ is Standard Brownian Motion.
Use Itô’s Formula to find the stochastic differential equation governing the process $Y (t) = W^{3} (t)$ and use the result to evaluate $\int_{0}^{t} W^{2} (t) dW(t) .$
Find the solution of the stochastic differential equation $dY (t) = Y (t) dt + 2 Y (t) dW .$
Find the solution of the stochastic differential equation $dY (t) = μ Y (t) dt + σ Y (t) dW .$
Find the solution of the stochastic differential equation $dY (t) = t Y (t) dt + 2 Y (t) dW$
Note the difference with the previous problem, now the multiplier of the $dt$ term is a function of time.
Find the solution of the stochastic differential equation $dY (t) = μ t Y (t) dt + σ Y (t) dW$
Note the difference with the previous problem, now the multiplier of the $dt$ term is a function of time.
Find the solution of the stochastic differential equation $dY (t) = μ (t) Y (t) dt + σ Y (t) dW$
Note the difference with the previous problem, now the multiplier of the $dt$ term is a general (technically, a locally bounded integrable) function of time.

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Books

Reading Suggestion:

References

[1] Linda J. S. Allen. An Introduction to Stochastic Processes with Applications to biology. Pearson Prentice-Hall, 2003.

[2] M. Baxter and A. Rennie. Financial Calculus: An introduction to derivative pricing. Cambridge University Press, 1996. HG 6024 A2W554.

[3] Desmond J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 43(3):525–546, 2001.

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Links

Outside Readings and Links:

Itô’s Lemma. Contains more general statements, applications and further links.

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