Stochastic Processes and Brownian Motion

Stochastic Processes and Brownian Motion

Of all the intellectual hurdles which the human mind has confronted and has overcome in the last fifteen hundred years, the one which seems to me to have been the most amazing in character and the most stupendous in the scope of its consequences is the one relating to the problem of motion.

— Herbert Butterfield (1900–1979)

Stochastic Processes

• A stochastic process

X = { X(t) } is a time series of random variables.

• X(t) (or X_t) is a random variable for each time t and is usually called the state of the process at time t.

• A realization of X is called a sample path.

Stochastic Processes (concluded)

• If the times t form a countable set, X is called a discrete-time stochastic process or a time series.

• In this case, subscripts rather than parentheses are usually employed, as in

X = { X_n }.

• If the times form a continuum, X is called a continuous-time stochastic process.

Random Walks

• The binomial model is a random walk in disguise.

• Consider a particle on the integer line, 0, ±1, ±2, . . . .

• In each time step, it can make one move to the right with probability p or one move to the left with

probability 1 − p.

– This random walk is symmetric when p = 1/2.

• Connection with the BOPM: The particle’s position denotes the number of up moves minus that of down moves up to that time.

20 40 60 80 Time

-8 -6 -4 -2 2 4

Position

Random Walk with Drift

X_n = μ + X_n−1 + ξ_n.

• ξ_n are independent and identically distributed with zero mean.

• Drift μ is the expected change per period.

• Note that this process is continuous in space.

Martingales

• { X(t), t ≥ 0 } is a martingale if E[ | X(t) | ] < ∞ for t ≥ 0 and

E[ X(t) | X(u), 0 ≤ u ≤ s ] = X(s), s ≤ t. (68)

• In the discrete-time setting, a martingale means

E[ X_n+1 | X₁, X₂, . . . , X_n ] = X_n. (69)

• X_n can be interpreted as a gambler’s fortune after the nth gamble.

• Identity (69) then says the expected fortune after the (n + 1)th gamble equals the fortune after the nth gamble regardless of what may have occurred before.

aThe origin of the name is somewhat obscure.

Martingales (concluded)

• A martingale is therefore a notion of fair games.

• Apply the law of iterated conditional expectations to both sides of Eq. (69) on p. 544 to yield

E[ X_n ] = E[ X₁ ] (70) for all n.

• Similarly,

E[ X(t) ] = E[ X(0) ] in the continuous-time case.

Still a Martingale?

• Suppose we replace Eq. (69) on p. 544 with E[ X_n+1 | X_n ] = X_n.

• It also says past history cannot affect the future.

• But is it equivalent to the original definition (69) on p. 544?^a

aContributed by Mr. Hsieh, Chicheng (M9007304) on April 13, 2005.

Still a Martingale? (continued)

• Well, no.^a

• Consider this random walk with drift:

X_i =

⎧⎨

⎩

X_i−1 + ξ_i, if i is even, X_i−2, otherwise.

• Above, ξ_n are random variables with zero mean.

aContributed by Mr. Zhang, Ann-Sheng (B89201033) on April 13, 2005.

Still a Martingale? (concluded)

• It is not hard to see that

E[ X_i | X_i−1 ] =

⎧⎨

⎩

X_i−1, if i is even, X_i−1, otherwise.

– It is a martingale by the “new” definition.

• But

E[ X_i | . . . , X_i−2, X_i−1 ] =

⎧⎨

⎩

X_i−1, if i is even, X_i−2, otherwise.

– It is not a martingale by the original definition.

Example

• Consider the stochastic process



Z_n =^Δ

n i=1

X_i, n ≥ 1

 ,

where X_i are independent random variables with zero mean.

• This process is a martingale because E[ Z_n+1 | Z1, Z2, . . . , Z_n ]

= E[ Z_n + X_n+1 | Z1, Z₂, . . . , Z_n ]

= E[ Z_n | Z1, Z₂, . . . , Z_n ] + E[ X_n+1 | Z1, Z₂, . . . , Z_n ]

Probability Measure

• A probability measure assigns probabilities to states of the world.^a

• A martingale is defined with respect to a probability measure, under which the expectation is taken.

• Second, a martingale is defined with respect to an information set.

– In the characterizations (68)–(69) on p. 544, the

information set contains the current and past values of X by default.

– But it need not be so.

aOnly certain sets such as Borel sets receive probabilities (Feller, 1972).

