Problem 1. Construct a filtered probability space on which a Brownian motion W and an adapted process X are defined and such that

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1 Stochatic Calculu Example heet 4 - Lent 5 Michael Tehranchi Problem. Contruct a filtered probability pace on which a Brownian motion W and an adapted proce X are defined and uch that dx t = X t t dt + dw t, X =. I X adapted to filtration generated by W? I W a Brownian motion in the filtration generated by X? [Hint: See example heet, problem 3. Solution. Let X be a Brownian motion, and let X W t = X t d. By example heet, the proce W i a Brownian motion. But ince X t and W t are independent, X i not meaurable with repect to the filtration generated by W. Note that the SDE doe not have uniquene in law, ince X (k) t = X t + kt i a olution for all k R. [Tirel on found a bounded function b uch that the SDE ha a weak olution but no trong olution. dx t = b(t, X t )dt + dw t Problem. (a) Prove Bihari inequality: Suppoe f i atifie the inequality f(t) a + k(f())d for all t for a contant a and where the function k i poitive and increaing and k f i locally integrable. Show that f(t) du t for all t. a k(u) (b) Show that if f(t) k(f())d for all t and r du = for all r > k(u) then f(t) = for all t. (c) Prove Ogood uniquene theorem: Suppoe for every n there exit a poitive, increaing function k n uch that b(x) b(y) k n ( x y ) for all x, y R d, x n, y n. Furthermore, uppoe that for all n, r du k n (u) = for all r >. Show that for each x R d there i at mot one olution to the ODE ẋ = b(x)

2 with x() = x. (d) Formulate and prove a imilar pathwie uniquene theorem for the SDE Solution. f(t) a du a+ t k(u) k(f())d du k(u) = t a dx t = b(x t )dt + σ(x t )dw t. k ( a + v k(f(v))dv ubt u = a + k(f())d) ince k i increaing and hence k(f(v)) k ( a + v k(f())d). (b) By part (a) f(t) du k(u) t. v k(f())d Now, if f(t) > for ome t, then the left-hand ide i +, a contradiction. (c) Let x and x be olution of the ODE, with the ame initial point. Let Then T n = inf{t : max{ x(t), x (t) } > n}. x(t) x (t) b(x()) b(x ()) d k n ( x() x () )d. Letting f(t) = x(t) x (t) in the above reult, yield x(t) = x (t) for all t T n. Now end n. (d) Here i one: Suppoe that there are function k n uch that (x y) [b(x) b(y) + σ(x) σ(y) k n ( x y ) for all x, y R d, x n, y n. and that for all n, r du k n (u) = for all r >. Then the SDE ha the trong uniquene property. Proof: Let X and X be olution with the ame initial point. Compute f(t) = E ( X t Tn X t T n ) via Itô formula, and proceed a in part (c). Problem 3. Let b be bounded and meaurable. Ue Giranov theorem to contruct a weak olution to the SDE dx t = b(x t )dt + dw t over the finite (non-random) time interval [, T.

3 Solution 3. GIven X, let X X be a Brownian motion on (Ω, F, P) generating a filtration F. Define a new meaure Q on (Ω, F T ) by ( ) dq dp = E b(x)dx. T Finally, let W t = X t X b(x )d. By Giranov theorem, W i a Q-Brownian motion. Hence (Ω, F T, Q, F, W, X) i a weak olution of the SDE. Problem 4. By inpecting the proof of Novikov condition, how that if there exit an increaing equence t n uch that E(e ( M tn M t n ) ) < then E(M) i a true martingale. Hence, how that if b : R d R d i meaurable and atifie the linear growth condition for a contant C >, then the SDE b(x) C( + x ) for all x R d dx t = b(x t )dt + dw t ha a weak olution over any finite time interval [, T. You may want to ue thi form of Jenen inequality: e au du e (t )au du. t Solution 4. Let Z be a poitive local martingale with Z =, and uppoe that there exit an increaing equence t n uch that [ Ztn E = Z tn for all n, then Z i a true martingale. Indeed, by problem 3 and 4from example heet, we have that the proce (Z t /Z tn ) t [tn,t n i a martingale, and hence ( [ ) Ztn E(Z tn ) = E Z tn E F tn = E(Z tn ) Z tn and o E(Z tn ) = for all n by induction. Finally ince Z i a upermartingale E(Z t ) = for all t and hence Z i a true martingale. Letting Z = E(M), Novikov condition ay that if then and Z i a martingale. E(e ( M tn M t n ) ) < [ Ztn E = Z tn 3

