On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$
UDC 519.21 We prove the solvability of Itô stochastic equations with uniformly nondegenerate bounded measurable diffusion and drift in $L_{d+1}(R^{d+1}).$Actually, the powers of summability of the drift in $x$ and $t$ could be different. Our results seem to be new even if the diffusion is constant....
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| author | Krylov, N. V. Krylov, N. V. Krylov, N. V. |
| author_facet | Krylov, N. V. Krylov, N. V. Krylov, N. V. |
| author_sort | Krylov, N. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2022-03-26T11:02:07Z |
| description | UDC 519.21
We prove the solvability of Itô stochastic equations with uniformly nondegenerate bounded measurable diffusion and drift in $L_{d+1}(R^{d+1}).$Actually, the powers of summability of the drift in $x$ and $t$ could be different. Our results seem to be new even if the diffusion is constant. The method of proving the solvability belongs to A. V. Skorokhod.Weak uniqueness of solutions is an open problem even if the diffusion is constant. |
| doi_str_mv | 10.37863/umzh.v72i9.6280 |
| first_indexed | 2026-03-24T03:26:54Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i9.6280
UDC 519.21
N. V. Krylov (Univ. Minnesota, Minneapolis, MN, USA)
ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS
WITH DRIFT IN \bfitL \bfitd +\bfone
ПРО НЕОДНОРIДНI ЗА ЧАСОМ СТОХАСТИЧНI РIВНЯННЯ IТО
З ПЕРЕНОСОМ В \bfitL \bfitd +\bfone
We prove the solvability of Itô stochastic equations with uniformly nondegenerate bounded measurable diffusion and drift
in Ld+1(\BbbR d+1). Actually, the powers of summability of the drift in x and t could be different. Our results seem to be new
even if the diffusion is constant. The method of proving the solvability belongs to A. V. Skorokhod. Weak uniqueness of
solutions is an open problem even if the diffusion is constant.
Доведено розв’язнiсть стохастичних рiвнянь Iто з рiвномiрно невиродженою та обмеженою матрицею дифузiї i з
переносом в Ld+1(\BbbR d+1). Справдi, показники iнтегровностi по x i t можуть вiдрiзнятися. Цей результат є новим
навiть коли дифузiя стала. Метод, який ми використовуємо, належить А. В. Скороходу. Питання про слабку єдинiсть
є вiдкритим навiть коли дифузiя стала.
1. Introduction. Let \BbbR d be a Euclidean space of points x = (x1, . . . , xd), d \geq 2. We fix some
p, q \in [1,\infty ] such that
d
p
+
1
q
\leq 1 (1.1)
with further restrictions on them to be specified later. The goal of this article is to study the solvability
of Itô’s stochastic equations of the form
xt = x(0) +
t\int
0
\sigma
\Bigl(
t(0) + s, xs
\Bigr)
dws +
t\int
0
b
\Bigl(
t(0) + s, xs
\Bigr)
ds, (1.2)
where wt is a d-dimensional Wiener process, \sigma is a uniformly nondegenerate, bounded, Borel
function with values in the set of symmetric (d \times d)-matrices, b is a Borel measurable \BbbR d- valued
function given on ( - \infty ,\infty )\times \BbbR d such that\int
\BbbR
\left( \int
\BbbR d
| b(t, x)| p dx
\right) q/p
dt <\infty (1.3)
if p \geq q or \int
\BbbR d
\left( \int
\BbbR
| b(t, x)| q dt
\right) p/q
dx <\infty
if p \leq q. If p = \infty or q = \infty we interpret this conditions in a natural way. Observe that the case
p = q = d + 1 is not excluded and in this case the condition becomes b \in Ld+1(\BbbR d+1). Under this
condition the solvability of (1.2) was proved in [17].
We are talking, of course, about weak solutions and prove their existence in Theorem 3.1. In
Theorem 6.1 we prove the existence of strong Markov processes corresponding to diffusion \sigma and
c\bigcirc N. V. KRYLOV, 2020
1232 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1233
drift b with the above properties. If b is bounded, as we know from [16], there exist strong Markov
and strong Feller processes with diffusion \sigma and drift b for which the Harnack inequality holds and
the caloric functions are Hölder continuous. We are far from proving such fine properties.
The main technical tools are collected in Section 4 where we prove new mixed norms estimates
for the distributions of semimartingales. The treatment there, actually, follows very closely the work
by A. I. Nazarov [11] written in terms of PDEs.
There is a vast literature about stochastic equations with irregular drift. Probably one of the
first authors starting this area was N. I. Portenko, see his book [12], where he constructed diffusion
processes with sufficiently regular \sigma and b \in Lp(\BbbR d+1), p > d + 2. This condition on b was later
refined in many articles with various ambitious goals in them to the requirement that b be such
that (1.3) holds not under condition (1.1) but rather
d
p
+
2
q
\leq 1. (1.4)
We refer the reader to the recent articles [2, 10, 15] and the references therein for the discussion
of many powerful results obtained under condition (1.4), when the case of equality is treated as
“critical”. It could be critical in some respects but not for obtaining our results, that seem to be the
first ones about the existence of solutions and Markov processes with our condition on the drift. Still
it is worth emphasizing that our condition is different if p \geq q (and, hence, p \geq d + 1) or p < q,
whereas there is no such distinction attached to (1.4).
We assume that d \geq 2 and denote
BR =
\bigl\{
x \in \BbbR d : | x| < R
\bigr\}
, Di =
\partial
\partial xi
, Dij = DiDj , \partial t =
\partial
\partial t
.
For p, q \in [1,\infty ], we introduce the space Lp,q as the space of Borel functions on \BbbR d+1 such that
\| f\| qp,q :=
\int
\BbbR
\left( \int
\BbbR d
| f(t, x)| p dx
\right) q/p
dt <\infty
if p \geq q or
\| f\| pp,q :=
\int
\BbbR d
\left( \int
\BbbR
| f(t, x)| q dt
\right) p/q
dx <\infty
if p \leq q with natural interpretation of these definitions if p = \infty or q = \infty . To better memorize these
formulas observe that p is associated with integration with respect to x, q with that with respect to
t and the interior integral is always elevated to the power \leq 1. In case p = q = d+ 1 we abbreviate
Ld+1,d+1 = Ld+1, \| \cdot \| d+1,d+1 = \| \cdot \| d+1.
2. An example of nonexistence.
Example 2.1. Suppose that numbers \alpha and \beta satisfy
0 < \alpha \leq \beta < 1, \alpha + \beta = 1, (2.1)
and set
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1234 N. V. KRYLOV
b(t, x) = - 1
t\alpha | x| \beta
x
| x|
I0<| x| \leq 1,t\leq 1.
Observe that if d/p+ 1/q = 1 + \varepsilon , \varepsilon > 0, one can take \beta = d/(p+ p\varepsilon ), \alpha = 1/(q + q\varepsilon ) and then
1\int
0
\left( \int
| x| \leq 1
| b(t, x)| p dx
\right)
q/p
dt <\infty ,
\int
| x| \leq 1
\left( 1\int
0
| b(t, x)| q dt
\right) p/q
dx <\infty .
Also note that if p \leq qd (say p = q), condition (2.1) is satisfied.
However, it turns out that no matter which \alpha , \beta we take satisfying (2.1) there is no solutions of
the equation dxt = dwt + b(t, xt) dt starting at zero, where wt is a d-dimensional Wiener process.
To prove this assume the contrary. Namely, assume the there is a stopping time \tau such that
P (\tau > 0) > 0 and for t \leq \tau there is xt such that
xt = wt +
t\int
0
b(s, xs) ds.