Probability Measure (continued)

• A stochastic process { X(t), t ≥ 0 } is a martingale with respect to information sets { I_t } if, for all t ≥ 0,

E[ | X(t) | ] < ∞ and

E[ X(u) | I_t ] = X(t) for all u > t.

• The discrete-time version: For all n > 0, E[ X_n+1 | I_n ] = X_n, given the information sets { I_n }.

Probability Measure (concluded)

• The above implies

E[ X_n+m | I_n ] = X_n for any m > 0 by Eq. (26) on p. 169.

– A typical I_n is the price information up to time n.

– Then the above identity says the FVs of X will not deviate systematically from today’s value given the price history.

Example

• Consider the stochastic process { Z_n − nμ, n ≥ 1 }.

– Z_n =^Δ _n

i=1 X_i.

– X1, X2, . . . are independent random variables with mean μ.

• Now,

E[ Z_n+1 − (n + 1) μ | X₁, X₂, . . . , X_n ]

= E[ Z_n+1 | X₁, X₂, . . . , X_n ] − (n + 1) μ

= E[ Z_n + X_n+1 | X1, X2, . . . , X_n ] − (n + 1) μ

= Z_n + μ − (n + 1) μ

− nμ.

Example (concluded)

• Define

I_n =^Δ { X1, X₂, . . . , X_n }.

• Then

{ Z_n − nμ, n ≥ 1 } is a martingale with respect to { I_n }.

Martingale Pricing

• Stock prices and zero-coupon bond prices are expected to rise, while option prices are expected to fall.

• They are not martingales.

• Why is then martingale useful?

• Recall a martingale is defined with respect to some information set and some probability measure.

• By modifying the probability measure, we can convert a price process into a martingale.

Martingale Pricing (continued)

• The price of a European option is the expected discounted payoff in a risk-neutral economy.^a

• This principle can be generalized using the concept of martingale.

• Recall the recursive valuation of European option via C = [ pC_u + (1 − p) C_d ]/R.

– p is the risk-neutral probability.

– $1 grows to $R in a period.

aRecall Eq. (37) on p. 265.

Martingale Pricing (continued)

• Let C(i) denote the value of the option at time i.

• Consider the discount process C(i)

Rⁱ , i = 0, 1, . . . , n

.

• Then,

E

 C(i + 1) Rⁱ⁺¹

 S(i)

= E[ C(i + 1) | S(i) ] Rⁱ⁺¹

= pC_u + (1 − p) C_d Rⁱ⁺¹

= C(i) .

Martingale Pricing (continued)

• It is easy to show that E

 C(k) R^k

 S(i)

= C(i)

Rⁱ , i ≤ k.

• This simplified formulation assumes:^a

1. The model is Markovian: The distribution of the

future is determined by the present (time i) and not the past.

2. The payoff depends only on the terminal price of the underlying asset^b (Asian options do not qualify).

aContributed by Mr. Wang, Liang-Kai (Ph.D. student, ECE, Univer- sity of Wisconsin-Madison) and Mr. Hsiao, Huan-Wen (B90902081) on May 3, 2006.

bRecall they are called simple claims.

Martingale Pricing (continued)

• In general, the discount process is a martingale in that^a E_i^π

 C(k) R^k

= C(i)

Rⁱ , i ≤ k. (71) – E_i^π is taken under the risk-neutral probability

conditional on the price information up to time i.

• This risk-neutral probability is also called the EMM, or the equivalent^b martingale (probability) measure.

aIn this general formulation, Asian options do qualify.

bTwo probability measures are said to be equivalent if they assign nonzero probabilities to the same set of states.

Martingale Pricing (continued)

• Equation (71) holds for all assets, not just options.

• When interest rates are stochastic, the equation becomes C(i)

M (i) = E_i^π

 C(k) M (k)

, i ≤ k. (72)

– M(j) is the balance in the money market account at time j using the rollover strategy with an initial

investment of $1.

– It is called the bank account process.

• It says the discount process is a martingale under π.

Martingale Pricing (continued)

• If interest rates are stochastic, then M(j) is a random variable.

– M(0) = 1.

– M(j) is known at time j − 1.^a

• Identity (72) on p. 560 is the general formulation of risk-neutral valuation.

aBecause the interest rate for the next period has been revealed then.

Martingale Pricing (concluded)

Theorem 16 A discrete-time model is arbitrage-free if and only if there exists an equivalent probability measure^a such that the discount process is a martingale.

aCalled the risk-neutral probability measure.