4 Now, to build a weak olution, we firt let X X be a Brownian motion, and define W t = X t X b(x )d a in the previou quetion. We need to check that Z = E ( b(x)dx ) i a true martingale. Now letting ˆX = X X we have and hence b(x u ) C ( + X u ) (C + 4 X ) + 4C ˆX u [ E e du b(xu) e (C + X )(t ) E [e C t ˆX u du [ + X )(t ) e(c E t = e(c + X )(t ) t e(c + X )(t ) 4C (t )t e (t )C ˆX u du 4C (t )u du Letting t k = k C we ee that t k and [ E e k t b(x k u) du < by the above computation, o the generalied verion of Novikov criterion implie Z i a true martingale. Problem 5. Show that the SDE dx t = 3X /3 t dt + 3X /3 t dw t, X = ha trong exitence, but not trong uniquene. Solution 5. The proce { if t T X t = (W t W T ) 3 if t > T i a olution to the SDE for any topping time T. Problem 6. Let X be the Markov proce aociated with the calar SDE dx t = b(x t )dt + σ(x t )dw t. Let the C function u : [, ) R R atify the PDE u t = b(x) u x + σ(x) u x + g(x) with boundary condition u(, x) = for all x. Auming that g i bounded, and that u i bounded on any trip [, t R, then how that [ u(t, x) = E g(x )d X = x. 4

5 Solution 6. Fix a contant T >, and let v(, x) = u(t, x). Then by Itô formula ( ) ( d g(x u )du + v(, X ) = g(x ) + v t + b(x ) v x + σ(x ) ) v d + σ(x x ) v x dw. By the PDE, the d integral vanihe, and we are left with the local martingale dw integral. But ince g(x u)du + v(, X ) i bounded on [, T by aumption, and bounded local martingale are true martingale, we can deduce that [ T g(x u )du + v(, X ) = E g(x u )du + V (T, X T ) F. Since v(t, x) = and g(x u)du i F -meaurable, we have the calculation [ T v(, X ) = E g(x u )du F [ T = E g(x u )du X = E [ T The reult follow from the time-homogeneity of X. g(x u+ )du X Problem 7. Find the unique trong olution of the SDE dx t = X tdt + + X t dw t, X = x. [Hint: conider the change of variable Y t = inh X t. Solution 7. There i trong exitence and uniquene ince the function b(x) = x and σ(x) = + x are globally Lipchitz. Now, let X be thi unique olution and let Y t = inh X t. By Itô formula dy t = ( + Xt ) dx X t / t ( + Xt ) d X 3/ t = dw t, yielding Y t = W t + inh x and thu X t = inh ( W t + inh x ). Problem 8. Let W and B be independent Brownian motion, and let ) X t = e (x Wt + e W db Show that there exit a Brownian motion Z uch that dx t = X tdt + + X t dz t, X = x. Ue the previou problem to find the denity function of the random variable ew db. 5

6 Solution 8. Itô formula, uing B, W = by independence, yield ( ) dx t = db t + X t dt dw t where = X tdt + + X t dz t, Z t = db X dw + X. The proce Z i a continuou local martingale, and it quadratic variation i Z t = t o Z i a Brownian motion by Lévy characteriation theorem. By the previou problem P ( x + and hence the denity f of ew db i f(y) = Problem 9. Let X be a olution of the SDE ) e W db > = P(X t > ) = P(inh x + Z t > ) ( πt( + y ) exp (inh y) t dx t = X t g(x t )dw t where g i bounded and non-random X >. (a) Show that P(X t > for all t ) =. [Hint: Apply Itô formula to E( g(x)dw ) X. (b) Show that E(X t ) = X for all t. (c) Fix a non-random time horizon T >. Show that there exit an equivalent meaure ˆP on (Ω, F T ) and a ˆP-Brownian motion Ŵ uch that where Y t = /X t. dy t = Y t g(/y t )dŵt Solution 9. (a) Let Z = E( g(x)dw ). By Itô formula, dx/z = and hence X = X Z. Since X and Z are poitive, o i X. (b) Since g i bounded, Novikov criterion enure that Z i a true martingale. (c) Let dˆp = Z dp T. But Giranov, W g(x)d i a ˆP Brownian motion. By ymmetry, o i Ŵ = (W g(x)d). Now apply Itô formula. Problem. (quare-root diffuion) Let W be an n-dimenional Brownian motion, and define an n-dimenional proce X to be the olution to the SDE dx t = X t dt + dw t with X = x R n. If R t = X t, how that there exit a calar Brownian motion Z uch that dr t = (n R t )dt + R t dz t. 6 ).

