Introduction to Stochastic calculus

Introduction to Stochastic Calculus


Dr. Ashwin Rao

Morgan Stanley, Mumbai

March 11, 2011

Dr. Ashwin Rao Introduction to Stochastic Calculus

Review of key concepts from Probability/Measure Theory
Lebesgue Integral

(Ω, F, P )
Lebesgue Integral: Ω X (ω)dP (ω) = EP X


Change of measure

Random variable Z with EP Z = 1
Deﬁne probability Q (A) = A Z (ω)dP (ω) ∀A ∈ F
EQ X = EP [XZ ]
EQ Y = EP Y
Z


Radon-Nikodym derivative

Equivalence of measures P and Q: ∀A ∈ F, P (A) = 0 iﬀ Q (A) = 0
if P and Q are equivalent, ∃Z such that EP Z = 1 and
Q (A) = A Z (ω)dP (ω) ∀A ∈ F
Z is called the Radon-Nikodym derivative of Q w.r.t. P and
denoted Z = dQdP


Simpliﬁed Girsanov’s Theorem

X = N (0, 1)
θ2
Z (ω) = e θX (ω)− 2 ∀ω ∈ Ω
Ep Z = 1

∀A ∈ F , Q = ZdP
A
EQ X = EP [XZ ] = θ


Information and σ-Alebgras
Finite Example

Set with n elements {a1 , . . . , an }
Step i: consider all subsets of {a1 , . . . , ai } and its complements
At step i, we have 2i +1 elements
∀i, Fi ⊂ Fi +1


Uncountable example

Fi = Information available after ﬁrst i coin tosses
i
Size of Fi = 22 elements
Fi has 2i ”atoms”


Stochastic Process Example

Ω = set of continuous functions f deﬁned on [0, T ] with f (0) = 0
FT = set of all subsets of Ω
Ft : elements of Ft can be described only by constraining function
values from [0, t ]


Filtration and Adaptation

Filtration: ∀t ∈ [0, T ], σ-Algebra Ft . foralls t, Fs ⊂ Ft
σ-Algebra σ(X ) generated by a random var X = {ω ∈ Ω|X (ω) ∈ B }
where B ranges over all Borel sets.
X is G-measurable if σ(X ) ⊂ G
A collection of random vars X (t ) indexed by t ∈ [0, T ] is called an
adapted stochastic process if ∀t, X (t ) is Ft -measurable.


Multiple random variables and Independence

σ-Algebras F and G are independent if P (A ∩ B ) = P (A) · P (B )
∀A ∈ F , B ∈ G
Independence of random variables, independence of a random
variable and a σ-Algebra
Joint density fX ,Y (x, y ) = P ({ω|X (ω) = x, Y (ω) = y })
∞
Marginal density fX (x ) = P ({ω|X (ω) = x }) = −∞ fX ,Y (x, y )dy

X , Y independent implies fX ,Y (x, y ) = fX (x ) · fY (y ) and
E [XY ] = E [X ]E [Y ]
Covariance(X , Y ) = E [(X − E [X ])(Y − E [Y ])]
Covariance (X ,Y )
Correlation pX ,Y = √
Varaince (X )Variance (Y )
1 −1
1 (x −µ)T
Multivariate normal density fX (x ) = √
¯ ¯ e − 2 (x −µ)C
¯ ¯ ¯ ¯
(2π)n det (C )

X , Y normal with correlation ρ. Create independent normal
variables as a linear combination of X , Y


Conditional Expectation

E [X |G] is G-measurable

E [X |G](ω)dP (ω) = X (ω)dP (ω)∀A ∈ G
A A


An important Theorem

G a sub-σ-Algebra of F
X1 , . . . , Xm are G-measurable
Y1 , . . . , Yn are independent of G

E [f (X1 , . . . , Xm , Y1 , . . . , Yn )|G] = g (X1 , . . . , Xm )

