Stochastic differential equations with singular drift

Statistics & Probability North-Holland

August

Letters 10 (1990) 225-229

STOCHASTIC

DIFFERENTIAL

EQUATIONS

WITH SINGULAR

1990

DRIFT

Marek RUTKOWSKI of Mathematics,

Institute Received

August

Technical

University of Warsaw, 00-661 Warszawa,

Poland

1989

Abstract: We study the pathwise uniqueness of solutions of one-dimensional stochastic differential equations involving local times, under the assumption that the diffusion coefficient satisfies the (LT) condition introduced by Barlow and Perkins (1984). We show that this condition is sufficient for the pathwise uniqueness in the case of SDE’s involving local times studied until now. In the final section a more general class of equations is introduced. Keywords:

Stochastic

differential

equations,

local times, pathwise

1. Introduction We intend to study the pathwise uniqueness of solutions of stochastic diferential equations (SDE’s) of the form X,=X,+

/0

‘n(s,

X,) dW,+/L;(X)u(do) w (1.1)

where W denotes the one-dimensional Wiener process, u stands for a signed Radon measure on Iw and u : Iw + X Iw + Iw is a Bore1 measurable function. Finally for a continuous semimartingale X we denote by Lp( X) its symmetric local time at a. The SDE’s of the form (1.1) were studied by several authors (see, e.g., Le Gall, 1984; Engelbert and Schmidt, 1985; Rutkowski, 1987) and different sufficient conditions for the pathwise uniqueness of solutions were provided. In the first part of this note a rather general result concerning the pathwise uniqueness of solutions of the equation (1.1) is established. In the final section a more general class of SDE’s involving local times is introduced and briefly studied.

2. Pathwise uniqueness of solutions We adopt the currently standard notation of the stochastic calculus based on the It6 integral with 0167-7152/90/$3.50

0 1990 - Elsevier Science Publishers

uniqueness.

respect to continuous semimartingales (see, e.g., Rogers and Williams, 1987). Let IJ : R +X R + R be a Bore1 measurable function. Let us recall the following definition due to Barlow and Perkins (1984). Definition 2.1. We say that u verifies the (LT) condition if, whenever X’, i = 1, 2, are processes satisfying dXi=a(t,

X:) dw+ddA:,

i=l,2,

where A’, i = 1, 2, are continuous processes of finite variation, then LF+( X’ - X2) = 0 for t E II3+’ Note that here and in what follows we denote by L:‘(X) (L:-(X) respectively) the right continuous version (the left continuous version respectively) of the local time of X. The main result of this section is the following: Theorem 2.2. Let u : Iw +X Iw 3 R be a Bore1 measurable function verifying the (LT) condition. Suppose that v is a signed Radon measure on 08 such that the measure U = IN,‘v, where N,” denotes the complement of the set N,=

n {x~lw: SEW,

B.V. (North-Holland)

a(s,

x)=0},

(2.1)

225

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Volume 10, Number 3

8t PROBABILITY

verifies IW~Hl
uniqueness

of

We precede the proof of Theorem 2.2 by the useful lemma which is easy to establish (cf. Lemma 4.2 in Rutkowski, 1989).

August 1990

LETTERS

Let X’ and X2 be arbitrary solutions of equation (2.2). By virtue of the (LT) condition haveL~f(X’-X2)=OfortEtR+.Takinginview the inequalities (2.4) one easily verifies Ouknine, 1988) that also Lp+(Y’ - Y2) = 0 t E R +. Since Y’ and Y2 are both solutions of SDE without drift (2.3) an application of Tanaka formula yields immediately Y’ = Y2 t E R +. Clearly this implies that X: = X: for R + and this ends the proof. 0

the we (cf. for the the for tE

Lemma 2.3. Suppose that u verifies the (LT) condition and X is a continuous semimartingale

with the

canonical decomposition Xr=XO+

J0

‘o(s,

X3) dW,+A,.

Then for each a E N, we have Lp+(X)=L~-(X)=0 and thus also L:(X)

fortElW+, = 0 for t E IF2+.