Futures Price under the BOPM

• Futures prices form a martingale under the risk-neutral probability.

– The expected futures price in the next period is^a pfF u + (1 − pf) F d = F

1 − d

u − d u + u − 1 u − d d



= F.

• Can be generalized to

F_i = E_i^π[ F_k ], i ≤ k, where F_i is the futures price at time i.

• This equation holds under stochastic interest rates, too.^b

aRecall p. 517.

Martingale Pricing and Numeraire

• The martingale pricing formula (72) on p. 560 uses the money market account as numeraire.

– It expresses the price of any asset relative to the money market account.^b

• The money market account is not the only choice for numeraire.

• Suppose asset S’s value is positive at all time.

aJohn Law (1671–1729), “Money to be qualified for exchaning goods and for payments need not be certain in its value.”

bLeon Walras (1834–1910).

Martingale Pricing and Numeraire (concluded)

• Choose S as numeraire.

• Martingale pricing says there exists a risk-neutral

probability π under which the relative price of any asset C is a martingale:

C(i)

S(i) = E_i^π

 C(k) S(k)

, i ≤ k.

– S(j) denotes the price of S at time j.

• So the discount process remains a martingale.^a

aThis result is related to Girsanov’s theorem (1960).

Example

• Take the binomial model with two assets.

• In a period, asset one’s price can go from S to S1 or S₂.

• In a period, asset two’s price can go from P to P1 or P2.

• Both assets must move up or down at the same time.

• Assume

S1

P1 < S

P < S2

P2 (73)

to rule out arbitrage opportunities.

Example (continued)

• For any derivative security, let C₁ be its price at time one if asset one’s price moves to S₁.

• Let C₂ be its price at time one if asset one’s price moves to S₂.

• Replicate the derivative by solving

αS₁ + βP₁ = C₁, αS2 + βP2 = C2,

using α units of asset one and β units of asset two.

Example (continued)

• By inequalities (73) on p. 566, α and β have unique solutions.

• In fact,

α = P₂C₁ − P1C₂

P2S1 − P1S2 and β = S₂C₁ − S1C₂ S2P1 − S1P2 .

• The derivative costs C = αS + βP

= P2S − P S2

P2S1 − P1S2 C1 + P S1 − P1S

P2S1 − P1S2 C2.

Example (continued)

• It is easy to verify that C

P = p C₁

P1 + (1 − p) C₂ P2

with

p =^Δ (S/P ) − (S₂/P₂) (S₁/P₁) − (S2/P₂).

• By inequalities (73) on p. 566, 0 < p < 1.

• C’s price using asset two as numeraire (i.e., C/P ) is a martingale under the risk-neutral probability p.

• The expected returns of the two assets are irrelevant.

Example (concluded)

• In the BOPM, S is the stock and P is the bond.

• Furthermore, p assumes the bond is the numeraire.

• In the binomial option pricing formula (39) on p. 269, S 

b(j; n, pu/R) uses stock as the numeraire.

– Its probability measure pu/R differs from p.

• SN(x) for the call and SN(−x) for the put in the Black-Scholes formulas (p. 299) use stock as the numeraire as well.^a

aSee Exercise 13.2.12 of the textbook.

Brownian Motion

• Brownian motion is a stochastic process { X(t), t ≥ 0 } with the following properties.

1. X(0) = 0, unless stated otherwise.

2. for any 0 ≤ t₀ < t₁ < · · · < t_n, the random variables X(t_k) − X(t_k−1)

for 1 ≤ k ≤ n are independent.^b

3. for 0 ≤ s < t, X(t) − X(s) is normally distributed with mean μ(t − s) and variance σ²(t − s), where μ and σ = 0 are real numbers.

aRobert Brown (1773–1858).

Brownian Motion (concluded)

• The existence and uniqueness of such a process is guaranteed by Wiener’s theorem.^a

• This process will be called a (μ, σ) Brownian motion with drift μ and variance σ².

• Although Brownian motion is a continuous function of t with probability one, it is almost nowhere differentiable.

• The (0, 1) Brownian motion is called the Wiener process.

• If condition 3 is replaced by “ X(t) − X(s) depends only on t − s,” we have the more general Levy process.^b

aNorbert Wiener (1894–1964). He received his Ph.D. from Harvard in 1912.

bPaul Levy (1886–1971).