We may assume that \tau \leq 1 and before \tau the process is in B1. Then, for t \leq \tau ,
dxt = - 1
t\alpha | xt| \beta
xt
| xt|
Ixt \not =0 dt+ dwt,
d| xt| 2 = - 2
| xt|
t\alpha | xt| \beta
dt+ d dt+ 2xt dwt.
(2.2)
We will be interested in | xt| 1+\beta = \xi
(1+\beta )/2
t , where \xi t = | xt| 2. By Itô’s formula for any \varepsilon > 0
we have
d(\xi t + \varepsilon )(1+\beta )/2 =
1 + \beta
2
(\xi t + \varepsilon )(\beta - 1)/2 d\xi t +
\beta 2 - 1
8
(\xi t + \varepsilon )(\beta - 3)/24| xt| 2 dt =
= It(\varepsilon ) dt+ Jt(\varepsilon ) dt+ (1 + \beta )(\xi t + \varepsilon )(\beta - 1)/2xt dwt, (2.3)
where
It(\varepsilon ) = - (1 + \beta )(\xi t + \varepsilon )(\beta - 1)/2 | xt| \alpha
t\alpha
,
Jt(\varepsilon ) =
1 + \beta
2
\bigl[
d+ (\beta - 1)(\xi t + \varepsilon ) - 1| xt| 2
\bigr]
(\xi t + \varepsilon )(\beta - 1)/2.
Since (\xi t + \varepsilon ) - \alpha /2| xt| \alpha \uparrow Ixt \not =0 as \varepsilon \downarrow 0, by the dominated convergence theorem
t\int
0
Is(\varepsilon ) ds\rightarrow - (1 + \beta )
t\int
0
Ixs \not =0
1
s\alpha
ds,
which is finite.
Furthermore, since | xs| \beta - 1xs is bounded on each trajectory, by the dominated convergence the-
orem
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1235
t\int
0
\bigm| \bigm| \bigm| (\xi s + \varepsilon )(\beta - 1)/2xs - | xs| \beta - 1xs
\bigm| \bigm| \bigm| 2 ds\rightarrow 0,
and we conclude from (2.3) that for t \leq \tau
| xt| 1+\beta = - (1 + \beta )
t\int
0
Ixs \not =0
1
s\alpha
ds+ \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \downarrow 0
t\int
0
Js(\varepsilon ) ds+ (1 + \beta )
t\int
0
| xs| \beta - 1xsIxs \not =0 dws (2.4)
and the above limit exists and is finite. Since 2Js(\varepsilon ) \geq (\xi s + \varepsilon )(\beta - 1)/2, it follows that
t\int
0
| xs| \beta - 1 ds = \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \downarrow 0
t\int
0
(\xi s + \varepsilon )(\beta - 1)/2 ds
and the left-hand side is finite. In particular,
\tau \int
0
Ixs=0 ds = 0. (2.5)
Now by the dominated convergence theorem (2.4) implies that
| xt| 1+\beta = - (1 + \beta )
t\int
0
1
s\alpha
ds+
+
1
2
(1 + \beta )
t\int
0
(d+ \beta - 1)| xs| \beta - 1 ds+ (1 + \beta )
t\int
0
| xs| \beta - 1xs dws.
Next, use \alpha \leq \beta and Hölder’s inequality to conclude that
t\int
0
| xs| - \alpha ds =
t\int
0
\biggl(
1
s\alpha | xs| \beta
\biggr) \alpha /\beta
s\alpha
2/\beta ds \leq
\left( t\int
0
1
s\alpha | xs| \beta
ds
\right) \alpha /\beta \left( t\int
0
s\alpha
2/(\beta - \alpha ) ds
\right) (\beta - \alpha )/\beta
.
Since \alpha 2/(\beta - \alpha ) + 1 = (\alpha 2 + 1 - 2\alpha )/(\beta - \alpha ) = \beta 2/(\beta - \alpha ),
t\int
0
| xs| - \alpha ds \leq N
\left( t\int
0
1
s\alpha | xs| \beta
ds
\right) \alpha /\beta t\beta ,
where N = N(\alpha , \beta ) (which is trivial if \alpha = \beta ). Thus,
| xt| 1+\beta + ct\beta \leq N1
\left( t\int
0
1
s\alpha | xs| \beta
ds
\right) \alpha /\beta t\beta + (1 + \beta )
t\int
0
| xs| \beta - 1xs dws,
where c > 0 is a constant. For equation (2.2) to make sense we should have
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1236 N. V. KRYLOV
\tau \int
0
1
s\alpha | xs| \beta
ds <\infty (a. s.).
Therefore,
\gamma := \tau \wedge \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ t \geq 0 : N1
\left( t\int
0
1
s\alpha | xs| \beta
ds
\right) \alpha /\beta \geq c/2
\right\} ,
is a stopping time such that P (\gamma > 0) = P (\tau > 0). It follows that for any t > 0
t\int
0
Is<\gamma | xs| \beta - 1xs dws \geq 0,
which is only possible if Is<\gamma | xs| \beta - 1xs = 0 for almost all (\omega , s). Then xs = 0 for s < \gamma and (2.5)
is only possible if P (\tau = 0) = 1.
3. An existence theorem. In this section, we state a result saying that in a wide class of cases
there exists a probability space and a Wiener process on this space such that a stochastic equation
having measurable coefficients as well as this Wiener process is solvable. In other words, according
to conventional terminology, we are talking here about “weak” solutions of a stochastic equation.
The main difference between “weak” solutions and usual (“strong”) solutions consists in the fact that
the latter can be constructed on any a priori given probability space on the basis of any given Wiener
process.
Let \sigma (t, x) be Borel d\times d symmetric matrix valued, b(t, x) be Borel \BbbR d-valued functions given
on \BbbR d+1 := ( - \infty ,\infty ) \times \BbbR d. We assume that the eigenvalues of \sigma (t, x) are between \delta and \delta - 1,
where \delta \in (0, 1) is a fixed number. The set of such matrices we denote by \BbbS \delta .
Next, fix numbers p, q \in (1,\infty ), \| b\| \in (0,\infty ) and let bn(t, x), n = 1, 2, . . . , be \BbbR d-valued
Borel functions on \BbbR d+1
+ and suppose that
\| b\| p,q, \| bn\| p,q \leq \| b\| , n = 1, 2, . . . ,
d
p
+
1
q
= 1,
and bn \rightarrow b as n\rightarrow \infty in Lp,q. Let \sigma n(t, x), n = 1, 2, . . . , be Borel functions on \BbbR d with values in
\BbbS \delta such that \sigma n \rightarrow \sigma as n\rightarrow \infty (\BbbR d+1-a.e.).
Theorem 3.1. Take (t0, x0) \in \BbbR d+1. (i) There exists a probability space (\Omega ,\scrF , P ), a filtration
of \sigma -fields \scrF t \subset \scrF , t \geq 0, a process wt, t \geq 0, which is a d-dimensional Wiener process relative
to \{ \scrF t\} , and an \scrF t-adapted process xt such that (a.s.) for all t \geq 0 equation (1.2) holds.
(ii) Furthermore, let (tn, xn) \in \BbbR d+1, n = 1, 2, . . . , and let (tn, xn) \rightarrow (t0, x0) as n \rightarrow \infty .
Assume that for each n = 1, 2, . . . there exists a probability space (\Omega n,\scrF n, Pn), a filtration of
\sigma -fields \scrF n
t \subset \scrF n, t \geq 0, a process wnt , t \geq 0, which is a d-dimensional Wiener process relative to
\{ \scrF n
t \} , and an \scrF n
t -adapted process xnt such that (a.s.) for all t \geq 0
xnt = xn +
t\int
0
\sigma n(tn + s, xns ) dw
n
s +
t\int
0
bn(tn + s, xns ) ds.
