5 Stochastic Calculus - Maa
Lecture FiveStochastic Calculus5 Stochastic Calculus5.1 Itô Integral for a Simple Integrand5.2 Properties for Simple Integrands5.3 Construction for General Integrands5.4 Example of an Itô Integral5.5 Itô’s Formula for One Process5.6 Solution to ExercisebySteven E. ShreveDepartment of Mathematical SciencesCarnegie Mellon Universityfor theMAA Short Course“Financial Mathematics”August 4-5, 2009Portland, Oregon1 / 372 / 37The Itô integral problemDefinitionLet W be a Brownian motion defined on a probability space(Ω, F, P). A process (s, ω), a function of s 0 and ω Ω, isadapted if the dependence of (s, ω) on ω is as a function of theinitial path fragment W (u, ω), 0 u s. In particular, (s) isindependent of W (t) W (s) whenever 0 s t.We want to make sense ofZ t (s) dW (s), 0 t T .5 Stochastic Calculus5.1 Itô Integral for a Simple Integrand0RemarkIf g (s) is a differentiable function, then we can defineZ tZ t (u)g ′ (s) ds. (s) dg (s) 03 / 370This won’t work for Brownian motion, however, because the pathsof Brownian motion are not differentiable.4 / 37
Simple IntegrandInterpretation of Simple IntegrandLet Π {t0 , t1 , . . . , tn } be a partition of [0, T ], i.e., 0 t0 t1 · · · t n T . Assume that (s) is constant in s on each subinterval [tk , tk 1 ).We call such a a simple process.Think of W (s) as the price per share of an asset at time s.Think of t0 , t1 , . . . , tn 1 as the trading dates in the asset.Think of (t0 ), (t1 ), . . . , (tn 1 ) as the number of sharesof the asset acquired at each trading date and held to thenext trading date.Gain from trading.I (t) (t0 )[W (t) W (t0 )] (t0 )W (t),0 t t1 ,I (t) (t0 )[W (t1 ) W (t0 )] (t1 )[W (t) W (t1 )],t1 t t2 ,t1t2t3t4sI (t) (t0 )[W (t1 ) W (t0 )] (t1 )[W (t2 ) W (t1 )] (t2 )[W (t) W (t2 )],One path of The process I is the Itô integral of the simple process , i.e.,Z tI (t) (s) dW (s), 0 t T .Example (s) W (tk ),t2 t t3 .tk s tk 105 / 376 / 37Expectation of Itô integralTheoremThe Itô integral of a simple process has expectation zero.Proof: By definition5 Stochastic Calculus5.2 Properties for Simple IntegrandsI (T ) n 1Xj 0 (tj ) W (tj 1 ) W (tj ) .Compute expectation term by term. Because (tj ) is independentof W (tj 1 ) W (tj ), we have E (tj ) W (tj 1 ) W (tj ) E (tj ) · E W (tj 1 ) W (tj ) E (tj ) · 0 0.7 / 378 / 37
Quadratic Variation of Itô IntegralExercise (5.1)Suppose Y (t), 0 t T , is a stochastic process (a function of tand ω) such that if 0 s t, then the increment Y (t) Y (s) isindependent of the path of Y up to time s and has expectationzero. Let { (s)}0 s T be a simple process adapted to Y , i.e.,there is a partition Π {t0 , t1 , . . . , tn } of [0, T ] such that (s) isconstant in s in each subinterval [tj , tj 1 ), and for each s [0, T ],the random variable (s) depends on ω only through the path ofY up to time s, and hence (s) is independent of Y (t) Y (s) forall t [s, T ]. Define the Itô integralI (T ) n 1Xj 0TheoremThe simple Itô integralI (t) Zt (u) dW (u)0has quadratic variation[I , I ](T ) ZT 2 (u) du(QV )0 (tj ) Y (tj 1 ) Y (tj ) .and variance E I 2 (T ) E(i) Show that EI (T ) 0.(ii) A simple arbitrage is a simple process such that I (T ) 0almost surely and P{I (T ) 0} 0. Show that there is nosimple arbitrage under the assumptions of this exercise.ZT 2 (u) du.