CONF CRS IN PROBABIL AND STAT
CONF CRS IN PROBABIL AND STAT M 394C
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Part II CONTINUOUS TIME STOCHASTIC PROCESSES Chapter 4 For an advanced analysis of the properties of the Wiener process see Reuus D and Yor M 39 Continuous martingales and Brownian Motion Karatzas I and Shreve S E 39 Brownian Motion and Stochastic Calculus Beginning from this lecture we study continuous time processes A stochastic process X is de ned in the same way as in Lecture 1 as a family of random variables X Xt t E T but now T 000 or T ab C R Our main examples still come from Mathematical Finance but now we assume that nancial assets can be traded continuously We assume the same simple situation as in Lecture 1 that is we assume that we have a simple nancial market consisting of bonds paying xed interest one stock and derivatives of these two assets We assume that the interest on bonds is compounded continuously hence the value of the bond at time t is Bt B06 We denote by St the time t price of a share and we assume now that S can also change continuously In order to introduce more precise models for S similar to discrete time models like exponential random walk we need rst to de ne a continuous time analogue to the random walk process which is called Wiener process or Brownian Motion 41 WIENER PROCESS DEFINITION AND BASIC PROPERTIES We do not repeat the de nitions introduced for discrete time processes in previous lectures if they are exactly the same except for the different time set Hence we assume that the de nitions of the ltration martingale are already known On the other hand some properties of the corresponding objects derived in discrete time can not be taken for granted De nition 41 A continuous stochastic process Wt t 2 0 adapted to the ltration is called an g Wiener process if 1 W00 D for every 0 S s S t the random variable Wt 7 W5 is independent of f5 00 for every 0 S s S t the random variable Wt 7 W5 is normally distributed with mean zero and variance t 7 s In many problems we need to use the so called Wiener process started at p It is a process de ned by the equation Wf x m The next proposition gives the rst consequences of this de nition Proposition 42 Let be an g Wiener process Then 1 ElVt 0 and EWslVt minst for all st 2 0 and in particular ElVtz t 2 The process is Gaussian 3 The Wiener process has independent increments that is ifO 3 t1 3 t2 3 t3 3 t4 then the random variables lVt2 7 114 and lVt4 7 lVta are independent Proof We shall prove 3 rst To this end note that by de nition of the Wiener process the random variable lVt4 7 lVt3 is independent of E2 and lVt2 7 1th is 2 measurable Hence 3 follows from Proposition 23 if we put f 039 lVt4 7 lVta and g E2 1 Because W0 0 and Wt Wt 7 W0 the rst part of 1 follows from the de nition of Wiener process Let s S t Then CovWsWt EWSWt EWsUVt 7 W5 EWf EWSEUVt 7 W5 EWSZ s minst 2 It is enough to show that for any n 2 1 and any choice of 0 3 t1 lt lt tn the random vector is normally distributed To this end we shall use Proposition 29 Note rst that by 3 the random variables VVtNlVt2 7 114 V14 7 VVtTH are independent and therefore the random vector Wm Wm 7 V141 Vth 7 VVI w i is normally distributed It is also easy to check that X AY with the matrix A given by the equation 1 0 0 0 1 1 0 0 A 1 1 1 1 1 Hence 2 follows from Proposition 29 l Note that the properties of Wiener process given in De nition 41 are similar to the properties of a random walk process We shall show that it is not accidental Let Sth be the process de ned in Exercise 18 Let us choose the size of the space grid a h related to the time grid by the condition a2h h 41 Then using the results of Exercise 18 we nd immediately that ESth 0 and mint1 t2 COVSt1h7St2h h Hence llim0 CovSt1h 52h mint1t2 For every t gt 0 and h gt 0 the random variable Sth is a sum of mutually independent and identically distributed random variables Moreover if 41 holds then all assumption of the 42 Central Limit Theorem are satis ed and therefore Sth has an approximately normal distribution for small h and any It mh with m large More precisely7 for any It gt 07 we nd a sequence tn 72 such that tn converges to t and for a xed 71 we choose hn Then we can show that y 1 22 73330 PltSthn S y e E dz39 It can be shown that the limiting process 5 is identical to the Wiener process introduced in De nition 41 Proposition 43 Let be m g Wiener process Then 1 is m ft martmgale 2 is a Markov process for 0 S s gt PW w Pm am 42 Proof 1 By