Example

• If { X(t), t ≥ 0 } is the Wiener process, then X(t) − X(s) ∼ N (0, t − s).

• A (μ, σ) Brownian motion Y = { Y (t), t ≥ 0 } can be expressed in terms of the Wiener process:

Y (t) = μt + σX(t). (74)

• Note that

Y (t + s) − Y (t) ∼ N (μs, σ²s).

Brownian Motion as Limit of Random Walk

Claim 1 A (μ, σ) Brownian motion is the limiting case of random walk.

• A particle moves Δx to the right with probability p after Δt time.

• It moves Δx to the left with probability 1 − p.

• Define

X_i =^Δ

⎧⎨

⎩

+1 if the ith move is to the right,

−1 if the ith move is to the left.

– X_i are independent with

Prob[ X_i = 1 ] = p = 1 − Prob[ X_i = −1 ].

Brownian Motion as Limit of Random Walk (continued)

• Recall

E[ X_i ] = 2p − 1,

Var[ X_i ] = 1 − (2p − 1)².

• Assume n = t/Δt is an integer.^Δ

• Its position at time t is

Y (t) = Δx (X^Δ ₁ + X₂ + · · · + X_n) .

Brownian Motion as Limit of Random Walk (continued)

• Therefore,

E[ Y (t) ] = n(Δx)(2p − 1), Var[ Y (t) ] = n(Δx)² 

1 − (2p − 1)²  .

• With Δx = σ^Δ √

Δt and p = [ 1 + (μ/σ)^Δ √

Δt ]/2,^a E[ Y (t) ] = nσ√

Δt (μ/σ)√

Δt = μt, Var[ Y (t) ] = nσ²Δt 

1 − (μ/σ)²Δt

→ σ²t, as Δt → 0.

aIdentical to Eq. (42) on p. 292!

Brownian Motion as Limit of Random Walk (concluded)

• Thus, { Y (t), t ≥ 0 } converges to a (μ, σ) Brownian motion by the central limit theorem.

• Brownian motion with zero drift is the limiting case of symmetric random walk by choosing μ = 0.

• Similarity to the the BOPM: The p is identical to the probability in Eq. (42) on p. 292 and Δx = ln u.

• Note that

Var[ Y (t + Δt) − Y (t) ]

=Var[ Δx X_n+1 ] = (Δx)² × Var[ X_n+1 ] → σ²Δt.

Geometric Brownian Motion

• Let X =^Δ { X(t), t ≥ 0 } be a Brownian motion process.

• The process

{ Y (t) = e^{Δ X(t)}, t ≥ 0 }, is called geometric Brownian motion.

• Suppose further that X is a (μ, σ) Brownian motion.

• By assumption, Y (0) = e⁰ = 1.

Geometric Brownian Motion (concluded)

• X(t) ∼ N(μt, σ²t) with moment generating function E



e^sX(t)



= E [ Y (t)^s ] = e^{μts+(σ2ts2/2)} from Eq. (27) on p 171.

• In particular,^a

E[ Y (t) ] = e^μt+(σ2t/2), Var[ Y (t) ] = E 

Y (t)² 

− E[ Y (t) ]²

= e^2μt+σ2t



e^σ2t − 1 .

0.2 0.4 0.6 0.8 1 Time (t) -1

1 2 3 4 5 6 Y(t)

A Case for Long-Term Investment

• Suppose the stock follows the geometric Brownian motion

S(t) = S(0) e^{N (μt,σ2t)} = S(0) e^{tN (μ,σ2/t )}, t ≥ 0, where μ > 0.

• The annual rate of return has a normal distribution:

N



μ, σ² t

 .

• The larger the t, the likelier the return is positive.

• The smaller the t, the likelier the return is negative.

aContributed by Dr. King, Gow-Hsing on April 9, 2015. See

Continuous-Time Financial Mathematics

A proof is that which convinces a reasonable man;

a rigorous proof is that which convinces an unreasonable man.

— Mark Kac (1914–1984) The pursuit of mathematics is a divine madness of the human spirit.

— Alfred North Whitehead (1861–1947), Science and the Modern World

Stochastic Integrals

• Use W =^Δ { W (t), t ≥ 0 } to denote the Wiener process.

• The goal is to develop integrals of X from a class of stochastic processes,^a

I_t(X) =^Δ

 _t

0 X dW, t ≥ 0.

• I_t(X) is a random variable called the stochastic integral of X with respect to W .

