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