Institute for Applied Mathematics SS2016 Massimiliano Gubinelli
Stochastic Analysis Course note 2
Version 1.1. Last revision 24/5/2016. Please send remarks and corrections to [email protected].
1 Girsanov transformation
Girsanov (Maruyama, Cameron and Martin, Ramer) transformation describes the relation between modifying semimartingales via a nite variation process and absolutely continuous changes of probability measures. In particular it describes the quasiinvariance of Wiener measure wrt. adapted translations with values in the CameronMartin space. We rst analyse general change of measures in presence of a ltration and a general form of Girsanov theorem. Then we prove some applications to SDEs and to the computation of conditional laws on path space. The main reference is [5, IV.38 and IV.39].
1.1 Change of measure on a ltered probability space
Let P; Q two probabilities on a ltered measure space ( ; ; ). We will assume that is F F F right continuous. If Q P on = t 0 t let F1 _ > F dQ Z : = 1 dP
F1 and note that for all t > 0, Q P on with Ft dQ EP[Z t] = =: Zt (1) 1jF dP t F
so the process (Zt)t>0 is a uniformly integrable martingale closed by Z . In what follows we will make the main assumption 1
(Zt)t>0 is a Pa.s. continuous martingale. (2)
Lemma 1.
i. If T is a stopping time then dQ = Z ; (3) dP T T F
ii. If Q P (i.e. Q P and P Q) th en P(Z > 0 for all t 0) = 1. t > iii. If Q P then for any t s and X ^ and X 0 we have (Bayes formula) > 2 Ft >
EP[X Zt s] EQ[X s] = jF Q and P a.s. (4) jF Zs
1 iv. If Q P then M is a Qmartingale MZ is a Pmartingale (5) M is a local Qmartingale () MZ is a local Pmartingale ()
Proof. (i) If A , by optional sampling : 2 FT
EP[IA] = EQ[Z IA] = EQ[EQ[Z T ]IA] = EQ[ZT IA]; 1 1jF since ZT is T measurable we proved the claim. (ii) Let T = inf t > 0: Zt = 0 , by continuity Z = 0 on thFe event A = T < . Then f g T f 1g 2 FT
Q(A) = EQ[IA] = EP[Z IA] = EP[ZT IA] = 0: 1
Since P Q we have P(A) = 0. (iii) For any t s 0 and A we have > > 2 Fs
EQ[ZtX s] EP[X IA] = EQ[Z X IA] = EQ[ZtX IA] = EQ[EQ[ZtX s] IA] = EP jF IA 1 jF Zs and the claim follows. Let us prove (iv). The rst equivalence is clear from Bayes formula (4). Let us consider the case of local martingales. Let T be a stopping time such that (M Z)T is a Q martingale. Then for all t s 0 and A we have > > 2 Fs
T T T EP[Mt IA] = EP[Mt IAIT 6s] + EP[Mt IAIT >s]:
T T Now EP[Mt IAIT 6s] = EP[Ms IAIT 6s] since here T 6 s 6 t. On the other hand IAIT >s is T s T t measurable, so by (3) we have F ^ F ^
T T EP[Mt IAIT >s] = EQ[ZT t MT tIAIT >s] = EQ[(Z M)s IAIT >s] ^ ^ since I I is measurable and (MZ)T a Qmartingale. But now A T >s Fs
T T EQ[(ZM)s IAIT >s] = EQ[ZsMsIAIT >s] = EP[MsIAIT >s] = EP[Ms IAIT >s] since M I I is also measurable. Then s A T >s Fs
T T T T EP[Mt IA] = EP[Ms IAIT 6s] + EP[Ms IAIT >s] = EP[Ms IA] which proves that M T is a Pmartingale. The reverse implication can be proven similarly. Taking a localizing sequence of stopping times (Tn)n allows to conclude.