Then the finite dimensional distributions of a subsequence of xn\cdot converge weakly to the corre-
sponding distributions of one of the solutions of (1.2) described in (i). Moreover, if p \geq q, the set of
distributions of xn\cdot on C
\bigl(
[0,\infty ),\BbbR d
\bigr)
is tight.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1237
The proof of this theorem, following a similar proof by A. V. Skorokhod, is given in Section 5,
after we make a crucial step in the next section where we prove, in particular, that for solutions
of (1.2), any Borel f \geq 0, and T \in (0,\infty )
E
T\int
0
f(t, xt) dt \leq N\| f\| p,q,
where N is independent of f and (t0, x0).
It is worth saying that deciding whether the solutions of (1.2) are weakly unique or not under our
conditions is a very challenging open problem even if \sigma ij \equiv \delta ij .
Remark 3.1. Theorem 3.1 is also true if d/p + 1/q < 1. This can be seen from its proof which
becomes somewhat more technical in that case because of the form of our main estimate (4.9). Also
the main interest in Theorem 3.1 is, of course, the lowest local integrability of b, when the condition
d/p+ 1/q = 1 is weaker than d/p+ 1/q < 1 due to Hölder’s inequality.
4. Estimates of the distributions of semimartingales. Here we first prove a version of
Lemma 5.1 of [6]. The proof given in [6] uses somewhat advanced knowledge of very powerful
results from the theory of fully nonlinear parabolic equations. We give a proof based on a simpler
fact which in turn was one of the cornerstones of that theory.
Let (\Omega ,\scrF , P ) be a complete probability space, let \scrF t, t \geq 0, be an increasing family of complete
\sigma -fields \scrF t \subset \scrF , t \geq 0, let mt be an \BbbR d-valued continuous local martingale relative to \scrF t, let At
be a continuous \scrF t-adapted nondecreasing process, let Bt be a continuous \BbbR d-valued \scrF t-adapted
process which has finite variation (a.e.) on each finite time interval. Assume that
A0 = 0, m0 = B0 = 0, d\langle m\rangle t \ll dAt
and that we are also given progressively measurable relative to \scrF t nonnegative processes rt and ct.
Finally, take an \scrF 0 measurable \BbbR d-valued x0 and introduce
xt = x0 +mt +Bt, \tau t =
t\int
0
rs dAs, \phi t =
t\int
0
cs dAs, aijt =
1
2
d\langle mi,mj\rangle t
dAt
.
Lemma 4.1. Let \gamma be an \scrF t-stopping time and set
A = E
\gamma \int
0
e - \phi t\mathrm{t}\mathrm{r} as dAt, B = E
\gamma \int
0
e - \phi t | dBt| .
Then, for any Borel f(t, x) \geq 0, we have
E
\gamma \int
0
e - \phi t(rt \mathrm{d}\mathrm{e}\mathrm{t} at)
1/(d+1)f(\tau t, xt) dAt \leq N(d)(B2 +A)d/(2d+2)\| f\| d+1. (4.1)
Proof. Without losing generality we may assume that A < \infty and B < \infty . Furthermore, just
stopping the processes At, mt, and Bt at time \gamma , we reduce the general case to the one in which
\gamma = \infty . In that case we also observe that, as usual, it suffices to prove (4.1) for f \in C\infty
0 (\BbbR d+1).
After these reductions we use Theorem 2.2.4 of [8] according to which, for any \lambda > 0 on \BbbR d+1,
there exists a nonnegative function v(t, x) such that
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1238 N. V. KRYLOV
(i) all Sobolev derivatives \partial tv, Div, Dijv exist and are bounded on \BbbR d+1 and v \leq Ne - | x| /N
for all t, x and a constant N ;
(ii) for any nonnegative symmetric (d\times d)-matrix \alpha and r \geq 0,
r\partial tv + \alpha ijDijv - \lambda (r + \mathrm{t}\mathrm{r}\alpha )v +
d+1
\surd
r \mathrm{d}\mathrm{e}\mathrm{t}\alpha f \leq 0,
\partial tv - \lambda v \leq 0, (\lambda v\delta ij - Dijv) \geq 0, | Dv| \leq
\surd
\lambda v (a. e.); (4.2)
(iii) for any y \in \BbbR d, t \in ( - \infty ,\infty ), we have
v(t, y)e - \lambda t \leq N(d)
1
\lambda d/(2d+2)
It, (4.3)
where
Id+1
t :=
\infty \int
0
ds
\int
\BbbR d
e - \lambda (d+1)(t+s)fd+1(t+ s, x) dx.
Take a nonnegative \zeta \in C\infty
0 (\BbbR d+1) with unit integral, for \varepsilon > 0 denote
\zeta \varepsilon (t, x) = \varepsilon - (d+1)\zeta (\varepsilon t, \varepsilon x)
and use the notation u(\varepsilon ) = u \ast \zeta \varepsilon . Then v(\varepsilon ) is infinitely differentiable and in light of (4.2), for any
nonnegative symmetric (d\times d)-matrix \alpha and r \geq 0,
r\partial tv
(\varepsilon ) + \alpha ijDijv
(\varepsilon ) - \lambda (r + \mathrm{t}\mathrm{r}\alpha )v(\varepsilon ) +
d+1
\surd
r \mathrm{d}\mathrm{e}\mathrm{t}\alpha f (\varepsilon ) \leq 0,
\partial tv
(\varepsilon ) - \lambda v(\varepsilon ) \leq 0,
\bigl(
\lambda v(\varepsilon )\delta ij - Dijv
(\varepsilon )
\bigr)
\geq 0,
\bigm| \bigm| Dv(\varepsilon )\bigm| \bigm| \leq \surd
\lambda v(\varepsilon ).
(4.4)
Next, by Itô’s formula the process
v(\varepsilon )(\tau t, xt)e
- \phi t - \lambda \tau t -
t\int
0
e - \phi s - \lambda \tau sDiv
(\varepsilon )(\tau s, xs) dB
i
s+
+
t\int
0
e - \phi s - \lambda \tau s
\Bigl(
(\lambda rs + cs)v
(\varepsilon ) - rs\partial tv
(\varepsilon ) - aijs Dijv
(\varepsilon )
\Bigr)
(\tau s, xs) dAs
is a local martingale. Here owing to (4.4)\Bigl(
(\lambda rs + cs)v
(\varepsilon ) - rs\partial tv
(\varepsilon ) - aijs Dijv
(\varepsilon )
\Bigr)
dAs - Div
(\varepsilon ) dBi
s \geq
\geq (rs \mathrm{d}\mathrm{e}\mathrm{t} as)
1/(d+1)f (\varepsilon ) dAs - \lambda \mathrm{t}\mathrm{r} asv
(\varepsilon ) dAs -
\surd
\lambda v(\varepsilon ) | dBs| .
Therefore, for
M \varepsilon = \mathrm{s}\mathrm{u}\mathrm{p}
t\geq 0,x\in \BbbR d
v(\varepsilon )(t, x)e - \lambda t,
the process
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1239
\kappa \varepsilon t := v(\varepsilon )(\tau t, xt)e
- \phi t - \lambda \tau t +
t\int
0
e - \phi s - \lambda \tau s(rs \mathrm{d}\mathrm{e}\mathrm{t} as)
1/(d+1)f (\varepsilon )(\tau s, xs) dAs -
-
t\int
0
e - \phi s
\Bigl(
\lambda \mathrm{t}\mathrm{r} as dAs +
\surd
\lambda | dBs|
\Bigr)
M \varepsilon
is a local supermartingale. In addition, it is bounded from below by a summable quantity (A,B <
<\infty ). Hence, it is a supermartingale and by Fatou’s lemma
Ev(\varepsilon )(0, x0) = \kappa \varepsilon 0 \geq E
\infty \int
0
e - \phi t - \lambda \tau s(rt \mathrm{d}\mathrm{e}\mathrm{t} at)
1/(d+1)f (\varepsilon )(\tau t, xt) dAt - M \varepsilon (\lambda A+
\surd
\lambda B).