(VAR)0RemarkBoth sides of (QV) are random, but the expressions in (VAR) arenot. (VAR) is called Itô’s Isometry.9 / 37Proof of (QV)10 / 37Proof of (VAR)For s [tj , tj 1 ], we have (s) (tj ) andhiI (s) I (tj ) (tj ) W (s) W (tj )hi I (tj ) (tj )W (tj ) (tj )W (s).I (T ) j 0 2Xj k E (tj ) (tk ) W (tj 1 ) W (tj ) W (tk 1 ) W (tk ) .We use independence to simplify the pure square terms: 2 2 E 2 (tj ) W (tj 1 ) W (tj ) E 2 (tj ) · E W (tj 1 ) W (tj )Z tj 1 2 E 2 (s) ds. E (tj ) · (tj 1 tj ) tjj 0n 1 hX 2 i E 2 (tj ) W (tj 1 ) W (tj )E I 2 (T ) k j 2 (tj )(tj 1 tj )Z tj 1 2 (s) ds. T (tj ) W (tj 1 ) W (tj ) .Squaring and taking expectations, we obtainOn this subinterval, quadratic variation of I comes from thequadratic variation of W , which is scaled by (tj ). Therefore [I , I ](tj 1 ) [I , I ](tj ) 2 (tj ) [W , W ](tj 1 ) [W , W ](tj )Summing over subintervals, we obtainn 1 ZX[I , I ](T ) [I , I ](tj 1 ) [I , I ](tj ) n 1Xtj 2 (s) ds.The sum of the pure square terms is E011 / 37RT0 2 (s)ds.12 / 37
Proof of (VAR) (continued)It remains to show that the cross-terms have zero expectation. Forj k, the incrementW (tk 1 ) W (tk ) is independent of (tj ) (tk ) W (tj 1 ) W (tj ) , and hence5 Stochastic Calculush iE (tj ) (tk ) W (tj 1 ) W (tj ) W (tk 1 ) W (tk )5.3 Construction for General Integrandsh i E (tj ) (tk ) W (tj 1 ) W (tj ) · E W (tk 1 ) W (tk )h i E (tj ) (tk ) W (tj 1 ) W (tj ) · 0 0.13 / 37Outline of construction for general integrands 14 / 37Outline of construction (continued)Given (s), 0 s T , satisfyingZ TE 2 (s) ds , 0construct an approximating sequence of simple processes n (s), 0 s T , such thatZ T 2lim E (s) n (s) ds 0.n Set In (T ) 0 0RT n (s) dW (s). Itô’s isometry implies thatZ T 2E In (T ) Im (T ))2 E n (s) m (s) ds.0 0 L2 (Ω, F, P) is complete, and so the sequence {In (T )} n 1 hasa limit I (T ) in this space.We defineZ T (s) dW (s) I (T ) lim In (T ).Because the sequence { n } n 1 converges inL2 (Ω [0, T ], F Borel([0, T ]), P Lebesgue), it is Cauchyin this space. Therefore, {In (T )} n 1 is Cauchy in L2 (Ω, F, P).15 / 37n This limit does not depend on the approximating sequence{ n } n 1 .By choosing approximating sequences that converge rapidly,we can in fact make the convergence of In (T ) to I (T ) bealmost sure (almost everywhere with respect to P) rather thanin L2 .With additional work, one can choose the approximatingsequence so that the paths of In (t), 0 t T , convergeuniformly in t [0, T ] almost surely. This guarantees thatthere is a limit I (t), 0 t T , that is a continuous functionof t [0, T ] for P-almost every ω.16 / 37
TheoremRTUnder the assumption E[ 0 2 (s) ds] , the Itô integralI (t) Zt (s) dW (s),0 t T,0is defined and continuous in t [0, T ]. We haveEI (t) 0,5 Stochastic Calculus0 t T.5.4 Example of an Itô IntegralThe quadratic variation of the Itô integral isZ t 2 (s) ds, 0 t T ,[I , I ](t) 0and the Itô integral satisfies Itô’s Isometry Z t 2 2 (s) ds ,Var[I (t)] E I (t) E0 t T.017 / 37RT018 / 37W (s) dW (s)By definition,Divide [0, T ] into n equal subintervals. Define jTjT(j 1)T n (s) Wfor s .nnnZTW (s) dW (s)0 limn n 1Xj 0W jTn (j 1)TjTW W.nnTo simplify notation, we denote Wj WW0 W (0) 0, Wn W (T ), andt1t2t3t4 TZs0TW (s) dW (s) limn n 1X jTn . ThenWj (Wj 1 Wj ).j 0One path of W (s) and 4 (s)19 / 3720 / 37