Lemma 2. (Stochastic exponential, Doleans) Let (Zt)t>0 be a continuous local Pmar- tingale such that Zt > 0 for all t > 0 Palmost surely then there exists a unique continuous local Pmartingale (Lt)t>0 such that
[L] Z = exp L t =: (L) (6) t t ¡ 2 E t
2 and given by the formula t dZ L = log Z + s : (7) t 0 Z Z0 s
Proof. (Uniqueness) Assume L~ is another such martingale. Then the local martingale D = ~ ~ L L satis es D0 = 0 and 2Dt = [L]t [L]t. Which implies that Dt = 0 for all t > 0 since D is a ¡local martingale of bounded variat¡ion starting at 0. (Existence) Let L be de ned by (7), 2 applying Itô formula to t log (Zt) (possible since Zt > 0 for all t > 0 and z log(z) is C away from z = 0) we get the cla7!im, since 7!
dZt 1 [Z]t 1 dlog(Zt) = 2 = Lt [L]t Zt ¡ 2 Zt ¡ 2
2 given that dZt/Zt = dLt and d[L]t = d[Z]t/Zt from eq. (7).
Remark 3. Uniqueness of L can be also checked via Itô formula. Let Dt = exp( Lt + [L]t/2) where L is de ned by (7). Applying Itô formula to D Z one has ¡
Z D d(D Z) = Z dD + D dZ + d[D; Z] = Z D ( dL + d[L] /2) + t t d[L] + D dZ D d[L; Z] ; t t t t t t t t ¡ t t 2 t t t ¡ t t
1 using dL = dZ /Z and dD = DdL we get d(D Z) = 0 so Z = (Z D )D¡ = exp( L + [L] /2) ¡ t t 0 0 t ¡ t t since D0Z0 = 1.
Theorem 4. (Girsanov) Let P Q and let Z be martingale de ned in eq. ( 1), (continuous according our assumptions). Let L be the local martingale such that Z = (L). If M is a E local P-martingale then M~ = M [L; M] is a local Qmartingale. In particular if B is a ( ; P)Brownian motion then B~ =¡B [B; L] is a ( ; Q)Brownian motion. F ¡ F
Proof. By Itô formula d(M~Z) = ZdM~ + M~ dZ + d[M~ ; Z] = ZdM Zd[L; M] + M~ dZ + Zd[M ; L] = ZdM + M~ dZ (8) ¡ since [M~ ;Z]= [M ;Z] due to the fact that M~ M is of bounded variation and d[M ;Z]= Zd[M ; L] since dZ = ZdL by the de nition (6) of ¡(L). Then M~Z is a local Pmartingale given that the r.h.s. of (8) is the sum of two stochastiEc integrals wrt. the local Pmartingales M and Z.
By (5) we conclude that M~ is a local Q-martingale. We remark that [B~; B~]t = [B; B]t = t for all t > 0 so by Levy's theorem B~ is a ( ; Q)Brownian motion. F
Remark 5. By Girsanov theorem two equivalent probability measures P and Q agree in classi ng the same process X as a semimartingale. Indeed if X is a Psemimartingale with decomposition X = X0 + M + V then X is also a Qsemimartingale with decomposition X = X0 + M~ + V~ where V~ = V + [L; M]. On the other hand, note that, again by Girsanov, L~ = L + [L] is a local martingale under Q and ¡
dP ~ 1 ~ ~ ~ t=Zt = Zt¡ = exp( L + [L]/2) = exp(L [L]/2) = (L) dQjF ¡ ¡ E
3 is a Qmartingale. Then if N~ is a local Qmartingale, the process N: =N~ [N~ ; L~] = N~ + [N~ ; ¡ L] = N~ + [N ; L] is a Pmartingale. In particular :
M is a local Pmartingale i M [M ; L] is a local Qmartingale. ¡
Remark 6. (Finite horizon) We can replace an in nite time horizon with a nite one, t < + 1 and require that P and Q are equivalent on t and (Zt)t [0;t ] is a continuous Pmartingale F 2 for which there exists a local Pmartingale (Lt)t [0;t ] such that Z = (L), etc... 2 E
Example 7. (Brownian motion with constant drift) Let X be a ddimensional PBrow- nian motion and Rd a xed vector. Consider the martingale 2 1 Z := exp :X 2t = ( :X) ; t 0 t t ¡ 2j j E t >
and for any t 0 the measure Pt de ned on ( ; t; ( s) s t) by dPt = Zt dP t. By Girsanov's > 6 6 F theorem we have that, the process F F j
X~ = X [L; X] = X [ :X; X] = X s; s [0; t] s s ¡ s s ¡ s s ¡ 2
is a ddimensional Pt Brownian motion, so under Pt the process X is a Brownian motion with drift . Note that the family of measures (( ; t; Pt ))t 0 is consistent: Pt s=Ps for all F > jF 0 6 s 6 t. By Kolmogorov extension theorem there exists a unique measure P on ( ; ) such F1 that P t=Pt for all t > 0. Now, the measurable event jF F1