de nition7 the process is adapted to the ltration and lt 00 because it has a normal distribution The de nition of the Wiener process yields also for s S t EWt 7 Wsl EWt 7 W9 0 which is exactly the martingale property 2 We have S 7 W5 l and putting X M4 7 W5 and Y W5 we obtain 42 from Proposition 24 l The Markov property implies that for every t 2 0 and y E R Sylel17ng 277Wsn n S ylen n for every collection 0 S 51 S S 5 S t and 1 an E R On the other hand7 the Gaussian property of the Wiener process yields for 5 lt t y 1 27 y7xn PWlt Wsnxn ex 7 dzltIgt t yl 00 27Tt7sn plt 2t 5n H Therefore7 if we denote by t s 7 ex p m p 2t 7 then pt 7 sy 7 m is the conditional density of W given W5 z for s lt t The density pt 7 sy 7 z is called a transition density of the Wiener process and is suf cient for determining all nite dimensional distributions of the Wiener process To see this let us consider the random variable lVt17 lVt2 for t1 lt t2 We shall nd its joint density For arbitrary z and y 26 yields Pltmltzm2sygt Pltmzsyim72gtpltthzgtdz w y 10 thy ZPt172d2 43 and therefore the joint density of VV VV2 is ft1t2zy pt2 7t1y 7 rooms The same argument can be repeated for the joint distribution ofthe random vector VV VVT for any n 2 1 The above results can be summarized in the following way If we know the ini tial position of the Markov process at time t 0 and we know the transition density then we know all nite dimensional distributions of the process This means that in principle we can cal culate any functional of the process of the form EFVVVV In particular for n 2 EF VVVV2 fix fix Fzypt2 7 t1y 7 zpt1zdz for t1 lt t2 The next proposition shows the so called invariance properties of the Wiener process Proposition 44 Let be an f Wiener process 1 For ped s 2 0 we de ne a new process V VVS 7 W5 Then the process is also a Wiener process with respect to the ltration 9 45 2 For a ped c gt 0 we de ne a new process U cVVc2 Then the process U is also a Wiener process with respect to the ltration H Cz Proof 1 Clearly Vb 0 and the random variable V is THE g measurable For any h gt 0 we have Vh 7 V VVHMS 7 VV5 and therefore Vh 7 V is independent of TH g with Gaussian N0 h distribution Hence V is a Wiener process 2 The proof of 2 is similar and left as an exercise I We shall apply the above results to calculate some quantities related to simple transformations of the Wiener process Example 41 Let T gt 0 be ped and let B WT 7 WT Then the process B is a Wiener process on the time interval 0 T with respect to the ltration g 0WT 7 WT5 s S t Proof Note rst that the process B is adapted to the ltration G We have also for 0 S s lt t S T Bt 7 B9 WT75 7 WT7t and because the Wiener process W has independent incrernents we can check easily that B 7 BS is independent of 9 Obviously B 7 BS has the Gaussian N0t 7 5 distribution and B0 0 Because B is a continuous process all conditions of De nition 41 are satis ed and B is a Wiener process I Example 42 Let S mt UVV The process S is called a Wiener process with drift in and variance 02 Clearly ES int and E S 7 mt2 ozt We shall determine the joint density of the random variables VVS2 Note that S is a Gaussian process and the random variable VVS2 VVmt2 UlV2 is jointly Gaussian as well can you show that By the de nition of the Wiener process we obtain CovVVmt2 UlV2 oEVVVV2 o mint1t2 44 Hence the covariance matrid is O lt t1 omint1t2 o mint1 t2 Uztg 0 t1 039 mint1 t2 ml StZ N mtg 7 lt omint1t2 Uztg 39 Example 43 Let Yt ze39rritc7Wt39 The process Y is called an exponential Wiener process and ubiquitous in mathematical nance This process is a continuous time analogue of the eccponential random walk Note rst that by Theorem 22 the process Y is adapted to the ltration By 7 EYt zeltm 2t We shall show now that the process Y is an g martingale if and only if Indeed for s S t we obtain E zemtaWt js E zemsmtisaWs 60Wt7WS YsEemtisaWtiWs Ys6mtisEeaWtis Ys6mtis02WtiWs and the claim follows In the next example we shall need the following Lemma 45 Let X be a stochastic process such that T E Xslds lt oo 0 T T E Xsds EXsds 0 0 Moreover if the process X is adapted to the ltration ft then the process Th en t YtXsds igT 0 is also adapted to Additionally if X is a Gaussian process then the process Y is also Gaussian 45 Proof We give only a very brief idea of the proof By assumption the process X is integrable on 07 T and hence we can de ne an approximating sequence kT k 7 1 T W XE 9 2n 2n 2n OltkT2quotgt for the process It is not dif cult to see that all