By sending \varepsilon \downarrow 0 and using (4.3) and Fatou’s lemma once more we obtain
E
\infty \int
0
e - \phi t - \lambda \tau s(rt \mathrm{d}\mathrm{e}\mathrm{t} at)
1/(d+1)f(\tau t, xt) dAt \leq N(d)
1
\lambda d/(2d+2)
\Bigl(
1 + \lambda A+
\surd
\lambda B
\Bigr)
I0.
We replace here e - \lambda tf by f and arrive at
E
\infty \int
0
e - \phi t(rt \mathrm{d}\mathrm{e}\mathrm{t} at)
1/(d+1)f(\tau t, xt) dAt \leq N(d)
1
\lambda d/(2d+2)
\Bigl(
1 + \lambda A+
\surd
\lambda B
\Bigr)
\| f\| d+1.
Now we use the arbitrariness of \lambda . If A < B2, then for \lambda = B - 2 we have
1
\lambda d/(2d+2)
\Bigl(
1 + \lambda A+
\surd
\lambda B
\Bigr)
\leq 3Bd/(d+1) \leq 3(B2 +A)d/(2d+2).
If A \geq B2 and A > 0, then for \lambda = A - 1 the above inequality between the extreme terms still holds.
Finally, if A = B = 0, then the left-hand side of (4.1) is zero.
The lemma is proved.
Lemma 4.2. In the notation of Lemma 4.1 for any Borel f(x) \geq 0 we have
E
\gamma \int
0
e - \phi t(\mathrm{d}\mathrm{e}\mathrm{t} at)
1/df(xt) dAt \leq N(d)(B2 +A)1/2\| f\| Ld(\BbbR d). (4.5)
Proof. We follow a probabilistic version of an argument in [11]. We again may concentrate on
the case of A + B < \infty , \gamma = \infty , and f \in C\infty
0 (\BbbR d). In that case observe that by Theorem 2.2.3 of
[8] there exists a nonnegative function v(x) defined on \BbbR d such that
(a) v \leq Ne - | x| /N for all x and a constant N ; the generalized derivatives Div and Dijv,
i, j = 1, . . . , d, are bounded on \BbbR d ;
(b) for any nonnegative symmetric (d\times d)-matrix \alpha (a. e.)
- \lambda v \mathrm{t}\mathrm{r} \alpha + \alpha ijDijv +
d
\surd
\mathrm{d}\mathrm{e}\mathrm{t}\alpha f \leq 0, | Dv| \leq
\surd
\lambda v,
(\lambda v\delta ij - Dijv) \geq 0; (4.6)
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1240 N. V. KRYLOV
(c) for any x \in \BbbR d
v(x) \leq N(d)\lambda - 1/2\| f\| Ld(\BbbR d). (4.7)
Then we closely follow the proof of Lemma 4.1. Take a nonnegative \zeta \in C\infty
0 (\BbbR d) with unit
integral, for \varepsilon > 0 denote \zeta \varepsilon (x) = \varepsilon - d\zeta (\varepsilon x) and use the notation u(\varepsilon ) = u\ast \zeta \varepsilon . Then v(\varepsilon ) is infinitely
differentiable and in light of (4.6), for any nonnegative symmetric (d\times d)-matrix \alpha ,
\alpha ijDijv
(\varepsilon ) - \lambda \mathrm{t}\mathrm{r}\alpha v(\varepsilon ) +
d
\surd
\mathrm{d}\mathrm{e}\mathrm{t}\alpha f (\varepsilon ) \leq 0,\Bigl(
\lambda v(\varepsilon )\delta ij - Dijv
(\varepsilon )
\Bigr)
\geq 0,
\bigm| \bigm| Dv(\varepsilon )\bigm| \bigm| \leq \surd
\lambda v(\varepsilon ).
(4.8)
Next, by Itô’s formula the process
v(\varepsilon )(xt)e
- \phi t -
t\int
0
e - \phi sDiv
(\varepsilon )(xs) dB
i
s +
t\int
0
e - \phi s - \lambda \tau s
\bigl(
csv
(\varepsilon ) - aijs Dijv
(\varepsilon )
\bigr)
(xs) dAs
is a local martingale. Here owing to (4.8)\Bigl(
csv
(\varepsilon ) - aijs Dijv
(\varepsilon )
\Bigr)
dAs - Div
(\varepsilon ) dBi
s \geq
\geq (\mathrm{d}\mathrm{e}\mathrm{t} as)
1/df (\varepsilon ) dAs - \lambda \mathrm{t}\mathrm{r} asv
(\varepsilon ) dAs -
\surd
\lambda v(\varepsilon ) | dBs| .
Therefore, for
M \varepsilon = \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR d
v(\varepsilon )(x)
the process
\kappa \varepsilon t := v(\varepsilon )(xt)e
- \phi t +
t\int
0
e - \phi sf (\varepsilon )(xs) dAs -
t\int
0
e - \phi s
\bigl(
\lambda \mathrm{t}\mathrm{r} as dAs -
\surd
\lambda | dBs|
\bigr)
M \varepsilon
is a local supermartingale. In addition, it is bounded from below by a summable quantity (A,B <
<\infty ). Hence, it is a supermartingale and by Fatou’s lemma
Ev(\varepsilon )(x0) = \kappa \varepsilon 0 \geq E
\infty \int
0
e - \phi t(\mathrm{d}\mathrm{e}\mathrm{t} at)
1/df (\varepsilon )(xt) dAt - M \varepsilon (\lambda A+
\surd
\lambda B).
By sending \varepsilon \downarrow 0 and using (4.7) and Fatou’s lemma once more, we obtain
E
\infty \int
0
e - \phi t(\mathrm{d}\mathrm{e}\mathrm{t} at)
1/df(xt) dAt \leq N(d)
1
\lambda 1/2
\Bigl(
1 + \lambda A+
\surd
\lambda B
\Bigr)
\| f\| Ld(\BbbR d).
Now we use the arbitrariness of \lambda . If A < B2, then for \lambda = B - 2 we have
1
\lambda 1/2
\Bigl(
1 + \lambda A+
\surd
\lambda B
\Bigr)
\leq 3B1/2 \leq 3(B2 +A)1/2.
If A \geq B2 and A > 0, then for \lambda = A - 1 the above inequality between the extreme terms still holds.
Finally, if A = B = 0, then the left-hand side of (4.5) is zero.
The lemma is proved.
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ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1241
Theorem 4.1. Assume the notation of Lemma 4.1 and let p, q \in [1,\infty ] be such that
\theta := 1 - d
p
- 1
q
\geq 0,
then, for any Borel f(t, x) \geq 0, we have
I(p, q, f) := E
\gamma \int
0
e - \phi t\kappa tf(rt, xt) dAt \leq N(d)(A+B2)d/(2p)\| f\| p,q, (4.9)
where \kappa t = r
1/q
t (\mathrm{d}\mathrm{e}\mathrm{t} at)
1/pc\theta t and for any \alpha \geq 0 we set \alpha 0 = 1 (say, if \theta = 0).
Proof. By Hölder’s inequality, if \theta > 0,
I(p, q, f) \leq
\Bigl(
I
\bigl(
p(1 - \theta ), q(1 - \theta ), f1/(1 - \theta )
\bigr) \Bigr) 1 - \theta
.
It follows that it suffices to concentrate on \theta = 0. Then we observe that if q = \infty , then p = d and
\| f\| pp,q =
\int
\BbbR d
\mathrm{s}\mathrm{u}\mathrm{p}
t\geq 0
fd(t, x) dx.