• The stochastic process { I_t(X), t ≥ 0 } will be denoted by 

X dW .

aKiyoshi Ito (1915–2008).

Stochastic Integrals (concluded)

• Typical requirements for X in financial applications are:

– Prob[ _t

0 X²(s) ds < ∞ ] = 1 for all t ≥ 0 or the stronger  _t

0 E[ X²(s) ] ds < ∞.

– The information set at time t includes the history of X and W up to that point in time.

– But it contains nothing about the evolution of X or W after t (nonanticipating, so to speak).

– The future cannot influence the present.

Ito Integral

• A theory of stochastic integration.

• As with calculus, it starts with step functions.

• A stochastic process { X(t) } is simple if there exist 0 = t₀ < t₁ < · · ·

such that

X(t) = X(t_k−1) for t ∈ [ t_k−1, t_k), k = 1, 2, . . . for any realization (see figure on next page).^a

aIt is right-continuous.

J J J J_! J_" J_# : J

= B

J

Ito Integral (continued)

• The Ito integral of a simple process is defined as I_t(X) =^Δ

n−1

k=0

X(t_k)[ W (t_k+1) − W (t_k) ], (75) where t_n = t.

– The integrand X is evaluated at t_k, not t_k+1.

• Define the Ito integral of more general processes as a limiting random variable of the Ito integral of simple stochastic processes.

Ito Integral (continued)

• Let X = { X(t), t ≥ 0 } be a general stochastic process.

• Then there exists a random variable I_t(X), unique almost certainly, such that I_t(X_n) converges in probability to I_t(X) for each sequence of simple

stochastic processes X1, X2, . . . such that X_n converges in probability to X.

• If X is continuous with probability one, then I_t(X_n) converges in probability to I_t(X) as

δ_n = max^Δ

1≤k≤n(t_k − tk−1)

Ito Integral (concluded)

• It is a fundamental fact that 

X dW is continuous almost surely.

• The following theorem says the Ito integral is a martingale.^a

Theorem 17 The Ito integral 

X dW is a martingale.

• A corollary is the mean value formula E

  _b

a

X dW



= 0.

aSee Exercise 14.1.1 for simple stochastic processes.

Discrete Approximation

• Recall Eq. (75) on p. 588.

• The following simple stochastic process { X(t) } can be used in place of X to approximate  _t

0 X dW ,

X(s) = X(t^Δ _k−1) for s ∈ [ t_k−1, t_k), k = 1, 2, . . . , n.

• Note the nonanticipating feature of X.

– The information up to time s,

{ X(t), W (t), 0 ≤ t ≤ s },

cannot determine the future evolution of X or W .

Discrete Approximation (concluded)

• Suppose, unlike Eq. (75) on p. 588, we defined the stochastic integral from

n−1

k=0

X(t_k+1)[ W (t_k+1) − W (t_k) ].

• Then we would be using the following different simple stochastic process in the approximation,

Y (s) = X(t^Δ _k) for s ∈ [ t_k−1, t_k), k = 1, 2, . . . , n.

• This clearly anticipates the future evolution of X.^a

aSee Exercise 14.1.2 of the textbook for an example where it matters.

:

J

:

J

;

= >

:

Ito Process

• The stochastic process X = { X_t, t ≥ 0 } that solves X_t = X₀ +

 _t

0 a(X_s, s) ds +

 _t

0 b(X_s, s) dW_s, t ≥ 0 is called an Ito process.

– X0 is a scalar starting point.

– { a(X_t, t) : t ≥ 0 } and { b(X_t, t) : t ≥ 0 } are stochastic processes satisfying certain regularity conditions.

– a(X_t, t): the drift.

– b(X_t, t): the diffusion.

Ito Process (continued)

• Typical regularity conditions are:^a

– For all T > 0, x ∈ Rⁿ, and 0 ≤ t ≤ T ,

| a(x, t) | + | b(x, t) | ≤ C(1 + | x |) for some constant C.^b

– (Lipschitz continuity) For all T > 0, x ∈ Rⁿ, and 0 ≤ t ≤ T ,

| a(x, t) − a(y, t) | + | b(x, t) − b(y, t) | ≤ D | x − y | for some constant D.

aØksendal (2007).

bThis condition is not needed in time-homogeneous cases, where a

Ito Process (continued)

• A shorthand^a is the following stochastic differential equation^b (SDE) for the Ito differential dX_t,

dX_t = a(X_t, t) dt + b(X_t, t) dW_t. (76) – Or simply

dX_t = a_t dt + b_t dW_t.