of the properties stated in the lemma hold if we replace Y with Y By de nition of the integral lim Yt Y for every t gt 0 and it remains to prove only that the properties stated in the lemma are preserved in the limit This part of the proof is omitted I t Yt Wsds 0 By Lemma 45 it is an g adapted Gaussian process We shall determine the distribution of the random variable Y for a ded t By Lemma 45 we have EYt 0 Now for s S t once more by Lemma 45 s t s t EYSYt E Wudu Wvdo EWqududo 0 0 0 0 s s s t EWqududU EWqududo 0 0 0 s s s s t s t minuodudo EWuWv7Wsdudi EWqududo 0 0 0 s 0 s l l 1 S 530t7sudug53 t7s52 Example 44 Consider the process because 5 t EWM Wu 7 W5 dudo 0 0 5 Hence 1 1 EYSYt g min53t3 t 7 s min52t2 42 WIENER PROCESS PROPERTIES OF SAMPLE PATHS We did not specify the sample space of the Wiener process yet In analogy with the discussion in Lecture 1 we identify the sample point with the whole Brownian path to trajectory t 7 Wt w for t 2 0 or t 3 T7 which is a continuous function of time by de nition We shall show that Brownian paths are extremely irregular We start with some introductory remarks Consider a function f 07 oo 7 R such that f0 0 and T watch lt oo 0 46 in which case t t d f f8 5 Let us calculate how fast are the oscillations of the function f on a xed interval 0T To this end for every n 2 1 we divide 07 T into a sequence of kn subintervals P1 07t7P t7t 77P 2241 The whole division will be denoted by P With every division P of 0T we associate its 77size77 de ned by the equation dP to 7 am where t3 0 and t T For a given division P we de ne the corresponding measure of the oscillation of the function f f tk 7 f 271M Then k Vim Z i1 tgi kn tgi T ready 2 lfWdy lfWdy tgiil i1 tgiil It follows that T W s mm dy lt oo 0 and this bound is independent of the choice of the division P If sup Vnf lt oo n21 then the limit Vf exists and is called a variation of the function f A function with this property is called a function of bounded variation For a given division P we shall calculate now the so called quadratic variation dualism W kn 7 f tki712 of the function f Using the same arguments as above we obtain by Schwartz inequality kn WW Zlt k1 kn 23 T n 7 n 2 2 n 7 n 2 ta m1 lf ml do to lfyl do i 2 kn ti Hindu S 2 t3 tZi1gt lf yl2dy k1 k it tquot tquot 71 iii M l Finally T WW dun 0 lf yl2dy 43 47 and therefore 1 W 0 dP1quotIna0 f We say that the function f with the square integrable derivative has zero quadratic variation We shall show that the behavior of brownian paths is very different from the behavior of differentiable functions described above De nition 46 A stochastic process Xt t 2 0 is of nite quadratic variation if there epists a process ltXgt such that for every t 2 0 i 2 df 1nII0E V7900 ltXgttl 0 where P denotes a division of the interval 0t Theorem 47 A Wiener process is of nite quadratic variation and ltWgtt t for every t 2 0 Proof For a cced t gt 0 we need to consider kn WNW Z Wt lVtz lgt2 i1 Note rst that kn EVENW 2 E Wally Z t 771 t i1 i1 and therefore E WWW 02 E VthQgt2 t t1gt239 i1 Because increments of the Wiener process are independent 2 kn 2 2 E WW 7 t E on 7 will i at 7 11 i1 Finally we obtain kn kn WWWgt702 22tlet211fsup2t7t12t7tll i1 is 11 2tdP and therefore ltWgtt t as desired l Corollary 48 Brownian paths are of in nite variation on any interval PVWoo 1 Proof Let 07 t be such an interval that for a certain sequence P of division h T 2 m 2 Wei t we H 48 Th en k 2 M E 7mg sup Willem31 E Wt iii31 i1 is 11 Now the left hand side of this inequality tends to t and continuity of the Wiener process yields lim su lthilth 0 V LHOO Z 271 Therefore necessarily kn lim E Wu 7 Wu 00 V LHOO Z 271 i1 l Theorem 49 Brownian paths are nowhere di erentiable Consider the process Mt VV 7 t which is obviously adapted We have also EltMt l A Ewing Em7W22WSWt7W etl EltWt7Wgt2lf2WEltmlfsgteW37t tis2WfinitMs and therefore the process Mt is an g martingale We can rephrase this result by saying that the process IlIt lVtz 7 ltWgtt is a martingale7 the property which will be important in the future 43 EXERCISES a Let Yt t 2 0 be a Gaussian stochastic process and let g 0700 a R be two functions Show that the process X ft gtYt is Gaussian Deduce that the process X z at TlVt where W is a Wiener process is Gaussian for any choice of a E R and o gt 0 b Show that X is a Markov process Find EX and CovXsXt c Find the joint distribution of Xt17Xt2 Compute E 5th th fOI t1 S t2 d Find the distribution of the random variable t Z Xsds 0 Compute ltXgtt Show