(Xs s) lim ¡ = ; s + s ! 1 has P probability 1 (e.g. for the law of iterated logarithm applied to the PBM X) while it has P probability 0 since s X s is a P Brownian motion. 7! s ¡
1.2 Doob's htransform
Fix a nite time horizon I = [0; t ] and consider a measure P on ( ; ; ( t)t I) with = t . d F F 2 F F Let (Xt)t I an R valued Itô process with drift b, diusion coe cient and generator = 2 L b: + 1 T : 2. We denote by B the mdimensional Brownian motion driving X (d; m are r 2 r arbitrary integers). A very simple and natural source of equivalent change of measures for P 1;2 d is obtained from functions h C (I R ; R>0) such that the process (Zt := h(t; Xt))t I is a 2 2 martingale. In this case we can de ne Q by dQ= ZdP. By Itô formula, the function Z satis es
dZ = T h(t; X ):dB + (@ + )h(t; X )dt t t r t t t L t but by the assumed martingale property the bounded variation part in the semimartingale decomposition of Z must vanish, so
(@ + )h(t; X ) = 0 t L t
4 for almost every t I, P-a.s. In order to have a properly normalized probability measure we need 2 that Z =h(0;X )=1. Now, dL =dZ /Z =(T h(t;X ))/h(t;X )dB =T log h(t;X )dB so 0 0 t t t t r t t t t r t t t 1 t Z = exp T log h(s; X )dB T log h(s; X ) 2ds t s r s s ¡ 2 j s r s j Z0 Z0 and in particular, by Girsanov theorem, under Q the (Rmvalued) process B is a semimartingale given by
B = B~ + [B; L] = B~ + T log h(t; X )dt t t t t t r t where B~ is an Rmvalued PBrownian motion (on the interval I). As a consequence the Itô process X has decomposition
dX = b dt + dB = (b + T log h(t; X )) dt + dB~ = b~ dt + dB~: t t t t t t r t t t t This shows that under P the process X remains an Itô process with the same diusion coe cient but a dierent drift b~ given by
b~ = b + T log h(t; X ); t I: t t t t r t 2 This construction is called Doob's h-transform and was originally introduced in the context of Markov processes. In the case of Itô diusions (i.e. where the drift and diusion coe cients bt; t are deterministic functions of Xt) the htransform gives a transformation of the associated martingale problems. Indeed from the previous discussion we see that if (X; P) is a solution T of the martingale problem MP(x0; b; ) then (X; Q) is a solution of the martingale problem T MP(x0; b~; ).
Exercise 1. Prove that, more generally, the h-transform gives a transformation of martingale problems from MP(x0; b; a) to MP(x0; b~; a). (That is, reproduce the above argument without relying on the Ito decomposition of X)
The Brownian motion with drift introduced in Example 7 is a particular case of this transform where the function h is given by
h(t; x) = exp( :x t/2); t 0; x Rd: ¡ j j > 2 Very interesting cases of htransforms arise if we allow the interval I = [0; t ) (or [0; + )) to be open at the right endpoint, i.e. if we do not require the measure Q to be equivalen1t to P on t but given by dQ t=Zt dQ t for all t I. In this case we ask that (Zt)t I be striclty F jF jF 2 2 positive continuous martingale in I but we do not pose any restriction on Zt . 1.3 Diusion bridges Using Doob's htransform we can describe very eectively the behavior of a Markovian Itô d d diusion X with values in R conditioned to reach some state x1 R at a given time (which we take to be 1). The interesting fact is that this kind of conditioni2ng is singular since usually the event is of probability 0 wrt. the law of the unconditioned diusion. Let I = [0; 1) and assume that the Itô diusion X is a Markov process with transition density
P(X dy X = x) = p(s; x; t; y)dy: t 2 j s
5 In this case we can de ne the function
p(s; x; 1; x1) gx1(s; x) := ; s I; x Rd p(0; x0; 1; x1) 2 2