– This is Brownian motion with an instantaneous drift a_t and an instantaneous variance b²_t.

• X is a martingale if a_t = 0.^c

aPaul Langevin (1872–1946) in 1904.

bLike any equation, an SDE contains an unknown, the process Xt.

cRecall Theorem 17 (p. 590).

Ito Process (concluded)

• From calculus, we would expect  _t

0 W dW = W (t)²/2.

• But W (t)²/2 is not a martingale, hence wrong!

• The correct answer is [ W (t)² − t ]/2.

• A popular representation of Eq. (76) is dX_t = a_t dt + b_t√

dt ξ, (77)

where ξ ∼ N (0, 1).

Euler Approximation

• Define t_n = nΔt.^Δ

• The following approximation follows from Eq. (77),

X(t n+1)

= X(tn) +a( X(tn), tn) Δt + b( X(tn), tn) ΔW (tn). (78)

• It is called the Euler or Euler-Maruyama method.

• Recall that ΔW (t_n) should be interpreted as W (t_n+1) − W (t_n),

not W (t_n) − W (t_n−1)!^a

aRecall Eq. (75) on p. 588.

Euler Approximation (concluded)

• With the Euler method, one can obtain a sample path X(t 1), X(t2), X(t3), . . .

from a sample path

W (t₀), W (t₁), W (t₂), . . . .

• Under mild conditions, X(t_n) converges to X(t_n).

More Discrete Approximations

• Under fairly loose regularity conditions, Eq. (78) on p. 598 can be replaced by

X(t _n+1)

= X(t_n) + a( X(t_n), t_n) Δt + b( X(t_n), t_n)√

Δt Y (t_n).

– Y (t0), Y (t₁), . . . are independent and identically distributed with zero mean and unit variance.

More Discrete Approximations (concluded)

• An even simpler discrete approximation scheme:

X(t _n+1)

= X(t_n) + a( X(t_n), t_n) Δt + b( X(t_n), t_n)√

Δt ξ.

– Prob[ ξ = 1 ] = Prob[ ξ = −1 ] = 1/2.

– Note that E[ ξ ] = 0 and Var[ ξ ] = 1.

• This is a binomial model.

• As Δt goes to zero, X converges to X.^a

aHe (1990).

Trading and the Ito Integral

• Consider an Ito process

dS_t = μ_t dt + σ_t dW_t.

– S_t is the vector of security prices at time t.

• Let φ_t be a trading strategy denoting the quantity of each type of security held at time t.

– Hence the stochastic process φ_tS_t is the value of the portfolio φ_t at time t.

• φ_t dS_t =^Δ φ_t(μ_t dt + σ_t dW_t) represents the change in the value from security price changes occurring at time t.

Trading and the Ito Integral (concluded)

• The equivalent Ito integral, G_T(φ) =^Δ

 _T

0 φ_t dS_t =

 _T

0 φ_tμ_t dt +

 _T

0 φ_tσ_t dW_t, measures the gains realized by the trading strategy over the period [ 0, T ].

Ito’s Lemma

A smooth function of an Ito process is itself an Ito process.

Theorem 18 Suppose f : R → R is twice continuously differentiable and dX = a_t dt + b_t dW . Then f (X) is the Ito process,

f (X_t)

= f (X₀) +

 _t

0 f^(X_s) a_s ds +

 _t

0 f^(X_s) b_s dW +1

2

 _t

0 f^(X_s) b²_s ds for t ≥ 0.

aIto (1944).

Ito’s Lemma (continued)

• In differential form, Ito’s lemma becomes df (X)

= f^(X) a dt + f^(X) b dW + 1

2 f^(X) b² dt (79)

= 

f^(X) a + 1

2 f^(X) b²

dt + f^(X) b dW.

• Compared with calculus, the interesting part is the third term on the right-hand side of Eq. (79).

• A convenient formulation of Ito’s lemma is df (X) = f^(X) dX + 1

f^(X)(dX)². (80)

Ito’s Lemma (continued)

• We are supposed to multiply out

(dX)² = (a dt + b dW )² symbolically according to

× dW dt

dW dt 0

dt 0 0

– The (dW )² = dt entry is justified by a known result.