that the process X ilVt is also a Wiener process AA i hCD VV 49 D 00 4 CT CT Brownian bridge Let be a Wiener process Let utx Ef x M4 for a certain bounded function f Use the de nition of expected value to show that 00 1 yZ u ta z ex 77 d m 10M plt2t 1 Using change of variables show that the function u is twice differentiable in s and differen tiable in t gt 0 Finally7 show that 3U i t atlt 7x 16211 a lm39 Let S be an exponential Wiener process starting from SO z gt 0 a Find the density of the random variable St for t gt 0 b Find the mean and variance of 5 c Let for z gt 0 ut EfSt for a bounded function f Show that u has the same differentiability properties as in 2 and u A 02 321i 1 3U 2 02M m Eaz a tw Let be a Wiener process and let denote its natural ltration a Compute the covariance matrix C for lVS7 b Use the covariance matrix C to write down the joint density ay for lVS7 c Change the variable in the density f to compute the joint density 9 for the pair of random variables lVS7 M4 7 W5 Note this density factors into a density for W5 and a density for M4 7 lVS7 which shows that these two random variables are independent d Use the density 9 in c to compute Mum Eexp ltqu M thexp nv jlt Let Xt t 2 0 be a continuous and a Gaussian process7 such that EX 0 and E Xth rnins7 t Show that the process X is a Wiener process with respect to the ltration Verify that Let Ly be arbitrary real numbers The Brownian bridge between z and y is de ned by the formula t t Vf ylt1iflVt7flVTiy th a Show that PthHwPMM 7tT Deduce that for all t S T PWt a 1304041 g a mu where fT denotes the density of WT b For anyt TandanyO tlgugtngTshowthat PM a1mnanlWTyPi4gtya17wigtyan for any real 117an c Compute and Gov for t1 t2 3 T and arbitrary Ly d Show that the process t S T is also a Brownian bridge process 7 Let be an g Wiener process and let X a Find P X S s and evaluate the density of the random variable Xt b Write down the formula for PXS S LXt S y as a double integral from a certain function and derive the joint density of X57Xt Consider all possible values of s t and x y c For 5 lt t nd P X S y l XS m in the integral form and compute the conditional density of X given XS x d ls X a Markov process Chapter 5 STRONG MARKOV PROPERTY AND REFLECTION PRINCIPLE A good reference for this chapter is Karatzas I and Shreve S E 39 Stochastic Calculus and Brownian Motion Let be a real valued ft Wiener process For b gt 0 we de ne a random variable Tbmint20IVtb 51 It is clear that Tb S t 2 b This identity shows that the event Tb S t is completely determined by the past of the Wiener process up to time t and therefore for every t 2 0 Tb S t E 7 that is7 Tb is a stopping time We shall compute now its probability distribution Note rst that PTbltt PTb lttWt gtbPn7 lttWt ltb PWt gtbPn7 lttWt ltb Using in a heuristic way the identity PTblttWt ltb PTb lttWt gtb PWt gtb we obtain e w22dz 001 P7ltt2PWgtb2 I t we Hence for every b gt 0 the stopping time Tb has the density 2 fbt Swap t 2 0 53 Let us recall the result Proposition 43 from Lecture 4 which says that the Wiener process starts afresh at any moment of time s 2 0 in the sense that the process Bi Wst 7 W5 is also a Wiener process The intuitive argument applied to derive 52 is based on the assumption that the Wiener process starts afresh if the xed moment of time s is replaced with a stopping time Tb that is7 it is implicitly assumed that the process WW 7 Wm is also a Wiener process The rigorous argument is based on the following strong Markov property P WW 3 z I r P WW 3 z I W 54 for all z E R In this de nition we assume that 739 is a nite stopping time7 that is P 739 lt oo 1 In the same way as in the case of the Markov property 54 yields E Wrt I f7 Wrt I Theorem 51 The Wiener process enjoys the strong Markov property Moreover ifr is any stopping time then the process B W7 7 WT is a Wiener process with respect to the ltration gt fprt and the process Bt is independent of go Corollary 52 Let be an g Wiener process and for b gt 0 let Tb be given by 51 Then the density of Tb is given by It follows from 52 that i i b PTb lt00tlir 10Prb St 733302 1 7ltlgt 71 It means that the Wiener process cannot stay forever below any xed level b Theorem 51 allows us to calculate many important probabilities related to the Wiener process Note rst that for the random variable W W3 we obtain for z 2 0 d E Hence the random variables W and have the same distributions PWt 3xP7 mgtt2ltlgt gt71Pth 3x Proposition 53 Fort gt 0 the random variable lVtJVf has the