Exercise 4. An optional time which is not a stopping time

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1 M5MF6, EXERCICE SET 1 We shall here consider a gien filtered probability space Ω, F, P, spporting a standard rownian motion W t t, with natral filtration F t t. Exercise 1 Proe Proposition 1.1.3, Theorem 1..9, and the examples/exercises in the lectre notes in Sections Answer of exercise 1 See the lectre notes. Exercise An optional time which is not a stopping time Consider the filtration G t t and the random time τ defined by { { {, Ω}, if t 1, 1, if ω, G t : and τ : Ω, if t > 1., if ω /, where is some non-triial sbset of Ω. Show that τ is a G-optional time, bt not a G-stopping time. Exercise 3 Gassian integral Define the process X t t pathwise by X t : φ sd, where φ is a deterministic fnction path. Proe that X is a Gassian process with mean zero and coariance strctre EX s X t s t φ sds. Answer of exercise 3 We first proe that, for any t, X t is a Gassian random ariable with mean zero and ariance φ sds. Itô s lemma yields.1 E e X t φ se e X s ds, for any real nmber. For any sch, define ψ t : E e Xt, so that differentiating both sides of.1 implies t ψ t 1 φ t ψ t, 1 which is easily soled as ψ t exp t φ sds. This proes the claim. The proof of the coariance strctre follows the same steps. Exercise 4 Pinned rownian motion Find a orel fnction ϕ sch that E W t ϕw t, for any s t. Date: Janary 3, 16. 1

2 M5MF6, EXERCICE SET 1 Answer of exercise 4 Fix s < t. We are looking here for a orel fnction ϕ sch that E W t ϕw t ; we can write this eqialently, for any orel sbset R, as dp t dx φxp t dx φxp t, xdx, where P t is the law of the rownian motion at time t, and p t the Gassian density at time t; frthermore, P t dx xp s, xp t s x, ydx dy R pt;, y R sy t p t, ydy. Taking the fnction φy sy/t concldes the proof. sy xp st s/t t, x dx dy Exercise 5 Spremm of rownian motion and Qadratic ariation Let M be a continos local martingale starting at the origin. Then, for all x, >, P sp M t > x, M exp x. t Hint: Fix some x >, and let τ : inf{t : M t x} be the first hitting time of the leel x. For any α R, introdce the process Z t t defined pathwise by Z t : exp αmt τ 1 α M τ t. Use then the optional sampling theorem: a martingale stopped at a stopping time remains a martingale. Answer of exercise 5 Fix some x >, and let τ : inf{t : M t x} be the first hitting time of the leel x. For any α R, introdce the process Z t t defined pathwise by Z t : exp αmt τ 1 α M τ t. Clearly, Z is a continos local martingale by Itô s formla, and satisfies the ineqality Z t expαx almost srely for all t. When α 1, Z is sally called the stochastic exponential of M, and is denoted by EM. It is therefore sqare integrable, and the optional sampling theorem implies EZ EZ 1. Marko s ineqality frther yields, for any >, P sp M t > x, M t P Z exp αx 1 α exp αx + 1 α. Since α is taken randomly, one can optimise oer it, and the maximm on the right-hand side is clearly attained at α x/, from which the reslt follows. Exercise 6 Martingale Representation Theorem will be coered on 5/1/16

3 M5MF6, EXERCICE SET 1 3 Write down the explicit form of the martingale representation theorem for the process M t t defined as i M t ds; ii M t W t ; iii M t W s ds; i M t sinw t ; M t expw t. Answer of exercise 6 i Clearly, M t d; ii Integration by parts immediately gies M t t sd; iii Itô s formla yields M t t + d EM t + d ; i Itô s formla applied to the fnction f : W, t tw yields so that M t tw t W s ds tw t Itô s formla yields t sinw t exp cosw exp t t + 1 t + dw + 1 cosw exp dw. W s ds + s d + sds s d d sd W, W s s d 1 t t s d EM t + Since EM t is clearly nll, the representation follows. sinw exp d 1 t s d sinw exp d W, W i Hint: apply Itô s formla to the fnction t, W t e Wt ft, for some smooth fnction f : [, R. Exercise 7 Martingale Representation Theorem will be coered on 5/1/16 Define the process M by M t : EW 3 T F t, for any t [, T ]. Proe that M t 3 T s + W s dws. Answer of exercise 7 M t E t W 3 T E t [W T W t +W t 3 ] E t [W T W t 3 ]+W 3 t +3W t E t [W T W t ]+3W t E t [W T W t ] W 3 t +3T tw t Applying Itô s lemma then yields dm t 3 W t + T t dw t,

4 4 M5MF6, EXERCICE SET 1 which concldes the proof. Exercise 8 t Proe that the process Y : W d is Gassian, and compte its expectation and ariance. t Answer of exercise 8 The integration is defined pathwise as a Riemann integral since the integrand is continos. Since W is Gassian, for any t, the random ariable Y t is clearly Gassian as limit of Riemann sms, and so is the process Y. Clearly EY t, and, for any s t, s E Y s Y t E W d W d s dd s 6 3t s. Consider the process Y defined, for all t, by Exercise 9 Y t : W t W d. Proe that Y is Gassian, and compte its expectation and ariance. Show that it is not an F t -martingale. Answer of exercise 9 The process Y is the sm of two Gassian, bt these are not independent. Howeer, the integral is a Gassian process by definition of sms in L of Riemannn integrals. Clearly ey t and, for any s, t, { } EY s Y t E W t d W d s W t { EW t E W t d W t E d } + E d W d s s t s t s E W t W d t d E W d + s d + s s EW W dd dd s t, which proes that the Gassian process Z is a rownian motion. Howeer, for any s < t, E Z t Z s Fs W t E W t d W W s d t F s W W d, which is clearly non zero almost srely. s Exercise 1 Clark-Ocone Formla Let f be a bonded C 1 fnction on R. Proe that there exists a fnction g : [, 1] R R sch that E fw 1 F t gt, W t, for any t [.1],

5 M5MF6, EXERCICE SET 1 5 and write down an Itô formla for g. Proe finally that the following eqality holds for all t [, 1]: gt, W t EfW 1 + Answer of exercise 1 E f W 1 F s d. It is easy to see that [ ] E fw 1 F t E fw 1 W t + W t F t E f Ŵ1 t + W t Ft gt, W t, where the fnction g is defined as gt, x E fx + Ŵ1 t F t. Note that the process gt, W t t [,1] is clearly a martingale, and hence gt, W t EfW 1 + and the reslt follows. x gs, d EfW 1 + ] E [f + Ŵ1 s F s d, Department of Mathematics, Imperial College London address: a.jacqier@imperial.ac.k