0

Proof of Theorem 2.2. By virtue of Lemma 2.3 the equation (1.1) is equivalent (we say that two equations are equivalent if any solution of one of them verifies also the second one) to the equation of the form X,=X,+

J0

‘e(s,

X,) dW,+jLf(X)B(da). R (2.2)

By the usual localization procedure we may and do assume that 1v”(R) / -c + 00. It is well known (see Le Gall, 1984; Rutkowski, 1987) that one may attach to i? the strictly increasing function F:IW-+R suchthat: (a) if X is a solution of the equation (2.2) then the process Y = F( X,) verifies the SDE without drift term r,=

Yo+

/0

‘a(.~, y) dW,,

(2.3)

where a”(s, y) = a(s, F-‘( y))F’(F-‘( y)) (F’ stands for the symmetric derivative of F); (b) for a sufficiently small E > 0, there exist positive constants c and C such that c(~,,

x2)

G F(x,)

if 0 < xi - x2 < e. 226

-F(+)

d C(X,

-x2)

(2.4)

Remarks. Several sufficient conditions which en-. sure the (LT) condition are known, perhaps the more general are the hypotheses (H,) and (Hb) given in Barlow and Perkins (1984). Observe however that if the function u = a,~,,, where the function a, (ut, respectively) verifies the hypothesis H, (Hb respectively), the problem whether CJ verifies the (LT) condition rests, to our knowledge, still open. Therefore Theorem 2.2 does not cover the pathwise uniqueness result established previously by the author (see Theorem 4.2 in Rutkowski, 1987). On the other hand it is well known that in the case of SDE’s with ‘ordinary’ drift dX,=u(t,

Xt) dw+b(t,

X,) dt

the (LT) condition imposed on the diffusion coefficient u is not, in general, sufficient for the pathwise uniqueness of solutions (even if the drift coefficient b is continuous). This suggests that the class of SDE’s involving local times considered until now (and known also under the name of SDE’s with generalized drift) is not sufficiently large to include all interesting cases. For this reason we intend to introduce in the next section a larger class of SDE’s with local times.

3. SDE’s with singular drift Let u : [w + R be a Bore1 measurable function. We shall assume throughout that u satisfies the (LT) condition. This assumption is not necessary in general, but it permits to simplify the exposition. Suppose that p is a signed Radon measure on Iw. Let us introduce the following class of Bore1 measurable functions:

STATISTICS

Volume 10, Number 3

D;=

{f:lw

+R:

f=Oon

N, and u-‘f

8c PROBABILITY

and

is

integrable with respect to p} (by convention u-‘f

August 1990

LETTERS

u,(f)=~IxI~‘f(x)

= 0 on the set N,).

Definition 3.1. For a signed Radon measure p we denote by ~1~the application p0 : 0; -+ Iw defined by the formula

dx

forfED;.

Thus the equation (3.1) is in this case equivalent to the equation x, = x, +

/0

‘lXs11/2 dW,+ jL;(X)/uI-*

da.

R

(3.3) We shall find further a much simpler form of this equation (see Example 3.5).

We shall study the SDE’s of the form 4 = X, + J c’e(x,)

dK + cL&;(X)),

(34

t E R +. A process X is said to be a solution of the equation (3.1) if it is a continuous semimartingale and the formula (3.1) gives its canonical decomposition. Observe that due to the (LT) condition imposed on u, the equation (3.1) is equivalent to X,=X,+@X,)

dW,+iL:(X)C2(o)p(da).

The next proposition shows that if the measure p is absolutely continuous with respect to the Lebesgue, then the equation (3.1) may be reduced to the SDE with ‘ordinary’ drift. Proposition 3.4. Suppose that the measure u is absolutely continuous with respect to the Lebesgue measure. Then the equation (3.1) is equivalent to the equation X, = X0 + j’u( X,) dW, + j’b( X,&(

(3.2) Actually if X is a solution of (3.1) then it verifies also (3.2) by virtue of Definition 3.1. On the other hand in order to show that any solution of (3.2) verifies (3.1) it suffices to verify that L;(X) belongs to 0,” for each t E R +, and this easily follows by Lemma 2.3. Remark 3.2. Suppose that u is a Radon measure on Iw such that the function a2 is locally integrable with respect to u. One may verify that for the Radon measure p = u2u the application p, coincides with the restriction of u” to the set 0,“. Moreover in this case the SDE (1.2) is equivalent to the equation (3.2) with p = u2u. Let us consider a simple example ideas.

to fix the

Example 3.3. Suppose that p is the Lebesgue measure and a(x) = 1x 1‘12, x E R. The clearly D,“={f:IW+IW:

f(O)=Oandx-if(x)

is Lebesgue integrable},

0

X,) ds,

0

(3.4) if the Radon-Nikodym spect to the Lebesgue tive version.

derivative b of p with remeasure admits a non-nega-

Proof. It suffices to observe that for a continuous semimartingale X verifying (X),=L’u’(X,)

ds

for tER+,

the density of occupation time formula applied to the Bore1 measurable non-negative function h = um2bIni yields

JwL;(X)uP2(a)b(a) =

da

L;(X)u-2(a)b(a)I,,,~(a)

J

= ;e-1(X.)b(Y)k(Xl

d(X)

0

=

1 0

fb(X,h~(X,)

da

ds,

where the first equality follows by Lemma 2.3.