density 22yim 1 7 2yim2 may t Mexplt 2 gt zfz waO 0 otherwise Proof Note rst that by the symmetry of Wiener process PbW5 aPWS Zbia PbW52 2bia Hence denoting B WWH 7 WW and using the strong Markov property of the Wiener process we obtain WSWW21 WtSMTySUPWSt yWtiWTy gzlrygtPTy t yBHy z Ty tPTy gt yBtTy 22y7xlry tPry t ylVtiWTy 22yizlry tPry t m22yi lW 2yPWi2y m 2 2y eyedVi 2a m 2 2y 7 96 1 co 2 7 72 2td 7 6 Z v27Tt xZyim PWWSLWSSyPVW 7PlWSLVWZi Hence and because 2 f7ymPWt 963 31 the proposition follows easily I Note that by the invariance property of the Wiener process P maXlVS 2 b P ltmax7Ws 2 b P min W5 3 7b Sgt Sgt Sgt Proposition 53 allows us to determine the distribution of the random variable W 7 M4 We start with simple observation that W 7 W maxlVS 7 MG max VVFS 7 Sgt t7520 Since the process BS M4 7 VVFS is a Wiener process on the time interval 07 t we nd that PlVt7lVt x PmltatXB5 x PlBtl 3x Note that we have proved that the random variables llth W5 and W 7 W have the same density 53 We end up with one more example of calculations related to the distribution of the maximum of the Wiener process Let X z W with z gt 0 We shall calculate PltXt ylmltinXsgt0 For y S 0 this probability is equal to zero Let y gt 0 By the de nition of the process X PltXt ylmltinXsgt0 PltVVt y7lmltinWsgt7x P 7 27yl7mlti1nVVS ltx M2z7ylm237Wslt l quotU Bt2x7ylmaXBslt Sgt P P B x 7 y maxsgt BS lt m 2 P maxsgt BS lt s where B 7lVt is a new Wiener process Now7 we have P B 2z7y7maths ltz PBt 27y 7PBt 2z7y7B 2x and by Proposition 53 00 co 2 2 7 2 7 2 PBt2x7yB2z Mexp dvdu 711 w 27Tt3 2t Note that 3 co 22117 207 Wig 7 gt 7 gt 7 yP B x y Bi s m 2 t3 exp 2t d1 Therefore the conditional density of X given that rninsgt Xx gt 0 is i 6 i fltylrgglstgt0 PXtylglgl Xsgt0 1 3 3 7PBgt777PBgt7Bgt PTmlttlt6y 117 y 3y 117 yl tin 1 i 7 We end up this lecture with some properties of the distribution of the stopping time 71 If is a Wiener process then the process V1407 1Wb2t is also a Wiener process We de ne a stopping tirne 75b rnint 2 0 1151 1 Proposition 54 We have Tb bzrgb and consequently Tb has the same density as the random 2 variable b279 Moreover the random variables Tb and have the same densities 1 Proof By de nition Tbmint20b 1m1 rninb2t20lVtb 1 b2 min t 2 0lVtb 1 b27517 In order to prove the remaining properties it is enough to put b 1 and show that 7391 has the same density as Note rst that lV has the same density as Therefore 1 P712xPW1P2Wf1P 12 95 W1 and this concludes the proof I 51 EXERCISES 1 Let X z TlVt where is a Wiener process and 0x gt 0 are xed Let T0rnint20Xt0 a Find the conditional density of X given XS z for s S t b Find the density of the stopping time To c Compute P X S y77390 gt t d Let 7 X lftgto Find P Y S y 1 Y5 p for s S t and the conditional density of 1 given XS x 2 Let Tb be the stopping time 51 Show that Erb oo C40 4 Cf 03 I Let W be a Wiener process For a xed to gt 0 we de ne the barrier i a 1ft 3 to bt b iftgtt0 where 0 lt a lt b Let W be a Wiener process and Trnint20lVt 2bt Compute P 739 S t for t 2 0 Let Tb be de ned by 51 Using the Strong Markov Property show that for 0 S a lt b TbiTQinft20WmtilVmbia Derive that the random variable Tb 7 Ta is independent of the U algebra fly Finally7 show that the stochastic process Ta a 2 0 has independent incrernents Show that the conditional density of the pair mslVts exp lt Find the joint distribution of the random variable lVS7 lVt for s 31 t given W a and W b is 22yizia 27Ts3 ay l cab 28 2ysa2gt39 Let W and B be two independent Wiener processes a Show that the random variables 5 and W1 have the same Cauchy density 1 7T1239 b Let Tb be the stopping tirne de ned by 51 for the process Apply Proposition 54 to show that the random variable B72 has the same distribution as the random variable V l l Bl and deduce from a the density of the random variable BM Chapter 6 MULTIDIMENSIONAL WIENER PRO CESS Reference Karatzas I and Shreue S E 39 Stochastic Calculus and Brownian Motion Let W l7W27Wd be a family of d independent Wiener processes adapted to the same ltration An Rd valued process Vth Wt 3 Vth is called a d dirnensional Wiener process For this process the following properties hold Proposition 61 Let be a d dimensional Wiener process adapted to the ltration Then 1 ElVt 0 and EWslVtT rninstI for all st 2 0 and in particular ElthVtT t 2 The process is Gaussian 3 The Wiener process has independent increments Proposition 62 Let be a d dimensional Wiener process adapted to the ltration Then 1 is an g martingale 