1;2 d x1 x1 and assume that g is C (I R ; R>0). Let Zt = g (t; Xt) for t I and observe that, by x1 2 construction, Zt is a martingale since
p(t; y; 1; x )p(s; X ; t; y) p(s; X ; 1; x ) E[Zx1 ] = E[Zx1 X ] = 1 s dy = s 1 = Zx1; 0 s t < 1; t jFs t j s p(0; x ; 1; x ) p(0; x ; 1; x ) s 6 6 Z 0 1 0 1 by the Markov property and the ChapmanKolmogorov equation. De ne the measure Px1 x1 x1 on 1 as the extension of the family (Pt )t I where dPt = Zt dP t and observe that for all F 2 jF 0 < t < < t < 1 and bounded test function f we have 1 n
f(X ; :::; X ; x )p(t ; X ; 1; x )dx = E [f(X ; :::; X ; X ) ] t1 tn 1 n tn 1 1 P t1 tn 1 jFtn Z by Markov property. Then taking expectation wrt P we obtain
x EP[f(Xt1; :::; Xtn; X1)] = EP 1[f(Xt1; :::; Xtn; x1)]p(0; x0; 1; x1)dx1 Z
d which shows that letting (y) := E y[f(X ; :::; X ; x )] for y R , we have f P t1 tn 1 2
EP[f(Xt1; :::; Xtn; X1)r(X1)] = EP[f(X1)r(X1)]
for any bounded measurable function r. In particular f(X1) = EP[f(Xt1; :::; Xtn; X1) X1] and x1 j f is a regular conditional probability for P given (X1). We conclude that P is obtained by htransforming P via the function g, in particular that under the measure Px1 the Ito diusion X has a new drift bx1 given by :
bx1(t; X ) = b(t; X ) + (T )(t; X ) log p(t; X ; 1; x ); t I: t t t r t 1 2
Usually the additional logarithmic derivative of the transition density become singular as t 1. ! This accounts for the fact that the diusion must satisfy Xt x1 when t 1 and the martingale part of the diusion has to be counterbalances by a the stro!ng drift in o!rder for this to happen with probability one.
Exercise 2. Compute the drift in the case of X being a Brownian motion in Rd.
1.4 Diusions conditioned to stay in a given domain
We want to give an idea of the method used by Pinsky [3] to study a Markovian diusion process X on Rd conditioned to stay in a given subdomain D Rd. For more details please refer to the original paper.
6 Let D Rd an open bounded and connected set and let = inf t 0: X D . Let (X; P ) D f > t 2 g x be a Markov process satisfying the SDE
dXt = b(Xt)dt + (Xt)dBt; X0 = x: where B in a Brownian motion in Rm. Let = b: + 1 T : 2 the generator of X. Fix T > 0 L r 2 r and de ne the measure QT ( ) = P ( > T ) for the process X conditioned to stay in D up to x x j D time T . We assume that the function
gT (x) = P ( > T ) x D x D 2 is in C2(D) and that
T g (x) '0(x) lim r T = r T g (x) '0(x) !1 uniformly on compact subsets of D. Here ' is the eigenfunction of with Dirichlet boundary 0 L conditions corresponding to the real simple eigenvalue 0. We will assume that '0 > 0 on D T T T t T and that '0 = 0 on @D. Let M be the process Mt := g ¡ (Xt ) / g (X0), by the Markov ^ D property of X we have
T T Ex[H I D>T ] Ex[H g (Xt)I D>t] Ex[H g (Xt D)] T E T [H] = = = ^ = Ex[H Mt ] Qx gT (x) gT (x) gT (x) for any t < T and any r.v. H bounded and measurable. Moreover Ft
T t T s T E[Mt ¡ s] = E[PXt( D > T t)I D>t s] = PXs ( D > T s) = g ¡ (Xs D) = Ms ; jF ¡ jF ^ D ¡ ^ t > s > 0:
T T T Note that under Qx we have Qx ( D 6 t) = 0 so that under Qx the process X takes values in T D. By Doob's h-transform this implies that, under Qx the process X is an Itô diusion in D with diusion matrix and drift jD
gT (x) b~T (x) = b(x) + r ; x D: gT (x) 2
Under some technical assumptions on the behaviour of gT as T it is possible to show ! 1 that on any bounded interval [0; S] the process (X; QT ) converges in law, as T , to a weak x ! 1 solution (X; Qx) of the SDE with diusion matrix and drift
'0(x) b~(x) = lim b~T (x) = b(x) + (T )(x)r ; x D: T '0(x) 2 !1
The corresponding law (X; Qx) can be understood as the law of original process conditioned never to leave D.
