introduction to stochastic calculus

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

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

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Page 1: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Introduction to Stochastic Calculus

Dr. Ashwin Rao

Morgan Stanley, Mumbai

March 11, 2011

Dr. Ashwin Rao Introduction to Stochastic Calculus

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Page 2: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Review of key concepts from Probability/Measure Theory

Lebesgue Integral

(Ω, F, P)

Lebesgue Integral:∫Ω X (ω)dP(ω) = EPX

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 3: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Review of key concepts from Probability/Measure Theory

Change of measure

Random variable Z with EPZ = 1

Define probability Q(A) =∫

A Z (ω)dP(ω) ∀A ∈ F

EQX = EP [XZ ]

EQYZ = EPY

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 4: Introduction to Stochastic calculus

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 EPZ = 1 andQ(A) =

∫A Z (ω)dP(ω) ∀A ∈ F

Z is called the Radon-Nikodym derivative of Q w.r.t. P anddenoted Z = dQ

dP

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 5: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Review of key concepts from Probability/Measure Theory

Simplified Girsanov’s Theorem

X = N(0, 1)

Z (ω) = eθX(ω)− θ2

2 ∀ω ∈ ΩEpZ = 1

∀A ∈ F, Q =

∫A

ZdP

EQX = EP [XZ ] = θ

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 6: Introduction to Stochastic calculus

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

Page 7: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Information and σ-Alebgras

Uncountable example

Fi = Information available after first i coin tosses

Size of Fi = 22ielements

Fi has 2i ”atoms”

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 8: Introduction to Stochastic calculus

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 functionvalues from [0, t]

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 9: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Information and σ-Alebgras

Filtration and Adaptation

Filtration: ∀t ∈ [0, T ], σ-Algebra Ft . foralls 6 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 anadapted stochastic process if ∀t, X (t) is Ft-measurable.

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 10: Introduction to Stochastic calculus

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 randomvariable 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) andE [XY ] = E [X ]E [Y ]

Covariance(X , Y ) = E [(X − E [X ])(Y − E [Y ])]

Correlation pX ,Y =Covariance(X ,Y )√

Varaince(X)Variance(Y )

Multivariate normal density fX(x) = 1√(2π)ndet(C)

e− 12 (x−µ)C−1(x−µ)T

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

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 11: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Conditional Expectation

E [X |G] is G-measurable

∫A

E [X |G](ω)dP(ω) =

∫A

X (ω)dP(ω)∀A ∈ G

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 12: Introduction to Stochastic calculus

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

Page 13: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Quadratic Variation and Brownian Motion

Random Walk

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

Mi =

i∑j=1

Xj

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 independentEach incerement has expected value of 0Each increment has a variance = number of steps (i.e.,variance of 1 per step)

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 14: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Quadratic Variation and Brownian Motion

Two key properties of the random walk

Martingale: E [Mi |Fj ] = Mj

Quadratic Variation: [M, M ]i =∑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

Page 15: Introduction to Stochastic calculus

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

For a fixed positive integer n, define W (n)(t) = 1√nMnt

Usual properties: independent increments with mean 0 and varianceof 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 bycentral limit theorem)

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 16: Introduction to Stochastic calculus

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 0and variance of 1 per unit of time.

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 17: Introduction to Stochastic calculus

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

Page 18: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Ito Calculus

Ito’s Integral

I (T ) =

∫T

0∆(t)dW (t)

Remember that Brownian motion cannot be differentiated w.r.t time

Therefore, we cannot write I (T ) as∫T

0 ∆(t)W ′(t)dt

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 19: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Ito Calculus

Simple Integrands

∫T

0∆(t)dW (t)

Let Π = t0, t1, . . . , tn be a partition of [0, t]

Assume ∆(t) is constant in t in each subinterval [tj , tj+1]

I (t) =

∫ t

0∆(u)dW (u) =

k−1∑j=0

∆(tj )[W (tj+1)−W (tj )]+∆(tk)[W (t)−W (tk)]

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 20: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Ito Calculus

Properties of the Ito Integral

I (t) is a martingale

Ito Isometry: E [I 2(t)] = E [∫t

0 ∆2(u)du]

Quadratic Variation: [I , I ](t) =∫t

0 ∆2(u)du

General Integrands

An example:∫T

0 W (t)dW (t)

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 21: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Ito Calculus

Ito’s Formula

f (T , W (T )) = f (0, W (0)) +

∫T

0ft(t, W (t))dt

+

∫T

0fx (t, W (t))dW (t) +

1

2

∫T

0fxx (t, W (t))dt

Ito Process: X (t) = X (0) +∫t

0 ∆(u)dW (u) +∫t

0Θ(u)dW (u)

Quadratic variation [X , X ](t) =∫t

0 ∆2(u)du

Dr. Ashwin Rao Introduction to Stochastic Calculus

Page 22: Introduction to Stochastic calculus

Introduction to Stochastic Calculus

Ito Calculus

Ito’s Formula

f (T , X (T )) = f (0, X (0)) +

∫T

0ft(t, X (t))dt +

∫T

0fx (t, X (t))dX (t)

+1

2

∫T

0fxx (t, X (t))d [X , X ](t)

= f (0, X (0)) +

∫T

0ft(t, X (t))dt +

∫T

0fx (t, X (t))∆(t)dW (t)

+

∫T

0fx (t, X (t))Θ(t)dt +

1

2

∫T

0fxx (t, X (t))∆2(t)dt

Dr. Ashwin Rao Introduction to Stochastic Calculus