2 is a Markov process for 0 S s S t and real numbers yl yd Pm1y1mdydlfsPm1y1mdyilws 61 Let X and be two real valued continuous stochastic processes For any division P of the interval 0t we de ne kn 2 i 39 39 Vi gt XX Z Xti a th K e XL 62 11 De nition 63 Let X and Y be two R ualued continuous processes We say that X and Y haue joint quadratic variation process ltX7Ygt if for every t 2 0 there epists random variables ltX7Ygtt such that 2 lim 0E W2XY a ltXYgtt 0 Pup IfX Y then we write ltXgt instead of ltX7Xgt and call it quadratic uariation process of X If X and are two Rd ualued continuous stochastic processes that is XT Xt l7 7Xtd and YtT Yt17Ytd such that ltXi7Y7gtt epists for all ij 17 7d then the matricc ualued process ltX7Ygtt ltXi7Yjgtijltd is called ajoint quadratic variation of the processes Xi and Hence for vector valued processes quadratic variation is a matrix valued stochastic process which can be de ned using rnatrix notation kn T ltX7Ygtt dby gnao Xti 7 thil Yti 7 Ytlii 7 6393 11 where P n 2 17 is a sequence of divisions of 0t Lemma 64 If XY7 Z are real valued processes of nite quadratic variation then ltX7Ygtt ltY7Xgtt ltX Ygtt 7 ltX 7 Ygtt 6394 1 1 4 4 and ltaXbYZgtt altXZgttbltYZgtt 65 Proof The symmetry ofjoint quadratic variation follows immediately from the de nition To show the second equality in it is enough to notice that for any real numbers ab 1 1 ab 1a b2 7 1a 7 b2 and to apply this identity to Equation 64 follows easily from I We shall nd the quadratic variation of the d dimensional Wiener process W lfi j then using 62 and the result for one dimensional Wiener process Wi we nd that W Wigtt t We shall show that for i 79739 I 1 W1 W7gtt 0 66 Note rst that EVTEZ lVi7 Wj 0 because WW and W7 are independent By de nition kn kn vs ltle W75 Z 2 win 7 wt mi 7 W6 Wit 7 W521 a l1 m1 Then independence of the increments and independence of Wiener processes WW and W7 yield 2 kquot 2 2 EM WW0 ZE Win 7w E W5 7mg 1 Z t 7 61 td Pr l1 and the last expression tends to zero Hence we proved that M M 67 where I denotes the identity matrix Consider an Rm valued process X BlVt7 where B is an arbitrary m gtlt d matrix and W is an Rd valued Wiener process In future we shall develop other tools to investigate more general processes7 but properties of this process can be obtained by methods already known First note that X is a linear transformation of a Gaussian process and hence is Gaussian itself We can easily check that EX BEWt 0 and CovXs Xi EXSXtT min m BBT 68 The process X is also an g martingale lndeed7 for s S t We will nd quadratic variation of the process ltXgt using 62 1 V752 XvX Z th 7 Xtia X1517 th71T l1 kn 7 B ml 7 mlil ml 7 ml71TBT 11 31752 W W BT Therefore using 66 we obtain ltXgtt tBBT 69 In particular7 if d 1 and X blVt then ltXgtt bzt 1 Note that it follows from 67 and 68 that two coordinates of the process Xt X and X say7 are independent if and only if ltXCX7Igtt 0 Example 61 Let and be two independent Wiener processes and let B3 W and BE elth dWE We will nd the joint quadratic variation ltB1 B2gt Lemma 64 yields 3132 ltaW1 bWZ CW1 dW2gtt ac W1 W2gtt adltW1 W2gtt be W1 W2gtt bdltW1 W2gtt ac bdt where the last equality follows from 66 61 EXERCISES Continuation of Example 61 Assume that a2 b2 1 Show that in this case the process B1 is also a Wiener process Find ltW1B1gtt H D Assume that W is a d dimensional Wiener process and let U be a d gtlt d unitary matrix7 that is UT U l Show that the process X UlVt is also a Wiener process 00 Let W be a d dimensional Wiener process a For any a 6 Rd show that the process aTlVt is a martingale T i 2 lta Wgtt i lal t7 where lal is the length of the vector a b Show that the process Mt WW2 7 td is a martingale c Let X BlVt7 where B is an m gtlt d matrix Show that the process Mt lthzitMBBT is a rnartingale7 where for any m gtlt m matrix C trC elTC39eL e Cem and e17 em are basis vectors in R that is 1 if 239 j T I 7 gig 0 ifz39y j Show also that the process Mt Xng i X2X7gtt is a martingale for any choice of 2397j S m Deduce that the rnatriX valued process XtXtT 7 ltXgtt is a martingale 4 Let W be a d dirnensional Wiener process and let X lx lthz7 where z is any starting point in Rd The process X is called a Bessel process starting from Using the de nition of X2 distribution or otherwise write down the density of the random variable Xto Using properties of the Wiener process show that the random variables X and Xm have the same distribution Chapter 7 STOCHASTIC INTEGRAL Reference Karatzas I