Introduction to Stochastic Calculus                         Introduction to Stochastic Calculus                           ...
Introduction to Stochastic Calculus   Review of key concepts from Probability/Measure Theory     Lebesgue Integral        ...
Introduction to Stochastic Calculus   Review of key concepts from Probability/Measure Theory     Change of measure        ...
Introduction to Stochastic Calculus   Review of key concepts from Probability/Measure Theory     Radon-Nikodym derivative ...
Introduction to Stochastic Calculus   Review of key concepts from Probability/Measure Theory     Simplified Girsanov’s Theo...
Introduction to Stochastic Calculus   Information and σ-Alebgras     Finite Example               Set with n elements {a1 ...
Introduction to Stochastic Calculus   Information and σ-Alebgras     Uncountable example               Fi = Information av...
Introduction to Stochastic Calculus   Information and σ-Alebgras     Stochastic Process Example               Ω = set of c...
Introduction to Stochastic Calculus   Information and σ-Alebgras     Filtration and Adaptation               Filtration: ∀...
Introduction to Stochastic Calculus   Information and σ-Alebgras     Multiple random variables and Independence           ...
Introduction to Stochastic Calculus   Conditional Expectation               E [X |G] is G-measurable                      ...
Introduction to Stochastic Calculus   An important Theorem               G a sub-σ-Algebra of F               X1 , . . . ,...
Introduction to Stochastic Calculus   Quadratic Variation and Brownian Motion     Random Walk               At step i, ran...
Introduction to Stochastic Calculus   Quadratic Variation and Brownian Motion     Two key properties of the random walk   ...
Introduction to Stochastic Calculus   Quadratic Variation and Brownian Motion     Scaled Random Walk               We spee...
Introduction to Stochastic Calculus   Quadratic Variation and Brownian Motion     Brownian Motion       Definition of Brown...
Introduction to Stochastic Calculus   Quadratic Variation and Brownian Motion     Key concepts               Joint distrib...
Introduction to Stochastic Calculus   Ito Calculus     Ito’s Integral                                                     ...
Introduction to Stochastic Calculus   Ito Calculus     Simple Integrands                                                  ...
Introduction to Stochastic Calculus   Ito Calculus     Properties of the Ito Integral                  I (t ) is a marting...
Introduction to Stochastic Calculus   Ito Calculus     Ito’s Formula                                                      ...
Introduction to Stochastic Calculus   Ito Calculus     Ito’s Formula                                                      ...
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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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Introduction to Stochastic calculus

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
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