7 Example 8. (Brownian motion conditioned to stay positive) Let X be a Brownian motion on R starting at x0 (0; L) for some L > 0. Let Q be the measure obtained by conditioning the Brownian motion2to hit L before 0:
IAdP dQ = jF1 ; jF1 P(A) where A = T = + or XT = L and T = inf t > 0: Xt (0; L) . Note that P(T = + ) = 0. f 1 g T f 2 g dQ 1 Letting h(x) = (x/x0) the process Zt = h(Xt ) is a martingale starting at 1 and =Zt for all dP jFt t > 0. By Girsanov theorem under Q the process X satis es the SDE (on a random interval, until Xt = L for the rst time) dt dXt = dX~t + ; t [0; T ] Xt 2 starting at x0, where X~ is a Q-Brownian motion on [0; T ]. In particular this solution to the SDE never touches 0. Moreover as L we have T Pp.s.. This suggests to de ne Q by dQ T0 ! 1 ! 1 T0 t=Xt /x0 > 0 for all t > 0 where T0 = inf t > 0: Xt < 0 . Now, X is a positive martingale dP jF f g (why?) so Q is well de ned, but singular wrt. P on since (by recurrence of the BM) P(T < ) = 1 while Q(T < ) = 0 by de nition. In thisFc1ase the process X satis es the SDE 0 1 0 1 ~ dt dXt = dXt + ; t > 0; Xt ~ where now X is a Q-Brownian motion on R>0.
Example 9. (Brownian motion conditioned to stay in D =(0;)) Let X be a Brownian motion under P. In this case the rst Dirichlet eigenfunction of the Laplacian in D is '0(x) = sin(x) t and = 1 the corresponding eigenvalue. Let h(t; x) = e '0(x). Then Zt = h(t; Xt)/h(0; x0) is a Pmartingale (by Itô formula) and we can de ne the measure Q with density Zt wrt P on t. Under Q the process X satis es the SDE F
cos(Xt) dXt = dBt + dt: sin(Xt)
Example 10. (Brownian motion in a Weyl chamber). Let X =(X1;:::;Xn) be a ndimensional Brownian motion starting at x S where S Rn is the Weyl chamber, that is the set 0 2 S = x = (x1; :::; xn) Rn: x1 < x2 < < xn : f 2 g Consider the function h(x) = (xj xi)/ (xj xi ) and let T = inf t 0: h(X ) = 0 . i
1 dXi = dt + dBi; i = 1; :::; n; t 0: t Xi X j t > j=/ i X ¡ i In particular this shows that under Q the processes (X )i never intersects and preserve their linear order. This process is called Dyson's Brownian motion [1] and coincide with the process describing the evolution of eigenvalues of a natural continuous diusion in the space of sym- metric n n matrices.
8 1.5 Exponential tilting
As seen in the previous examples and in many applications we are required to study the measures PL on ( ; ) obtained from P via exponential tilting with a given local continuous Pmartingale L (staFr1ted at 0):
L dP t= (L)t dP t; t > 0: (9) jF E jF In order for this construction to make sense and PL be well de ned, we need ( (L) ) to be a E t t>0 (true) martingale and not just a local one (otherwise we cannot even guarantee that PL( )=1). L Moreover the fact that the family of measures dPt = (L)t dP t on ( ; t) indenti es a unique L L L E jF F measure P on ( ; ) via P t=Pt is due to KolmogorovDaniell extension theorem: indeed F1 L jF L the family is consistent Pt =Ps i (L) is a martingale. jFs E In a nite horizon I = [0; t ] we need to ensure that E[ (L)t ] = 1: E
Lemma 11. A positive local martingale M such that E[M ] < is a supermartingale and it 0 1 converges a:s: as t to a limit M > 0 such that E[M ] 6 E[M0]. If E[M ] = E[M0] then M is a UI marting!ale1. 1 1 1
Proof. Let (T ) a reducing family for (M M ) , then for all 0 s t n n>0 t ¡ 0 t>0 6 6
E[M Tn M ] = E[M Tn ] M = M Tn M : t ¡ 0jFs t jFs ¡ 0 s ¡ 0
By the conditional form of Fatou's lemma (and the integrability of M0)
Tn Tn E[Mt s] = E liminf Mt s 6 liminf E[Mt s] = Ms: jF n F n jF h i
We will assume (without proof) that a positi ve continuous supermartingale converges a.s. The next exercise completes the proof.