arid Shreue S E 39 Stochastic Calculus arid Brownian Motiori Let t be xed and let P be any division of 0t We de ne kn Mn Z Wag Wan i1 Note rst that by simple algebra kn 2 W3 lt2 i1 kn 2 kn Z Wt Wig QZWtL Wt lVtZT1gt i1 i1 Hence k 1 1 quot 2 Mn 5W3 5 Z War i1 but the second term on the right hand side is known to converge to the quadratic variation of the Wiener process7 and nally 2 1 E MniEUVfit 0 lim Pipe Therefore we are able to determine the limit of the integral sums M which can justi ably be called the integral t Wde5 3 WE it 0 2 In general the argument is more complicated but it repeats the same idea For an arbitrary stochastic process X we de ne an integral sum kn In ZXtiLI will i1 determined by a division P of the interval 0t Theorem 71 Assume that X is a process adapted to the ltration arid such that t stds lt 00 71 0 Theri there eists ari ft measurable random variable t I XsdlVS 0 such that for every 8 gt 0 lim PlIn71l gt8 0 If moreover t Edes lt 00 72 0 thert lirn E in i n2 0 If 71 holds for every t S T thert the above theorem allows us to de rte art adapted stochastic 17700688 t Mt Xdes 0 Theorem 72 If X is art adapted process artd T stds lt 00 0 thert the stochastic lrttegral ertjoys the following properties 2 forallOStlgtggT t1 t2 t2 XsdlVS XdeS XdeS 0 t1 0 3 if is another adapted process such that T Yszds lt 00 0 then for all t S T t t t aXS bYs dWS a XdeS b stWs 0 0 0 4 if moreover7 T E des lt 00 0 then the process t M XsdlVS 0 de ned for t S T is a continuous martingale with respect to the ltration and t EMS Edes 0 It will be important to know what is the quadratic variation of the martingale de ned by a stochastic integral The answer is provided by the next theorem Theorem 73 If Mt t XdeS thert 0 t ltMgtt des 0 Corollary 74 Let MtthdW51 and NttstWf 0 0 where Y5 X5 are two ft adapted processes such that t X y3dsltoo 0 and MG are two g adapted Wterter processes Thert t ltMNgtt XsstltW1W2gts 0 Proof We prove the corollary for W1 W2The general case is left as an exercise Invoking the result from chapter 5 we nd that ltMNgtt ltMNMNgttiltM7NM7Ngtt OVX stdseOWXSimzds t Xssts 0 and the corollary follows I Having de ned a stochastic integral we can introduce a large class of processes called semi martingales In these notes a semimartingale is a stochastic process X adapted to a given ltration such that 1 Z 1 Z t t XtX0 asds bdeS 73 0 0 where X0 is a random variable measurable with respect to the U algebra f0 and 17 b are two adapted processes with the property T lasi bi ds lt oo 0 We often write this process in a differential form dXt atdt btth Equation 73 is called a decomposition of the semimartingale X into the nite variation part it At 1st 0 and the martingale part if Mt bde5 0 For continuous semimartingales this decomposition is unique If T T Yszbids lt 00 and leasl d5 lt oo 0 0 then we can de ne an integral stdXS of one semimartingale with respect to another t t t stXs Ysasd5 YsbdeS igT 0 0 0 and the result is still a semimartingale Because processes of nite variation have zero quadratic variation7 the quadratic variation of a semimartingales X is the same as the quadratic variation of its martingale part ltXgtt ltMgtt Ot bids It follows that7 for any semimartingale Xi7 if ltXgtt 0 for every t 2 07 then its martingale part is zero and t X X0 asds 0 If the process X adapted to the ltration is a semimartingale7 then its decomposition 73 into a martingale and a process of bounded variation is called the semimartingale representa tion of Xt We consider now d semimartingales adapted to the same ltration t t X XgAg Mg39 Xg aid5 bgdwg 74 0 0 where i 17 7d and are g Wiener processes Theorem 75 Ito s formula Let F Rd a R be a function with two continuous derivatives and let the semimartingales th X be given by 74 Then d t M l FXt17Xf FX5XgO XjXfdAs d t d t 2 6F 1 d i l 1 d i j 6XsXsdMs2iO hiaijXsXsdltXXgts In a more erplieit form the Ito s formula can be written as d t FXt1Xtd FX3Xg ZO Z X Xjads t1 l t d t 2 5F X37Xzgtbdw1 LFXLx5bibdltwawjgt 7 3 2H1 0 Ml3w 9 64 Corollary 76 Let X X0 f5 asds fot bsdlVS and let F 07 00 gtlt R 7 R have two continuous deriuatiues Then taF taF FtX FOX 7 XS SCl i 7X9 9d 7 mow065e as06s gtas 16F 1 62F 2 0 E5Xsbde5 O w ngbsds Proof It is enough to put in Theorem 75 d 27 X t and X3 Xt I It is often convenient to write the lto7s formula in its in nitesimal form M M 1 262F oF dF mg E mg WE mg in mm dt bta mg aw Example 71 Let X Then t t 1 t 7 1 t Xt an ldW9 5710171 WZ stn Wf ldWs Wf zds 0 0 0 0 and therefore t 