In that case (4.12) follows from Lemma 4.2. If p = \infty , then q = 1, and
I(p, q, f) = E
\gamma \int
0
rtf(\tau t, xt) dAt \leq E
\gamma \int
0
\mathrm{s}\mathrm{u}\mathrm{p}
x
f(\tau t, x) d\tau t \leq
\infty \int
0
\mathrm{s}\mathrm{u}\mathrm{p}
x
f(t, x) dt = \| f\| p,q.
In the third simple situation when q = p = d+1 estimate (4.12) follows from Lemma 4.1. We prove
the lemma in the remaining cases by interpolating between the above ones.
If p > q (and hence p > d + 1) we take a nonnegative function h(t) such that (hf)/h = f
(0/0 := 0) and use
r
1/q
t (\mathrm{d}\mathrm{e}\mathrm{t} at)
1/pf =
\Bigl(
r
1/q - 1/p
t h - 1
\Bigr) \Bigl(
(r \mathrm{d}\mathrm{e}\mathrm{t} at)
1/pfh
\Bigr)
along with Hölder’s inequality. By performing simple manipulations we find
I(p, q, f) \leq IJ :=
:=
\Bigl(
I
\bigl(
\infty , 1, h - p/(p - d - 1)
\bigr) \Bigr) (p - d - 1)/p \Bigl(
I
\bigl(
d+ 1, d+ 1, (hf)p/(d+1)
\bigr) \Bigr) (d+1)/p
. (4.10)
Here
I \leq
\left( \infty \int
0
h - p/(p - d - 1)(t) dt
\right) (p - d - 1)/p
.
Also
J \leq N(d)(B2 +A)d/(2p)
\bigm\| \bigm\| \bigm\| (hf)p/(d+1)
\bigm\| \bigm\| \bigm\| (d+1)/p
d+1
=
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1242 N. V. KRYLOV
= N(d)(B2 +A)d/(2p)
\left( \infty \int
0
\left( \int
\BbbR d
fp(t, x) dx
\right) hp(t) dt
\right) 1/p
.
We now choose h so that
h - p/(p - d - 1)(t) =
\left( \int
\BbbR d
fp(t, x) dx
\right) hp(t).
Then both quantities become\left( \int
\BbbR d
fp(t, x) dx
\right) q/p
, J \leq N(d)(B2 +A)d/(2p)\| f\| q/pp,q , I \leq \| f\| q(p - d - 1)/p
p,q
and coming back to (4.10) we get (4.9).
In the remaining case q > p (and q > d+ 1) we use
r
1/q
t (\mathrm{d}\mathrm{e}\mathrm{t} at)
1/pf =
\Bigl(
(\mathrm{d}\mathrm{e}\mathrm{t} at)
1/p - 1/qh - 1
\Bigr) \Bigl(
(r \mathrm{d}\mathrm{e}\mathrm{t} at)
1/qfh
\Bigr)
.
This time for h = h(x)
I(p, q, f) \leq IJ :=
:=
\Bigl(
I
\bigl(
d,\infty , h - q/(q - d - 1)
\bigr) \Bigr) (q - d - 1)/q \Bigl(
I
\bigl(
d+ 1, d+ 1, (hf)q/(d+1)
\bigr) \Bigr) (d+1)/q
. (4.11)
Here
I \leq N(d)(B2 +A)(d/p - d/q)(1/2)
\left( \int
\BbbR d
h - qd/(q - d - 1)(x) dx
\right) (q - d - 1)/(qd)
,
J \leq N(d)(B2 +A)d/(2q)
\left( \int
\BbbR d
hq(x)
\left( \infty \int
0
f q(t, x) dt
\right) dx
\right) 1/q
.
We choose h so that
h - qd/(q - d - 1)(x) = hq(x)
\left( \infty \int
0
f q(t, x) dt
\right)
and then easily come to (4.12).
The theorem is proved.
Corollary 4.1. Introduce a measure (Green’s measure) on Borel subsets \Gamma of \BbbR d+1 by the formula
G(\Gamma ) = E
\gamma \int
0
e - \phi t\kappa tI\Gamma (\tau t, xt) dAt.
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ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1243
Assume that A,B <\infty and set p\prime = p/(p - 1), q\prime = q/(q - 1). Then G(\Gamma ) is absolutely continuous
and its density G(t, x) is such that, if p \geq q,\left( \infty \int
0
\left( \int
\BbbR d
Gp
\prime
(t, x) dx
\right) q\prime /p\prime
dt
\right)
1/q\prime
and, if p \leq q, \left( \int
\BbbR d
\left( \infty \int
0
Gq
\prime
(t, x) dt
\right) p\prime /q\prime
dx
\right)
1/q\prime
is dominated by
N(d)(B2 +A)(1 - \theta )d/(2p).
Theorem 4.2. Under the assumptions of Theorem 4.1 let p0 \in [1,\infty ] and q0 \in [1,\infty ) be such
that
\theta 0 := 1 - d
p0
- 1
q0
\geq 0.
Also assume that d| Bt| \ll dAt and there exists a Borel h(t, x) such that (P (d\omega )\times dAt-a.e.)
| bt| \leq \kappa 0th(\tau t, xt),
where bt = dBt/dAt and \kappa 0t = r
1/q0
t (\mathrm{d}\mathrm{e}\mathrm{t} at)
1/p0c\theta 0t . Then for any Borel f(t, x) \geq 0 we have
I(p, q, f) := E
\gamma \int
0
e - \phi t\kappa tf(\tau t, xt) dAt \leq N(d, p0, q0)C\| f\| p,q, (4.12)
where
\kappa t = r
1/q
t (\mathrm{d}\mathrm{e}\mathrm{t} at)
1/pc\theta t , C =
\Bigl(
A+ \| h\| 2p0/(p0 - d)p0,q0
\Bigr) d/(2p)
and for any number \alpha \geq 0 we set \alpha 0 = 1 (say, if \theta = 0).
Proof. Observe that p0 > d since q0 < \infty . Then, we may assume that A < \infty and \| h\| p0,q0 <
<\infty . Using stopping times we easily reduce the general situation to the one in which B <\infty . After
that, in light of Theorem 4.1, we need only prove that
B \leq N(d, p0, q0)
\Bigl(
A1/2 + \| h\| p0/(p0 - d)p0,q0
\Bigr)
. (4.13)
By Theorem 4.1
B = E
\tau \int
0
e - \phi t | dBt| \leq I(p0, q0, h) \leq N(d)(A+B2)d/(2p0)\| h\| p0,q0 .
Here if B2 \leq A, estimate (4.13) holds. If A \leq B2, then the above inequality yields
B \leq N(d)Bd/p0\| h\| p0,q0 , B(p0 - d)/p0 \leq N(d)\| h\| p0,q0
and we obtain (4.13) again.
The theorem is proved.
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1244 N. V. KRYLOV
Remark 4.1. In the case of q = \infty , p = d an estimate of B in terms of \| h\| p,q is given in
Theorem 5.2 of [6] if \gamma is the first exit time of xt from a ball and in Theorem 2.17 of [9] if At = t
and ct = \lambda \mathrm{t}\mathrm{r} at, where \lambda > 0 is a number (and \gamma = \infty ).
Remark 4.2. As in [11] we note that estimate (4.12) also, obviously, holds if
| bt| \leq
n\sum
k=1
\kappa kt hk(\tau t, xt),
where \kappa kt = r
1/qk
t (\mathrm{d}\mathrm{e}\mathrm{t} at)
1/pkc\theta kt , pk \in [1,\infty ], qk \in [1,\infty ), \theta k = 1 - d/pk - 1/qk \geq 0, and hk are
nonnegative Borel functions. In that case the constant C depends only on d, p, q, pk, qk, \| hk\| pk,qk ,
k = 1, . . . , n, in a somewhat complicated way.