Exercise 3. Let M be a positive continuous supermartingale such that E[M ] < . Let 0 1 M = limt Mt (assumed to exist Pa.s.). Show that if E[M ] = E[M0] then M is a UI 1 !1 1 martingale. [Hint: prove that E[M t] 6 Mt and that E[Mt] = E[M0] and conclude.] 1jF The next results describes su cient conditions under which we can ensure the crucial property E[ (L) ] = 1 when L0 = 0. As just shown in Lemma 11 this implies that ( (L)t)t 0 is a 1 > UIEmartingale. Condition (11) below is called Novikov's condition. Condition (E10) is due to Krylov [2] from which the proof below is taken.
Theorem 12. (Novikov, Krylov) Let L be a local martingale starting at 0 and assume that
lim " log E[exp((1 ")[L] /2)] = 0 (10) " 0 ¡ 1 # then E[ (L) ] = 1. In particular this holds if E 1 E[exp([L] /2)] < : (11) 1 1
9 Proof. We will start by proving the following weaker statement:
p0 > 1: E[exp(p0[L] /2)] < + = E[ (L) ] = 1: 9 1 1 ) E 1
Let (Tn)n>1 be a family of stopping times reducing L and such that Tn 6 n for all n > 1. Then for any r > 1 we have
E[( (LTn) )r] = E[exp(rLTn r [LTn] /2)] = E[exp(r LTn r2 p[LTn] /2)exp((r2p r)[LTn] / E 1 1 ¡ 1 1 ¡ 1 ¡ 1 2)]
1 1 by Hölder inequality (with p; q > 1 and p¡ + q¡ = 1)
1/p 1/q E[( (LTn) )r] 6 (E[ (rpLTn) ]) (E[exp( (r; p)[LTn] /2)]) ; E 1 E 1 1 where r2p r rp 1 (r; p) := q(r2p r) = ¡ = p r ¡ > 1: ¡ 1 p 1 p 1 ¡ ¡ ¡
When r 1 + , (r; p) p + so we can choose 1 < p < p0 and r = 1 + " for su ciently small ! ! Tn " > 0 to ensure that (r; p) 6 p0. With this choice, using the fact that (rpL ) is a martingale, Tn E that [L ] [L]n [L] and the hypothesis of (11), we get 1 6 6 1
Tn r 1/q 1/q E[( (L ) ) ] 6 (E[exp( (r; p)[L] /2)]) 6 (E[exp(p0[L] /2)]) < + : E 1 1 1 1
Tn r Tn Since r > 1, the fact that supnE[( (L ) ) ] < + implies that the family ( (L ) )n is 1 1 E 1 E Tn uniformly integrable for any t 0. From this we conclude that E( (L) ) = E[limn (L ) ] = > 1 1 Tn E E limnE[ (L ) ] = 1 so (11) is proven. E 1 2 Let us go back to the proof of the Lemma. Take " (0;1) and observe that letting p0 =(1+") >1 we have 2
2 E[exp(p0[(1 ")L] /2)] = E[exp((1 " )[L] /2)] < : ¡ 1 ¡ 1 1 which by (11) implies that E[ ((1 ")L) ] = 1. By Hölder inequality, E ¡ 1 1 = E[ ((1 ")L) ] = E[exp((1 ")(L [L] /2))exp((1 ")"[L] /2)] E ¡ 1 ¡ 1 ¡ 1 ¡ 1
(1 ") " 6(E[ (L) ]) ¡ (E[exp((1 ")[L] /2)]) : E 1 ¡ 1
Letting " 0 we obtain 16E[ (L) ] due to (10). This concludes our argument since we already ! E 1 know that E[ (L) ] 6 1 (Fatou). E 1
Remark 13. The conditions (10) and (11) are not necessary for (L) to be a martingale. Another well known su cient condition is Kazamaki condition E
E[exp(L /2)] < + ; (12) 1 1 which is weaker than Novikov's. For more details refer to Revuz and Yor [4].