How do we evaluate this conditional expectation ?
Treat X1 , . . . , Xm as constants
Y1 , . . . , Yn should be integrated out since they don’t care about G

g (x1 , . . . , xm ) = E [f (x1 , . . . , xm , Y1 , . . . , Yn )]


Quadratic Variation and Brownian Motion
Random Walk

At step i, random variable Xi = 1 or -1 with equal probability

i
Mi = Xj
j =1

The process Mn , n = 0, 1, 2, . . . is called the symmetric random walk
3 basic observations to make about the ”increments”
Independent increments: for any i0 < i1 < . . . < in ,
(Mi1 − Mi0 ), (Mi2 − Mi1 ), . . . (Min − Min−1 ) are independent
Each incerement has expected value of 0
Each increment has a variance = number of steps (i.e.,
variance of 1 per step)


Two key properties of the random walk

Martingale: E [Mi |Fj ] = Mj
i
Quadratic Variation: [M, M ]i = j =1 (Mj − Mj −1 )2 = i

Don’t confuse quadratic variation with variance of the process Mi .


Scaled Random Walk

We speed up time and scale down the step size of a random walk
1
For a ﬁxed positive integer n, deﬁne W (n )(t ) = √
n
Mnt
Usual properties: independent increments with mean 0 and variance
of 1 per unit of time t
Show that this is a martingale and has quadratic variation
As n → ∞, scaled random walk becomes brownian motion (proof by
central limit theorem)


Brownian Motion

Deﬁnition of Brownian motion W (t ).
W ( 0) = 0
For each ω ∈ Ω, W (t ) is a continuous function of time t.
independent increments that are normally distributed with mean 0
and variance of 1 per unit of time.


Key concepts

Joint distribution of brownian motion at speciﬁc times
Martingale property
Derivative w.r.t. time is almost always undeﬁned
Quadratic variation (dW · dW = dt)
dW · dt = 0, dt · dt = 0


Ito Calculus
Ito’s Integral

T
I (T ) = ∆(t )dW (t )
0

Remember that Brownian motion cannot be diﬀerentiated w.r.t time
T
Therefore, we cannot write I (T ) as 0 ∆(t )W (t )dt


Ito Calculus
Simple Integrands

T
∆(t )dW (t )
0

Let Π = {t0 , t1 , . . . , tn } be a partition of [0, t ]
Assume ∆(t ) is constant in t in each subinterval [tj , tj +1 ]

t k −1
I (t ) = ∆(u )dW (u ) = ∆(tj )[W (tj +1 )−W (tj )]+∆(tk )[W (t )−W (tk )]
0 j =0


Ito Calculus
Properties of the Ito Integral

I (t ) is a martingale
t 2
Ito Isometry: E [I 2 (t )] = E [ 0 ∆ (u )du ]
t 2
Quadratic Variation: [I , I ](t ) = 0 ∆ (u )du

General Integrands
T
An example: 0 W (t )dW (t )


Ito Calculus
Ito’s Formula

T
f (T , W (T )) = f (0, W (0)) + ft (t, W (t ))dt
0
T T
1
+ fx (t, W (t ))dW (t ) + fxx (t, W (t ))dt
0 2 0

t t
Ito Process: X (t ) = X (0) + 0 ∆(u )dW (u ) + 0 Θ(u )dW (u )
t 2
Quadratic variation [X , X ](t ) = 0 ∆ (u )du


Ito Calculus
Ito’s Formula

T T
f (T , X (T )) = f (0, X (0)) + ft (t, X (t ))dt + fx (t, X (t ))dX (t )
0 0
T
1
+ fxx (t, X (t ))d [X , X ](t )
2 0
T T
= f (0, X (0)) + ft (t, X (t ))dt + fx (t, X (t ))∆(t )dW (t )
0 0
T T
1
+ fx (t, X (t ))Θ(t )dt + fxx (t, X (t ))∆2 (t )dt
0 2 0


Introduction to Stochastic calculus

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