n 1n 1n 1n 1j 0j 0j 0j 0From the previous page, we haveX1X 21X1X 2(Wj 1 Wj )2 Wj 1 WjWj Wj 1 222 nn 1n 1k 1j 0j 0n 1X1X 2 X1X 2Wk WjWj Wj 1 22j 0n 1n 1n 1k 0j 0j 0n 1j 0j 0T01111W (s) dW (s) W 2 (T ) [W , W ](T ) W 2 (T ) T .2222If g is a differentiable function with g (0) 0, thenZX1Wj (Wj Wj 1 ). Wn2 2j 0j 0ZRemark1 2 X 2 XWj Wj Wj 1W 2 nn 1n 1Xj 0Letting n , we get1 2 1X 2 X1X 2Wk Wj Wj 1 WjWn 222n 1n 11X1(Wj 1 Wj )2 .Wj (Wj 1 Wj ) Wn2 22g (s) dg (s) 0ZT01g (s)g ′ (s) ds g 2 (s)2T01 g 2 (T ).2RTThe extra term 12 T in 0 W (s)dW (s) comes from the nonzeroquadratic variation of Brownian motion.n 111XWj (Wj 1 Wj ) Wn2 (Wj 1 Wj )2 .22Tj 021 / 3722 / 37Exercise (5.2)Show thatlimn n 1XWj 0 (j 1)Tn W (j 1)Tn W jTn5 Stochastic Calculus 5.5 Itô’s Formula for One Process11 2W (T ) T .2223 / 3724 / 37
Along the path of a Brownian motion, we want to “differentiate”f (W (t)), where f (x) is a differentiable function. If the path of theBrownian motion W (t) could be differentiated with respect to t,then the ordinary chain rule would giveRemarkThe mathematically meaningful form of Itô’s formula is Itô’sformula in integral form:f (W (T )) f (W (0)) df (W (t)) f ′ (W (t))W ′ (t),dttf ′ (W (t)) dW (t) 012ZTf ′′ (W (t)) dt.0This is because we have definitions for both integrals appearing onthe right-hand side. The first,which could be written in differential notation asdf (W (t)) f ′ (W (t)) W ′ (t) dt f ′ (W (t)) dW (t).ZBecause W has nonzero quadratic variation, the correct formulahas an extra term, namely,1df (W (t)) f ′ (W (t)) dW (t) f ′′ (W (t))2Zdt {z}f ′ (W (t)) dW (t)0is an Itô integral. The secondZ.dW (t)dW (t)This is Itô’s formula in differential form.TTf ′′ (W (t)) dt0is a Riemann integral with respect to time, computed path by path.25 / 37Application of Itô’s Formula26 / 37Derivation of Itô’s FormulaConsider f (x) 12 x 2 , so thatf ′ (x) x,Consider f (x) 12 x 2 , so thatf ′′ (x) 1.f ′ (x) x,Itô’s formula in integral form f W (T ) f W (0) becomes1 2W (T ) 2ZTZT0 1f ′ W (s) dW (s) 21W (s) dW (s) 20Z0T1 ds Z0TZT0Let xj 1 and xj be numbers. Taylor’s formula implies f ′′ W (s) ds1f (xj 1 ) f (xj ) (xj 1 xj )f ′ (xj ) (xj 1 xj )2 f ′′ (xj ).21W (s) dW (s) T ,2or equivalently,Z0Tf ′′ (x) 1.11W (u) dW (u) W 2 (T ) T .2227 / 37In this case, Taylor’s formula to second order is exact because f isa quadratic function.In the general case, the above equation is only approximate,and the error is of the order of (xk 1 xk )3 . The total error willhave limit zero in the last step of the following argument (seeExercise 4.6(iii) of Lecture 4).28 / 37
From the previous page, we haveFix T 0 and let Π {t0 , t1 , . . . , tn } be a partition of [0, T ].f (W (T )) f (W (0))"#2"#n 1n 1X1XW (tj 1 ) W (tj ) . W (tj ) W (tj 1 ) W (tj ) 2f (W (T )) f (W (0)) n 1Xj 0 ""n 1X#j 0f (W (tj 1 )) f (W (tj ))We let kΠk 0, to obtain#f (W (T )) f (W (0))Z T1 W (s) dW (s) [W , W ](T ){z} 20W (tj 1 ) W (tj ) f ′ (W (tj ))j 0"#2n 11XW (tj 1 ) W (tj ) f ′′ (W (tj )) 2 j 0 n 1Xj 0#Z0"#2n 11XW (tj 1 ) W (tj ) .W (tj ) W (tj 1 ) W (tj ) 2"k 0T1f ′ (W (s)) dW (s) 2TTZ0f ′′ (W (s)) ds. {z }1This is Itô’s formula in integral form for the special casej 01f (x) x 2 .229 / 3730 / 37y f (x)Exercise (5.3)f ′ (W (tj ))(W (tj 1 ) W (tj ))Let u R be constant and define f (x) e ux . Use Itô’s formulaapplied to f (W (t)) to obtain the moment-generating functionformula1 2Ee uW (T ) e 2 u T .