Remark 4.3. The main case of applications of Theorem 4.2 in this article is when p = p0 <
<\infty , q = q0 <\infty , \theta = \theta 0 = 0, \gamma = T, where T is a fixed number, rt = 1, ct = 0, At = t \wedge T,
| bt| \leq (\mathrm{d}\mathrm{e}\mathrm{t} at)
1/ph(t, xt)It\leq T .
In that case 2p/(p - d) = 2q and estimate (4.12) becomes
E
T\int
0
(\mathrm{d}\mathrm{e}\mathrm{t} at)
1/pf(t, x)dt \leq N(d, p)
\bigl(
T + \| hI(0,T )\| 2qp,q
\bigr) d/(2p) \| f\| p,q.
We finish the section with somewhat unrelated result which we use later in Section 6 and which
would be a simple consequence of Theorem 4.5.1 of [14] if we assumed that b is bounded.
Lemma 4.3. Let xt, t \geq 0, be an \BbbR d-valued process on a probability space (\Omega ,\scrF , P ). Define
\scrF t as the completion of the \sigma -field generated by xs, s \leq t. Let \sigma t be an \BbbS \delta -valued and b be an
\BbbR d-valued processes which are progressively measurable with respect to \{ \scrF t\} . Suppose that for any
T \in (0,\infty )
T\int
0
| bt| dt <\infty (a. s.)
and for any C\infty
0 (\BbbR d+1)-function u(t, x) the process
u(t, xt) -
t\int
0
Lsu(s, xs) ds (4.14)
is a local martingale with respect to \{ \scrF t\} , where for a = \sigma 2
Ltu(t, x) = \partial tu(t, x) +
1
2
aijt Diju(t, x) + bitDiu(t, x).
Then there exists a d-dimensional Wiener process (wt,\scrF t), t \geq 0, such that
xt = x0 +
t\int
0
\sigma s dws +
t\int
0
bs ds.
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ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1245
Proof. First observe that by using cut-off functions one easily shows that (4.14) is a local martin-
gale for any twice continuously differentiable function u. Then, we claim that the following processes
are local martingales:
Xt := xt -
t\int
0
bs ds,
Bt := xtx
\ast
t -
t\int
0
\bigl(
as + bsx
\ast
s + xsb
\ast
s
\bigr)
ds,
At := XtX
\ast
t -
t\int
0
as ds.
Indeed, the first two processes are obtained from (4.14) for u = x, xx\ast . Concerning the last one
introduce \gamma R as the minimum of \tau R = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t \geq 0 : | xt| \geq R
\bigr\}
and
\mathrm{i}\mathrm{n}\mathrm{f}
\left\{ t \geq 0 :
t\int
0
| bs| ds+ | Bt| \geq R
\right\} .
Also let
\Phi t =
t\int
0
bsIs<\gamma R ds.
Observe that Xt\wedge \gamma R and \Phi t are bounded and simple manipulations yield
At\wedge \gamma R =
t\int
0
Xs\wedge \gamma R d\Phi
\ast
s - Xt\wedge \gamma R\Phi
\ast
t +
t\int
0
\bigl(
d\Phi s
\bigr)
X\ast
s\wedge \gamma R - \Phi tX
\ast
t\wedge \gamma R +Bt\wedge \gamma R ,
which by the Lemma from Appendix 2 of [5] shows that At\wedge \gamma R is a martingale.
By the above claim the quadratic variation process of the local martingale Xt is
t\int
0
as ds.
After that our assertion follows directly from Theorem III.10.8 of [7].
The lemma is proved.
5. Proof of Theorem 3.1. Introduce
B(t) = \| bI( - \infty ,t)\| qp,q.
Lemma 5.1. Suppose that p \geq q and let xt be a solution of (1.2). Then, for 0 \leq s < t < s+1 <
<\infty and n = 1, 2, . . . , we have
E| xt - xs| n \leq N
\bigl(
t - s+B2(t0 + t) - B2(t0 + s)
\bigr) nd/(2p)
, (5.1)
where N = N
\bigl(
n, d, \delta , p, \| b\|
\bigr)
.
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1246 N. V. KRYLOV
Proof. We may assume that t0 = 0. Then observe that for any integer n = 1, 2, . . .
In+1 := E
\left( t\int
s
b(u, xu) du
\right) n+1
=
= (n+ 1)!E
\int
s\leq u1\leq ...\leq un
b(u1, xu1) . . . b(un, xun)E
\left( t\int
un
b(u, xu) du | \scrF un
\right) du1 . . . dun,
where the conditional expectation we can estimate by using Remark 4.3.
Then we get
In+1 \leq N(n+ 1)In
\Bigl(
t - s+ \| bI(s,t)\| 2qp,q
\Bigr) d/(2p)
\| b\| p,q,
where N depends only on d, p, and \delta . Here
\| bI(s,t)\| 2qp,q =
\Bigl(
B(t) - B(s)
\Bigr) 2
\leq B2(t) - B2(s).
Therefore,
In+1 \leq N(n+ 1)In
\Bigl(
t - s+B2(t) - B2(s)
\Bigr) d/(2p)
\| b\| p,q.
The induction on n yields
In \leq Nnn!
\bigl(
t - s+B2(t) - B2(s)
\bigr) nd/(2p) \| b\| np,q.
Also, as is well-known,
E
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
s
\sigma (u, xu) dwu
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n
\leq N(n, \delta )(t - s)n/2.
It follows that the left-hand side of (5.1) is less than a constant N times
(t - s)n/2 +
\Bigl(
t - s+B2(t) - B2(s)
\Bigr) nd/(2p)
,
which less than twice the factor of N in (5.1) because p > d and t - s \leq 1.
The lemma is proved.
Lemma 5.2. Under the assumptions in Theorem 3.1 (ii) the set of distributions of xn\cdot on
C
\bigl(
[0,\infty ),\BbbR d
\bigr)
is tight if p \geq q.
Proof. Define
Bn(t) =
\bigm\| \bigm\| bnI( - \infty ,tn+t)
\bigm\| \bigm\| q
p,q
and let \phi n(s) be the inverse function of \psi n(t) := tn + t + B2
n(t
n + t). By Lemma 5.1 and Kol-
mogorov’s criteria the set of distributions of yn\cdot := xn\phi n(\cdot ) on C
\bigl(
[0,\infty ),\BbbR d
\bigr)
is tight.
Observe that, as n \rightarrow \infty , \psi n(t) converges to t0 + t + B2(t0 + t) which is continuous and
monotone. By Polya’s theorem the convergence is uniform on any finite time interval, and hence, the
functions \psi n(t) are equicontinuous on any finite time interval. Now define
\Phi (s) = \mathrm{i}\mathrm{n}\mathrm{f}
n\geq 1
\phi n(s)
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ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1247
and take S \in (0,\infty ). By tightness, for any \varepsilon > 0 there is a compact set K\varepsilon in C
\bigl(
[0, S],\BbbR d
\bigr)
such
that Pn(\{ yns , s \leq S\} \in K\varepsilon ) \geq 1 - \varepsilon for all n. Due to the uniform continuity of \psi n and of the
elements of K\varepsilon , the elements of
\^K\varepsilon :=
\Bigl\{ \bigl\{
f(\psi n(t)), t \leq \Phi (S)
\bigr\}
: \{ f(s), s \leq S\} \in K\varepsilon , n = 1, 2, . . .
\Bigr\}
are uniformly continuous and, of course, uniformly bounded, so that \^K\varepsilon is a compact set in
C
\bigl(
[0,\Phi (S)],\BbbR d
\bigr)
and
P
\Bigl( \bigl\{
yn\psi n(t), t \leq \Phi (S)
\bigr\}
\in \^K\varepsilon
\Bigr)
\geq 1 - \varepsilon .