1 7 1 t Wf ldWs 7m 7 n WZ st 75 0 n 2 0 For n 2 we recover in a simple way the fact already known that t 1 Wde5 7 WE 7 t 0 2 t t Xt X0 asds bde91 0 0 t t Y 7Y0 usds 71de 0 0 be two semimartingales with possibly dependent Wiener processes and We will nd the semimartingale representation of the process Zt Xth The to formula applied to the function F Ly xy yields immediately t Xth XOYO XsusYgas ds 0 t t t XsodeSZ Ysbde91 bsosdltW1W2gts 0 0 0 We can write this formula in a compact form t t Xdes 7 Xth 7 XOYO 7 stXs 7 ltXYgtt 0 0 which can be called a stochastic integration by parts formula For t M Xde57 0 cortsider the process Nt exp 1 7 um We will apply the to formula to the process F XLXtZ where t t X ltMgtt Xids X3 M Xdes 0 0 artd Fz1x2 em wl Thert 6F 1 61172 5 17M 6F PF 6 17M 67 17M FWiJz artd therefore the to formula yields t 1 t 1 t t M 1 7 Nstds NsXdeS E Nsstds 1 NsXdes 0 0 0 0 As a by product we rtd that t ltNgtt N X ds 0 The process N will be important in many problems Below are the rst applications Example 72 Let S be art eccporterttial Wierter process St SO exp mt olVt Clearly St Soemt o2tethi o2t Soemt c72tjvtl rt this case usirtg the eccample from Lecture 5 we rtd that T 02 Estds lt oo 0 artd therefore the process N is a martirtgale Hertce we obtairt the semimartirtgale represerttatiort of the eccporterttial Wierter process t 1 t st so m E02 Ssds Usdes 0 0 artd ifm 702 thert as we already krtow St is a martingale with the represerttatiort t St SO oSdes 0 rt rtartcial applicatiorts the coe ciertt m is usually writtert irt the form and then t t S SO rSsds oSdes 0 0 Let us consider a simple but important case ofa stochastic integral with the deterministic integrand f such that T f2sds lt 00 It follows from Theorem 72 that the procOess Mt finsqu t g T 0 is a continuous martingale but in this case we can say more Proposition 77 Let Mt t Xdes 0 The process is Gaussian if and only if its quadratic uariation process t ltMgtt Xids 0 is deterministic or equivalently X5 is a deterministic function Theorem 78 Levy Theorem Let Mt be an Rd ualued continuous g martingale starting from zero with the quadratic variation process Then is a Wiener process if and only if ltM gt t a lt76 t 0 ifi 7s j 39 Proof If is a Wiener process then its quadratic variation is given by 76 Assume now that the process Mt satis es the assumptions of the theorem We need to show that Mt is a Wiener process We shall apply the theorem from chapter 2 Let7 for every a 6 Rd F a7 d em be a function de ned on Rd Then 6F 62F Ema iajF a7 x mm gt iajakF a7 z By the lto formula d t 39 T E ag e Mudu S d t 1 FaMt ewTMs HE aj ewTMudMu i 5 11 5 ij1 Hence eiaTMt7Ms 77 7 139 d t iaTMu Ms 71 d 2 t iaTMu7MS igaJs e dMu 21131ajS e du By the properties of stochastic integrals t E ewTMudMu f5 0 1 t EGMWFMmQEme wtIWWHMJMUgtW Let A 6 75 Then If we denote I T f E m M LQ then we obtain an equation 2 t a WFPWfww 01 1 Ho Qfm with the initial condition f s PA It is easy to show that the unique solution to this differential equation is given by the formula wPMMmlt2lttsO Finally7 taking into account the de nition of conditional expectation we proved that E eiaTMquotMSIAgt E JAE emTMquotMS E lt1Aexp t i sgtgt and therefore E eiaTMtiMs f5 equizgisyz and therefore the random variable Mt 7 M5 is independent of f5 and has the normal N 07 t 7 sI distribution and the proof is nished I We will apply this theorem to quickly show the following Corollary 79 Let be an Rd Ualued Wiener process and let the matricc B be unitary BT B l Then the process X BlVt is also art g Wiener process Proof We already know that X is a continuous rnartingale with quadratic variation ltXgtt tBBT Now7 by assumption ltXgtt t and the Levy Theorem concludes the proof I H D 7 CT 71 EXERCISES Find the semimartingale representation of the process t X eatx eat eiasb dlVS7 0 where 17 b are arbitrary constants Show that t Xtxa Xsdsbm 0 Use the lto7s Formula to write a semimartingale representation of the process Yt cos Next7 apply the properties of stochastic integrals to show that the function mt EYt satis es the equation 1 t mt17 0 msds Argue that m t ilmt and 7710 1 2 Show that mt 642 Let t X bde5 0 where b i 1 if W 2 0 t 71 if m lt 0 Show that X is a Wiener process Let be an g Wiener process and let F 07 oo gtlt R a R satis es the assumptions of the lto7s Lemma Find conditions on the function F under which the process Yt F t7 M4 is a martingale Prove the general version of Corollary 74
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