10 1.6 Tilting via a Brownian local martingale
Let ( ; = ; ; P) carry a ddimensional Brownian motion B and let b an adapted Rd- valued pFrocesFs1forFw hich
t b 2ds < + ; P a:s: t 0: j sj 1 ¡ 8 > Z0 Let Lb be the scalar continous local martingale
t b Lt := bs:dBs; t > 0; Z0 assume that (Lb) is a martingale and de ne Pb := PL as in eq. (9). The process B satis es the equation E
dBt = bt dt + dWt; t > 0; where W is a PbBrownian motion.
Example 14. (Solutions to SDEs) Exponential tilting via the martingale Lb is useful to construct (weak) solutions to SDEs with a general class of drift coe cients. It will be convenient d to assume that = C(R>0; R ), that P is the ddimensional Wiener measure and that X is the canonical process on and that the ltration is generated by X. We consider a predictable F Rd-valued process b^ given by a function b: R Rd via b^ (!) = b (X(!)) and we denote it >0 ! t t b^=:b(X). By tilting P via Lb(X) we obtain that, under Pb the process X is a solution of the SDE
dXt = bt(X) + dWt; t > 0; provided (Lb(X)) is a martingale. A natural condition on b which guaratees the martingale property iEs
bt(x) 6 Ct(1 + x ;[0;t]); t > 0; x (13) j j k k1 2 where C < + for all t 0 (see next exercise). t 1 >
Exercise 4.
a) Prove that if
b (x) C (1 + x ); t 0; x ; j t j 6 t j tj > 2
then Novikov's condition holds for all t > 0, i.e.
t E exp b (X) 2ds < + ; t 0: j s j 1 > Z0 [Hint: show that the condition holds for t small enough and then use the Markov property to extend to all t]
11 b) Prove that
r2/2t P( X > r) Ce¡ t 0; r 0: k k[0;t] 6 > >
i [Hint: use Doob's inequality for the submartingale eXt and optimize over > 0]
c) Prove the same result as in (a) under the more general assumption (13). [Hint: for small time use (b) to estimate the size of the maximum of the Brownian motion, extend to all times via the appropriate Markov process]
More generally, solutions to SDEs with coe cients b; (in general path dependent) are trans- formed to solutions:
Corollary 15. (Drift transformation of SDEs) If (X; B; P) is a weak solution to
dXt = bt(X)dt + t(X)dBt and C = c (X) is such that (LC) is a martingale then (X; B~ ; PC) is a weak solution to t t E
dXt = b~t(X)dt + t(X)dB~t where the new drift is given by b~= b + c and dB~ = dB c (X)dt. t t ¡ t
Exercise 5. Generalise Girsanov transformation to a martingale problem MP(x0; b; a).
Exercise 6. Use Girsanov transform to prove the uniqueness in law of the weak solution of the SDE
dXt = bt(X)dt + dBt where b: R C(R ; Rd) Rd is a bounded, previsible drift. >0 >0 !
Bibliography
[1] Freeman J. Dyson. A BrownianMotion Model for the Eigenvalues of a Random Matrix. Journal of Math- ematical Physics, 3(6):11911198, nov 1962. [2] Nicolai Krylov. A simple proof of a result of A. Novikov. ArXiv:math/0207013 , jul 2002. ArXiv: math/0207013. [3] Ross G. Pinsky. On the Convergence of Diusion Processes Conditioned to Remain in a Bounded Region for Large Time to Limiting Positive Recurrent Diusion Processes. The Annals of Probability, 13(2):363378, may 1985. [4] Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer, 3rd edition, dec 2004. [5] L. C. G. Rogers and David Williams. Diusions, Markov Processes and Martingales: Volume 2, Itô Cal- culus. Cambridge University Press, Cambridge, U.K. ; New York, 2 edition edition, sep 2000.
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