It only remains to observe that yn\psi n(t) = xnt , S is arbitrary, and \Phi (S) \rightarrow \infty as S \rightarrow \infty .
The lemma is proved.
This takes care of part of assertion (ii) of Theorem 3.1. To deal with the rest we rely on the
following results due to A. V. Skorokhod (see Ch. 1, \S 6 and Ch. 2, \S 3 in [13]).
Lemma 5.3. Suppose that d1-dimensional random processes \xi nt (t \geq 0, n = 1, 2, . . .) are
defined on some probability spaces. Assume that for each T > 0 and \varepsilon > 0
\mathrm{l}\mathrm{i}\mathrm{m}
c\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
n
\mathrm{s}\mathrm{u}\mathrm{p}
t\leq T
Pn
\bigl(
| \xi nt | > c
\bigr)
= 0, (5.2)
\mathrm{l}\mathrm{i}\mathrm{m}
h\downarrow 0
\mathrm{s}\mathrm{u}\mathrm{p}
n
\mathrm{s}\mathrm{u}\mathrm{p}
t1,t2\leq T
| t1 - t2| \leq h
Pn
\bigl(
| \xi nt1 - \xi nt2 | > \varepsilon
\bigr)
= 0. (5.3)
Then one can choose a sequence of numbers n\prime \rightarrow \infty , a probability space, and random processes
\~\xi t, \~\xi
n\prime
t defined on this probability space such that all finite-dimensional distributions of \~\xi n
\prime
t coincide
with the corresponding finite-dimensional distributions of \xi n
\prime
t and
P
\bigl(
| \~\xi t - \~\xi n
\prime
t |
\bigr)
\rightarrow 0
as n\prime \rightarrow \infty for any \varepsilon > 0 and t \geq 0.
Lemma 5.4. Suppose the assumptions of Lemma 5.3 are satisfied and \xi nt are defined on the
same probability space. Also, suppose that d1-dimensional Wiener processes (wnt ,\scrF n
t ) are defined
on this probability space. Assume that the functions \xi nt (\omega ) are bounded on [0,\infty )\times \Omega uniformly in
n and that the stochastic integrals
Int :=
t\int
0
\xi ns dw
n
s
are defined for t \geq 0. Finally, let
\xi nt \rightarrow \xi 0t , wnt \rightarrow w0
t (5.4)
in probability as n\rightarrow \infty for each t \geq 0. Then Int \rightarrow I0t in probability as n\rightarrow \infty for each t \geq 0.
Remark 5.1. As it follows from the proof of Lemma 5.4 given in [13] we need conditions (5.2),
(5.3), and (5.4) to hold only for t, t1, t2 restricted to a set of full measure in order for the assertion
of the lemma to be true.
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1248 N. V. KRYLOV
Lemma 5.5. Let \BbbR 2d-valued processes (xit, w
i
t), t \geq 0, i = 1, 2, defined on perhaps different
probability spaces have the same finite-dimensional distributions. Define \scrF i
t as the completion of
\sigma (xis, w
i
s : s \leq t) and assume that w1
t is a Wiener process with respect to \scrF 1
t . Also suppose that
(a.s.) for all t \geq 0
x1t =
t\int
0
\sigma (s, x1s) dw
1
s +
t\int
0
b(s, x1s) ds. (5.5)
Then x2t , w
2
t have modifications (called again x2t , w
2
t ) such that w2
t is a Wiener process with respect
to \scrF 2
t and (a.s.) for all t \geq 0
x2t =
t\int
0
\sigma (s, x2s) dw
2
s +
t\int
0
b(s, x2s) ds. (5.6)
Proof. Fix T \in (0,\infty ) and \varepsilon \in (0, 1). Since the trajectories of (x1t , w
1
t ) are continuous, there
exists a compact set K \subset C
\bigl(
[0, T ],\BbbR 2d
\bigr)
such that
P
\bigl(
(x1\cdot \wedge T , w
1
\cdot \wedge T ) \in K
\bigr)
\geq 1 - \varepsilon .
Hence, there is a constant N and a continuous function w(t), t \in [0, T ], such that w(0) = 0 and
with probability larger than 1 - \varepsilon for any s, t \in [0, T ]\bigm| \bigm| (x1s, w1
s)
\bigm| \bigm| \leq N,
\bigm| \bigm| (x1s, w1
s) - (x1t , w
1
t )
\bigm| \bigm| \leq w
\bigl(
| t - s|
\bigr)
. (5.7)
It follows that (5.7) holds for rational s, t if we replace (x1, w1) with (x2, w2). Then by conti-
nuity (x2t , w
2
t ) is extended to all t \in [0, T ]. The extensions coincide with the original ones (a.s.) for
any t because of the stochastic continuity of the original (x2t , w
2
t ). This is done on events whose pro-
babilities tend to one. Because of the arbitrariness of T we may assume that (x2t , w
2
t ) is continuous
in t with probability one.
By Remark 4.3 and by the coincidence of finite dimensional distributions (and by the measura-
bility of x2t due to its continuity) for any T \in [0,\infty ), Borel f(t, x) \geq 0,
E
T\int
0
f(t, x2t ) dt \leq N\| fI(0,T )\| p,q, (5.8)
where N is independent of f.
Furthermore, if \alpha (t, x) is a continuous d \times d symmetric matrix-valued, \beta (t, x) is a continuous
\BbbR d-valued, then the distributions of\left( xit, t\int
0
\alpha (s, xis) dw
i
s,
t\int
0
\beta (s, xis) ds
\right) , i = 1, 2,
coincide, because the integrals can be approximated by integral sums. This coincidence also holds
for \alpha = \sigma and \beta = b due to (5.8) and the possibility of approximation. Hence for each t with
probability one (5.6) holds due to (5.5). But then with probability one it holds for all t, because both
sides of (5.6) are continuous.
The lemma is proved.
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ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1249
Proof of Theorem 3.1. Due to the possibility to use mollifiers we see that assertion (ii) implies
(i). In the proof of (ii), thanks to Lemma 5.2, we need only prove the assertion concerning the
convergence of finite dimensional distributions.
Having in mind Lemma 5.3 define for M > 0
\xi nt =
t\int
0
bn(tn + s, xns ) ds, \xi nMt =
t\int
0
bn(tn + s, xns )I| bn(tn+s,xns )| \leq M ds.
Since the derivative of \xi nMt is bounded, both conditions (5.2) and (5.3) are satisfied for \xi nMt .
Furthermore,
Pn
\left( T\int
0
| bn(tn + s, xns )| I| bn(tn+s,xns )| \geq M ds > \varepsilon
\right) \leq \varepsilon - 1N\| bnI| bn| \geq M\| p,q,
where N is independent of n and \varepsilon . Since bn \rightarrow b in the \| \cdot \| p,q -norm, the latter quantity can be
made as small as we like on the account of choosing M large enough. Therefore, Lemma 5.3 is
applicable to \xi nt . It is, obviously, also applicable to
\eta nt = xn +
t\int
0
\sigma n(tn + s, xns ) dw
n
s .
Hence, there is a subsequence, which by common abuse of notation we identify with the original one,
a probability space and random \BbbR 2d-valued processes (\~xnt , \~w
n
t ), (\~x
0
t , \~w
0
t ) defined on this probability
space such that all finite-dimensional distributions of (\~xnt , \~w
n
t ) coincide with the corresponding finite-
dimensional distributions of (xnt , w
n
t ) and
P
\bigl(
| (\~xnt , \~wnt ) - (\~x0t , \~w
0
t )| \geq \varepsilon
\bigr)
\rightarrow 0 (5.9)
as n \rightarrow \infty for any \varepsilon > 0 and t \geq 0. Furthermore, for any T \in (0,\infty ) there exists a continuous
function w(t), t \in [0, T ], such that w(0) = 0 and for all n \geq 0, s, t \leq T,
E
\bigm| \bigm| \phi (\~xnt ) - \phi (\~xns )
\bigm| \bigm| \leq w
\bigl(
| t - s|
\bigr)
, (5.10)
where \phi (x) = x/
\bigl(
1 + | x|
\bigr)
.