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Volume 10, Number 3

STATISTICS

8z PROBABILITY

Remark. Observe that the assertion of the above proposition remains valid under the assumption that the Radon-Nikodym derivative b admits a version such that the function 6 = bINz is nonnegative (or locally bounded). Example 3.5. By virtue of Proposition equation (3.3) is equivalent to x, = x, +

J 0 tlx,l

“’

dW,

+

@a\&s)

3.4 the

ds.

August 1990

LETTERS

seems to us that the problem of existence solution of such equation was not studied now (cf., e.g., Yan, 1988, where the case of degenerate diffusion coefficient and a singular term is considered).

of a until nondrift

We end this note by a slightly more general example. Suppose that a(x) = I x ( A where X > :, and p(da) = I a I *‘+’ da for any fixed c > -2X - 1. Then by virtue of Proposition 3.4 the equation (3.1) admits the equivalent form

(3.5)

Let us consider the Clearly the process X of the above equation. (see Rutkowski, 1989) mits also a non-trivial c(al(Xs)

ds=O

initial condition X0 = 0. = 0 is a particular solution However it is well known that the equation (3.5) adsolution satisfying

X,=X,+

10

‘]XJ1/*

of the SDE with con-

JorlxI

“* dW, +

L;(X) JR

I a I = da,

which belongs to our class of equations c > - 2, has by virtue of Proposition equivalent form real

t]X,]’

dW,

ds. /otlx, I 2h+CIR\~o~(&)

Corollary SDE

3.6. Suppose

x, = x0 +

Jbl~I’dW’+~LP(X)l~lCd~

dW,+t

verifies the considered equation (3.5). We conclude that in the case of SDE’s of the form (3.1) the (LT) condition (clearly verified by a(x) = (x I ‘/*) is not sufficient for the pathwise uniqueness of solutions. Observe also that even if the Radon measure p is absolutely continuous, the SDE (3.1) does not necessarily reduce to the equation with regular coefficients. Actually the equation of the form x,=X,+

+

J0

(3.6)

By applying the standard results concerning the SDE’s with classical drift (see, for instance, Example 5.2 in Rutkowski, 1987) one gets the following:

for tER+,

more precisely any solution tinuous coefficients

Xt=X,,+

for any 3.4 the

(i)

possesses

that

a unique

X E [i, 1). Then

the

strong solution for c E

C-1, +m); (ii) admits a solution, but the puthwise uniqueness of solutions does not hold, if c E [ - 2h, - 11. 0 Observe that for c E (-2X - 1, - 2A) the problem of existence and uniqueness of solutions of the equation (3.6) remains, to our knowledge, open. Still a more interesting open problem occurs if one wishes to study the equation (3.2) under the assumption that the Radon measure ~1 is not absolutely continuous with respect to the Lebesgue with measure and a-* is not locally integrable respect to p.

x, = X0 + ot I X, I “* dW, J +

/0

‘1 X, ]‘+‘&,;(XJ

ds.

in particular for c = - 1 we get the SDE with the drift coefficient 6(x) = I x I -1’21R,coj. It Thus

228

References Barlow, M.T. and E. Perkins (1984), One-dimensional stochastic differential equations involving a singular increasing process, Stochastics 12, 229-249.

Volume 10, Number 3

STATISTICS

& PROBABILITY

Engelbert, H.J. and W. Schmidt (1985) On one-dimensional stochastic differential equations with generalized drift, Z_ecture Notes in Control and Information Sci. 69, 143-155. Le Gall, J.F. (1984), One-dimensional stochastic differential equations involving the local times of the unknown process, Lecture Notes in Math. 1095,51-82. O&nine, Y. (1988), GCnCralisation dun lemme de S. Nakao et applications, Stochastic 23, 149-153.

LE’ITERS

August 1990

Rogers, L.C.G. and D. Williams (1987), Diffusion, Markov Processes, and Martingales, Vol. II (Wiley, Chichester). Rutkowski, M. (1987), Strong solutions of stochastic differential equations involving local times, Stochastics 22,201-218. Rutkowski, M. (1989), Fundamental solutions of stochastic differential equations with drift. Stochastics 26, 193-204. Yan, J. (1988) On the existence of diffusions with singular drift coefficient, Acta Math. Appl. Sinica 4, 23-29.

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