(Compare with Exercise 4.4 of Lecture 4.)W (tj )W (tj 1 ) f W (tj 1 ) f W (tj ) f ′ W (tj ) W (tj 1 ) W (tj )Exercise (5.4)Let f (x) x 4 . Use Itô’s formula applied to f (W (t)) to obtain thefourth-moment formula Small Error f W (tj 1 ) f W (tj ) f ′ W (tj ) W (tj 1 ) W (tj ) 21 f ′′ W (tj ) W (tj 1 ) W (tj )2 Smaller ErrorWe need the higher accuracy before summing. Otherwise, theaccumulated small errors do not vanish as the step-size goes tozero.EW 4 (T ) 3T 2 .(Compare with Exercise 4.5 of Lecture 4.)31 / 3732 / 37
Solution to Exercise 5.1(i) As in the proof of the theorem preceding the exercise, we useindependence to compute EI (T ) term by term: E (tj ) Y (tj 1 ) Y (tj ) E (tj ) · E Y (tj 1 ) Y (tj )5 Stochastic Calculus5.7 Solution to Exercise E (tj ) · 0 0.(ii) If I (T ) 0 almost surely and P{I (T ) 0} 0, thenEI (T ) 0. This contradicts part (i).33 / 37Solution to Exercise 5.2Let tj jTn .Solution to Exercise 5.3We have f ′ (x) uf (x) and f ′′ (x) u 2 f (x). Therefore, Itô’sformula becomesZ TZ T1e uW (t) dt.e uW (T ) e uW (0) ue uW (t) dW (t) u 2200The quadratic variation result for Brownian motion islimn n 1Xj 0 2W (tj 1 ) W (tj ) T .Taking expectations and using the fact that the expectation of theItô integral is zero, we obtainZ T1Ee uW (t) dt.Ee uW (T ) 1 u 220The Example shows thatlimn n 1Xj 0 11W (tj )(W (tj 1 ) W (tj ) W 2 (T ) T .22We differentiate both sides with respect to T to obtainAdding these two equations, we obtainlimn n 1Xj 034 / 37d1Ee uW (T ) u 2 Ee uW (T ) .dT2 11W (tj 1 ) W (tj 1 ) W (tj ) W 2 (T ) T .22The unique solution to this ordinary differential equation satisfyingEe uW (0) 1 is1 2Ee uW (T ) e 2 u T .This is the desired result.35 / 3736 / 37
Exercise 5.4With f (x) x 4 , we have f ′ (x) 4x 3 and f ′′ (x) 12x 2 .Therefore, Itô’s formula becomesZ TZ TW 2 (t) dt.W 3 (t) dW (t) 6W 4 (T ) 400Taking expectations of both sides and using the fact that the Itôintegral has expectation zero, we obtainEW 4 (T ) 6ZTEW 2 (t) dt0 6ZTt dt02 3T .37 / 37
5 Stochastic Calculus 5.1 Itˆo Integral for a Simple Integrand 5.2 Properties for Simple Integrands 5.3 Construction for General Integrands 5.4 Example of an Itˆo Integral 5.5 Itˆo’s Formula for One Process 5.6 Solution to Exercise 2/37 5 Stochastic Calculus 5.1 Itˆo Integral for a Simple Integrand 3/37 The Itˆo integral problem Definition
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Ntorot of Divine Law and Balance are Maa and Maat. The term mmara (maa – ra) meaning ‘law’ in Akan is the same term maa meaning ‘law’ in Kamit. This is precisely why the monkey holding the divination bowl for the Obosomfo (High Priest – representative of Tehuti) in Akan is named
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Introduction In this unit we shall try to know about Aristotle and his life and works and also understand about the relationship between Criticism and Creativity. We shall see how criticism is valued like creative writings. We shall know the role and place given to 'the critic' in the field of literary criticism.