7 Solution. Let f(x) = x o that f = x i and f = if i = j and otherwie. By x i x i x j Itô formula n n dr t = XtdX i t i + d X i t The concluion now follow by defining i= = (n R t )dt + Z t = i= n i= X dw X X i tdw i t and noting that Z i a Brownian motion by Lévy characteriation theorem, ince Z t = t. Problem. By finding the tationary olution of the Fokker Plank PDE, find a formula for the invariant denity of the calar SDE dx t = b(x t )dt + σ(x t )dw t, auming it exit. Apply your formula to the proce R in the previou quetion. Comment of your anwer in light of example heet 3, problem 9. Solution. We mut olve L q = (bq) + (σ q) = A candidate olution i q(x) = C σ(x) e x b(y) a σ(y) dy for ome contant a and C uch that q(x)dx =. In the previou problem b(x) = n x and σ(x) = x. Plugging thi into the formula yield q(x) = Γ(n/) xn/ e x, x. Let X N n (, I). By example heet 3, problem 9, an invariant meaure for thi multivariate Orntein Uhlenbeck proce ha the ame ditribution a X and hence an invariant meaure of thi quare-root proce ha the ditribution of in agreement with our formal calculation. X Gamma(n/, ), Problem. Conider the SDE dx t = Xt dw t. (a) Ue example heet 3, problem[ 5 to ( contruct ) a weak olution. (b) Verify that both u (t, x) = x Φ and u (t, x) = x olve the PDE x t u t = x4 u x, u(, x) = x (c) Which of thee olution correpond to u(t, x) = E(X t X = x)? 7

8 X Solution. (a) Let Z be a three dimenional Brownian motion, and let X t = where X Z t u u = (,, ). Then X olve the SDE by Itô formula with an Brownian motion contructed by Lévy theorem. (b) Thi i jut calculu. A ueful identity i Φ (x) = xφ (x). (c) Note that ince σ(x) = x i locally Lipchitz, the SDE ha the path-wie uniquene property, and hence the uniquene in law property by the Yamada Watanabe theorem. So we can calculate with the weak olution from part (a) by witching to pherical coordinate: E(X t X = x) = (π) 3/ xe z / z / z 3 / (x dz dz dz 3 tz ) + x tz + x tz3 = x(π) 3/ = x(π) / π r= θ= π r= = x(πt) / θ= π φ= re r / r= r in θe r / x tr x dφ dθ dr t co θ + r in θe r / x tr x dθ dr t co θ + π x tr x t co θ + = x(πt) / (r {r>(x t) / } + x tr {r (x t) })e / r / dr = x r= (x t) / = u (t, x). e r / π dr Problem 3. (a) Suppoe X i a weak olution of the SDE dx t = b(x t )dt + σ(x t )dw t. Show that the proce [ f(x t ) b(x )f (X ) + σ(x ) f (X ) d i a local martingale for all f C. (b) Let X be a calar, continuou, adapted proce uch that [ f(x t ) b(x )f (X ) + σ(x ) f (X ) d i a local martingale for each f C. Suppoe σ i continuou and σ(x) > for all x. Show that there exit a Brownian motion uch that dx t = b(x t )dt + σ(x t )dw t. [Hint: Conider example heet 3 problem 3. Solution 3. By Itô formula (a) [ f(x t ) b(x )f (X ) + σ(x ) f (X ) d = f(x ) + θ= dr σ(x )f (X )dw and the right-hand ide i a local martingale, a the tochatic integral with repect to a Brownian motion. 8

9 (b) Define M by M t = X t X b(x )d Thi i a local martingale by letting f(x) = x in the aumption. By Itô formula we have X t = X + X dm + X b(x )d + M t Now define N another local martingale by letting f(x) = x in the aumption: o that N t = X t X M t [X b(x ) + σ(x ) d, σ(x ) d = N t X dm. Since the left-hand ide i of finite variation, and the right-hand ide i a continuou local martingale, both ide are contant and equal to zero. In particular, So define a local martingale W by d M t = σ(x t ) dt. W t = dm σ(x ). By contruction W t = t, o W i a Brownian motion. Finally, by the tochatic chain rule, dx t = b(x t )dt + dm t = b(x t )dt + σ(x t )dw t. Problem 4. (Brownian bridge) Let W be a tandard Brownian motion. (a) Let B t = W t tw. Thi i called a Brownian bridge. Can you ee why? Show that (B t ) t [, i a continuou, mean-zero Gauian proce. What i the covariance c(, t) = E(B B t )? (b) I B adapted to the filtration generated by W? (c) Let Verify that X t = ( t) exit a Brownian motion Z uch that dx t = X t t dt + dw t, X =. dw for t <. Show X t a t. [Hint: how that there dw = Z t/( t) and apply the Brownian trong law of large number. (d) Show that X i a continuou, mean-zero, Gauian proce with the ame covariance a B, i.e. X i a Brownian bridge. Solution 4. (a) c(, t) = E[(W W )(W t tw ) = t t. (b) No. Let (F t ) t be the filtration generated by W. Then E[B t F t = E[( t)w t + t(w t W ) F t = ( t)w t B t where we have ued the independence of W t W and F t. (c) A hinted, let u +u dw Z u =. 9

10 There are a couple of way to how Z i a Brownian motion. One i to note that Z i a continuou Gauian proce with covariance u +u d E[Z u Z v = ( ) = u +u = u if u v. Now lim t X t = lim t ( t)z t/( t) = lim u + u Z u = by the Brownian trong law of large number. (d) E[X X t = E[( )Z /( ) ( t)z t/( t) = min{t( ), ( t)} = t t