Talk given at Morgan Stanley and at the IITs to introduce students to stochastic calculus

1. Introduction to Stochastic Calculus
Introduction to Stochastic Calculus
Dr. Ashwin Rao
Morgan Stanley, Mumbai
March 11, 2011
Dr. Ashwin Rao Introduction to Stochastic Calculus

2. 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
Dr. Ashwin Rao Introduction to Stochastic Calculus

3. Introduction to Stochastic Calculus
Review of key concepts from Probability/Measure Theory
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
Dr. Ashwin Rao Introduction to Stochastic Calculus

4. Introduction to Stochastic Calculus
Review of key concepts from Probability/Measure Theory
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
Dr. Ashwin Rao Introduction to Stochastic Calculus

5. Introduction to Stochastic Calculus
Review of key concepts from Probability/Measure Theory
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
Dr. Ashwin Rao Introduction to Stochastic Calculus

6. Introduction to Stochastic Calculus
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
Dr. Ashwin Rao Introduction to Stochastic Calculus

7. Introduction to Stochastic Calculus
Information and σ-Alebgras
Uncountable example
Fi = Information available after first i coin tosses
i
Size of Fi = 22 elements
Fi has 2i ”atoms”
Dr. Ashwin Rao Introduction to Stochastic Calculus

8. Introduction to Stochastic Calculus
Information and σ-Alebgras
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 ]
Dr. Ashwin Rao Introduction to Stochastic Calculus

9. Introduction to Stochastic Calculus
Information and σ-Alebgras
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.
Dr. Ashwin Rao Introduction to Stochastic Calculus

10. Introduction to Stochastic Calculus
Information and σ-Alebgras
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
Dr. Ashwin Rao Introduction to Stochastic Calculus

11. Introduction to Stochastic Calculus
Conditional Expectation
E [X |G] is G-measurable
E [X |G](ω)dP (ω) = X (ω)dP (ω)∀A ∈ G
A A
Dr. Ashwin Rao Introduction to Stochastic Calculus

12. Introduction to Stochastic Calculus
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 )]
Dr. Ashwin Rao Introduction to Stochastic Calculus

13. Introduction to Stochastic Calculus
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)
Dr. Ashwin Rao Introduction to Stochastic Calculus

14. Introduction to Stochastic Calculus
Quadratic Variation and Brownian Motion
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 .
Dr. Ashwin Rao Introduction to Stochastic Calculus

15. Introduction to Stochastic Calculus
Quadratic Variation and Brownian Motion
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)
Dr. Ashwin Rao Introduction to Stochastic Calculus

16. Introduction to Stochastic Calculus
Quadratic Variation and Brownian Motion
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.
Dr. Ashwin Rao Introduction to Stochastic Calculus

17. Introduction to Stochastic Calculus
Quadratic Variation and Brownian Motion
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
Dr. Ashwin Rao Introduction to Stochastic Calculus

18. Introduction to Stochastic Calculus
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
Dr. Ashwin Rao Introduction to Stochastic Calculus

19. Introduction to Stochastic Calculus
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
Dr. Ashwin Rao Introduction to Stochastic Calculus

20. Introduction to Stochastic Calculus
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 )
Dr. Ashwin Rao Introduction to Stochastic Calculus

21. Introduction to Stochastic Calculus
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
Dr. Ashwin Rao Introduction to Stochastic Calculus

22. Introduction to Stochastic Calculus
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
Dr. Ashwin Rao Introduction to Stochastic Calculus