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This is a very good presentation on Stochastic Calculus by Rajeeva L.Karandikar

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  • 1. Introduction to Stochastic Calculus Rajeeva L. Karandikar Director Chennai Mathematical Institute rlk@cmi.ac.in rkarandikar@gmail.comRajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 1
  • 2. The notion of Conditional Expectation of a random variable given a -eld confuses many people who wish to learn stochastic calculus. Let X be a random variable. Suppose we are required to make a guess for the value of X , we would like to be as close to X as possible. Suppose the penalty function is square of the error. Thus we wish to minimize E[(X a)2 ] (1) where a is the guess. The value of a that minimizes (??) is the mean = E[X ].Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 2
  • 3. Let Y be another random variable which we can observe and we are allowed to use the observation Y while guessing X , i.e. our guess could be a function of Y . We should choose the function g such that E[(X g (Y ))2 ] (2) takes the minimum possible value. When Y takes nitely many values, the function g can be thought of as a look-up table. Assuming E [X 2 ] < , it can be shown that there exists a function g (Borel measurable function from R to R) such that E[(X g (Y ))2 ] E[(X f (Y ))2 ]. (3)Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 3
  • 4. H = L2 (, F , P) - the space of all square integrable random variables on (, F , P) with norm Z = E[Z 2 ] is an Hilbert space and K = {f (Y ) : f measurable, E[(f (Y ))2 ] < } is a closed subspace of H and hence given X H , there is a unique element in g K that satises E[(X g (Y ))2 ] E[(X f (Y ))2 ]. Further, E[(X g (Y ))f (Y )] = 0 f K .Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 4
  • 5. For X H , we dene g (Y ) to be the conditional expectation of X given Y , written as E [X | Y ] = g (Y ). One can show that for X , Z H and a, b R, one has E[a X + b Z | Y ] = a E[X | Y ] + b E[Z | Y ] (4) and X Z implies E[X | Y ] E[Z | Y ]. (5) Here and in this document, statements on random variables about equality or inequality are in the sense of almost sure, i.e. they are asserted to be true outside a null set.Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 5
  • 6. If (X , Y ) has bivariate Normal distrobution with means zero, variances 1 and correlation coecient r , then E[X | Y ] = rY . Thus g (y ) = ry is the choice for conditional expectation in this case.Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 6
  • 7. Now if instead of one random variable Y , we were to observe Y1 , . . . , Ym , we can similarly dene E[X | Y1 , . . . Ym ] = g (Y1 , . . . Ym ) where g satises E[(X g (Y1 , . . . Ym ))2 ] E[(X f (Y1 , . . . Ym ))2 ] = 0 measurable functions f on Rm . Once again we would need to show that the conditional expectation is linear and monotone.Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 7
  • 8. Also if we were to observe an innite sequence, we have to proceed similarly, with f , g being Borel functions on R , while if were to observe an Rd valued continuous stochastic process {Yt : 0 t < }, we can proceed as before with f , g being Borel functions on C ([0, ), Rd ). In each case we will have to write down properties and proofs thereof and keep doing the same as the class of observable random variables changes.Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 8
  • 9. For a random variable Y , (dened on a probability space (, F , P)), the smallest -eld (Y ) with respect to which Y is measurable (also called the -eld generated by Y ) is given by (Y ) = A F : A = {Y B}, B B(R) . An important fact: A random variable Z can be written as Z = g (Y ) for a measurable function g if and only if Z is measurable with respect to (Y ). In view of these observations, one can dene conditional expectation given a sigma eld as follows.Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 9
  • 10. Let X be a random variable on (, F , P) with E[|X |2 ] < and let G be a sub- eld of F . Then the conditional expectation of X given G is dened to be the G measurable random variable Z such that E[(X Z )2 ] E[(X U)2 ], U G -measurable r.v. (6) The random variable Z is characterized via E[X 1A ] = E[Z 1A ] A G (7) It should be remembered that one mostly uses it when the sigma eld G is generated by a bunch of observable random variables and then Z is a function of these observables.Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 10
  • 11. We can see that information content in observing Y or observing Y 3 is the same and so intutively E[X | Y ] = E[X | Y 3 ] This can be seen to be true on observing that (Y ) = (Y 3 ) as Y is measurable w.r.t. (Y 3 ) and Y 3 is measurable w.r.t (Y ).Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 11
  • 12. We extend the denition of E[X | G ] to integrable X as follows. For X 0, let X n = X n. Then X n is square integrable and hence Z n = E[X n | G ] is dened and 0 Z n Z n+1 as X n X n+1 . So we dene Z = lim Z n and since E[X n 1A ] = E[Z n 1A ] A G monotone convergence theorem implies E[X 1A ] = E[Z 1A ] A G and it follows that E[Z ] = E[X ] < and we dene E[X | G ] = Z .Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 12
  • 13. Now given X such that E[|X |] < we dene E[X | G ] = E[X + | G ] E[X + | G ] where X + = max(X , 0) and X = max(X , 0). It follows that E E[X G ]1A = E X 1A . A G (8) The property (??) characterizes the conditional expectation.Rajeeva L. Karandikar Director, Chennai Mathematical InstituteIntroduction to Stochastic Calculus - 13
  • 14. Let X , Xn , Z be integrable random variables on (, F , P) for n 1 and G be a sub- eld of F and a, b R. Then we have (i) E[a X + b Z | G ] = a E[X | G ] + b E[Z | G ]. (ii) If Y is G measurable and bounded then E(XY | G ) = Y E(X | G ). (iii) X Z E(X | G ) E(Z | G ). (iv) E E[X | G ] = E[X ]. (v) For a sub-sigma eld H with G H F one has E E[X | H ] | G = E[X | G ] (vi) |E[X | G ]| E[|X | | G ] (vii) If E[|Xn X |] 0 then E |E[Xn | G ] E[X | G ]| 0.Rajeeva L. Karandikar

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