• Hence (dX)² = (a dt + b dW )² = b² dt in Eq. (80).

• This form is easy to remember because of its similarity to the Taylor expansion.

Ito’s Lemma (continued)

Theorem 19 (Higher-Dimensional Ito’s Lemma) Let W1, W2, . . . , W_n be independent Wiener processes and

X = (X^Δ ₁, X₂, . . . , X_m) be a vector process. Suppose

f : R^m → R is twice continuously differentiable and X_i is an Ito process with dX_i = a_i dt + _n

j=1 b_ij dW_j. Then df (X) is an Ito process with the differential,

df (X) =

m i=1

f_i(X) dX_i + 1 2

m i=1

m k=1

f_ik(X) dX_i dX_k,

where f_i = ∂f /∂X^Δ _i and f_ik = ∂^{Δ 2}f /∂X_i∂X_k.

Ito’s Lemma (continued)

• The multiplication table for Theorem 19 is

× dW_i dt

dW_k δ_ik dt 0

dt 0 0

in which

δ_ik =

⎧⎨

⎩

1, if i = k, 0, otherwise.

Ito’s Lemma (continued)

• In applying the higher-dimensional Ito’s lemma, usually one of the variables, say X₁, is time t and dX₁ = dt.

• In this case, b1j = 0 for all j and a₁ = 1.

• As an example, let

dX_t = a_t dt + b_t dW_t.

• Consider the process f(X_t, t).

Ito’s Lemma (continued)

• Then df

= ∂f

∂X_t dX_t + ∂f

∂t dt + 1 2

∂²f

∂X_t² (dX_t)²

= ∂f

∂X_t (a_t dt + b_t dW_t) + ∂f

∂t dt +1

2

∂²f

∂X_t² (a_t dt + b_t dW_t)²

=

 ∂f

∂X_t a_t + ∂f

∂t + 1 2

∂²f

∂X_t² b²_t



dt + ∂f

∂X_t b_t dW_t. (81)

Ito’s Lemma (continued)

Theorem 20 (Alternative Ito’s Lemma) Let W₁, W₂, . . . , W_m be Wiener processes and

X = (X^Δ 1, X2, . . . , X_m) be a vector process. Suppose

f : R^m → R is twice continuously differentiable and X_i is an Ito process with dX_i = a_i dt + b_i dW_i. Then df (X) is the following Ito process,

df (X) =

m i=1

f_i(X) dX_i + 1 2

m i=1

m k=1

f_ik(X) dX_i dX_k.

Ito’s Lemma (concluded)

• The multiplication table for Theorem 20 is

× dW_i dt

dW_k ρ_ik dt 0

dt 0 0

• Above, ρ_ik denotes the correlation between dW_i and dW_k.

Geometric Brownian Motion

• Consider geometric Brownian motion Y (t) = e^{Δ X(t)}. – X(t) is a (μ, σ) Brownian motion.

– By Eq. (74) on p. 573,

dX = μ dt + σ dW.

• Note that

∂Y

∂X = Y,

∂²Y

= Y.

Geometric Brownian Motion (continued)

• Ito’s formula (79) on p. 605 implies dY = Y dX + (1/2) Y (dX)²

= Y (μ dt + σ dW ) + (1/2) Y (μ dt + σ dW )²

= Y (μ dt + σ dW ) + (1/2) Y σ² dt.

• Hence

dY

Y = 

μ + σ²/2

dt + σ dW. (82)

• The annualized instantaneous rate of return is μ + σ²/2 (not μ).^a

aConsistent with Lemma 9 (p. 297).

Geometric Brownian Motion (continued)

• Alternatively, from Eq. (74) on p. 573, X_t = X₀ + μt + σ W_t, an explicit (strong) solution.

• Hence

Y_t = Y0 e^{μt+σ Wt},

a strong solution to the SDE (82) where Y₀ = e^X0.

Geometric Brownian Motion (concluded)

• On the other hand, suppose dY

Y = μ dt + σ dW.

• Then X(t) = ln Y (t) follows^Δ dX = 

μ − σ²/2

dt + σ dW.

Exponential Martingale

• The Ito process

dX_t = b_tX_t dW_t is a martingale.^a

• It is called an exponential martingale.

• By Ito’s formula (79) on p. 605, X(t) = X(0) exp

−1 2

 _t

0 b²_s ds +

 _t

0 b_s dW_s

.

aRecall Theorem 17 (p. 590).