For n \geq 0 introduce \~\scrF n
t as the completion of \sigma (\~xns , \~w
n
s , s \leq t). It is easy to see, using Kol-
mogorov’s continuity criterion, that \~w0
t admits a continuous modification \^w0
t such that \{ \^w0
t ,
\~\scrF 0
t \} is
a Wiener process.
By Lemma 5.5, for each n \geq 1, the process (\~xnt , \~w
n
t ) admits a continuous modification denoted
by (\^xnt , \^w
n
t ) such that ( \^wnt , \~\scrF n
t ) is a Wiener process and (a.s.) for all t \geq 0
\^xnt = xn +
t\int
0
\sigma n(tn + s, \^xns ) d \^w
n
s +
t\int
0
bn(tn + s, \^xns ) ds. (5.11)
In light of (5.9) and (5.10) we have
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1250 N. V. KRYLOV
P
\bigl(
| (\^xnt , \^wnt ) - (\~x0t , \~w
0
t )| \geq \varepsilon
\bigr)
\rightarrow 0 (5.12)
as n\rightarrow \infty for any \varepsilon > 0 and, t \geq 0 and for all n \geq 1, s, t \leq T,
E
\bigm| \bigm| \phi (\^xnt ) - \phi (\^xns )
\bigm| \bigm| \leq w(| t - s| ). (5.13)
Now the fact that \~x0t may be not measurable in t causes some problems. However, observe
that, owing to (5.12), \phi (\^xnt ) form a Cauchy sequence in L1(\Omega \times [0, T ]) and, hence, converges in
that space to \phi (\^x0t ), where \^x0t is measurable with respect to (\omega , t). By Fubini’s theorem there is
a set \scrS \subset [0,\infty ) of full measure such that, for any t \in \scrS , \^x0t = \~x0t (a.s.). Without losing the
above properties we set \^x0t = 0 for t \not \in \scrS and then, for any s, t \geq 0, \^w0
t+s - \^w0
t is indepenent of
(\^x0r , \^w
0
r), r \leq t.
Now we note that (5.13) remains valid for n = 0 and (5.12) remains valid if we replace (\~x0t , \~w
0
t )
by (\^x0t , \^w
0
t ) and restrict the ranges of t, s to t, s \in \scrS . This is done to accommodate Remark 5.1.
Then, by Lemma 5.4 for any t \geq 0 and continuous d\times d symmetric matrix-valued \alpha (t, x), we have
t\int
0
\alpha (s, \^xns ) d \^w
n
s \rightarrow
t\int
0
\alpha (s, \^x0s) d \^w
0
s (5.14)
as n \rightarrow \infty in probability. We want to use this to pass to the limit in the stochastic term in (5.11).
But first observe that by Remark 4.3 for any T \in [0,\infty ), Borel f(t, x) \geq 0, and n \geq 1
E
T\int
0
f(t, \^xnt ) dt \leq N\| fI(0,T )\| p,q, (5.15)
where N is independent of f and n. The convergence in probability implies that (5.15) holds for
n = 0 as well with the same constant N, first for nonnegative f \in C\infty
0 (\BbbR d+1) and then, due to
general measure-theoretic arguments, for any Borel nonnegative f.
We claim that on the account of (5.15), if Borel functions gn converge to g in the \| \cdot \| p,q -norm,
then
E
T\int
0
\bigm| \bigm| gn(t, \^xnt ) - g(t, \^x0t )
\bigm| \bigm| dt\rightarrow 0. (5.16)
To prove (5.16) take \varepsilon > 0 and g\varepsilon \in C\infty
0 (\BbbR d+1) such that
\| g - g\varepsilon \| p,q \leq \varepsilon .
For g\varepsilon in place of g, (5.16) follows from the convergence in probability of \^xnt to \^x0t for t \in \scrS . After
that it only remains to observe that the limit of the error of the substitution in (5.16) is less than 2N\varepsilon
owing to (5.15). It follows, in particular, that in probability
\mathrm{s}\mathrm{u}\mathrm{p}
t\leq T
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
0
bn(tn + s, \^xns ) ds -
t\int
0
b(t0 + s, \^x0s) ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \rightarrow 0. (5.17)
Coming back to the stochastic part note that for any t \geq 0 and c \in (0,\infty )
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ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1251
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
E
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
0
\sigma n(tn + s, \^xns ) d \^w
n
s -
t\int
0
\alpha (s, \^xns ) d \^w
n
s
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2
=
= \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
E
t\int
0
\| \sigma n(tn + s, \^xns ) - \alpha (s, \^xns )\| 2 ds \leq
\leq N \mathrm{s}\mathrm{u}\mathrm{p}
n
t\int
0
P (| \^xns | > c) ds+N \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bigm\| \bigm\| \bigl( \sigma n(tn + \cdot , \cdot ) - \alpha (\cdot , \cdot )
\bigr)
I[0,t]\times Bc
\bigm\| \bigm\|
p,q
=
= N \mathrm{s}\mathrm{u}\mathrm{p}
n
t\int
0
P (| \^xns | > c) ds+N
\bigm\| \bigm\| \bigl( \sigma (t0 + \cdot , \cdot ) - \alpha (\cdot , \cdot )
\bigr)
I[0,t]\times Bc
\bigm\| \bigm\|
p,q
,
where the constants N are independent of t and c. The last quantity also dominates
E
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
0
\sigma (t0 + s, \^x0s) d \^w
0
s -
t\int
0
\alpha (s, \^x0s) d \^w
0
s
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2
.
This and (5.14) show how, for any given \varepsilon , \delta > 0, to choose c and a continuous \alpha in order to have
that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
P
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
t\int
0
\sigma n(tn + s, \^xns ) d \^w
n
s -
t\int
0
\sigma (tn + s, \^x0s) d \^w
0
s
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| > \varepsilon
\right) \leq \delta .
Upon combining this with (5.17) and coming back to (5.11) we conclude that for any t (a.s.)
\~x0t = x0 +
t\int
0
\sigma (t0 + s, \^x0s) d \^w
0
s +
t\int
0
b(t0 + s, \^x0s) ds =: yt.
In particular, this means that \~x0t admits a continuous modification yt. In turn, it allows us to replace
in the above equation \^x0s with yt, because for any s \in \scrS , \^x0s = \~x0s = ys (a.s.) and therefore \^x0s = ys
for almost all (\omega , s).
The theorem is proved.
6. Markov processes corresponding to \bfitsigma , \bfitb . We are going to use the results in [4] applied in
the case when the semicompactum E is \BbbR d+1, that is when the t-variable is considered just as one
of coordinates of points (t, x) \in \BbbR d+1.
Let \Omega be the set of \BbbR d+1-valued continuous function (t0+ t, xt), t0 \in \BbbR , defined for t \in [0,\infty ).
For \omega =
\bigl\{
(t0 + t, xt), t \geq 0
\bigr\}
, define \sanst t(\omega ) = t0 + t, xt(\omega ) = xt, and set \scrN t = \sigma ((\sanst s, xs), s \leq t),
\scrN = \scrN \infty . Denote by \sansT the set of stopping times relative to \{ \scrN t\} . In the following theorem we use
the terminology from [3].
Theorem 6.1. On \BbbR d+1 there exists a strong Markov process
X =
\bigl\{
(\sanst t, xt),\infty ,\scrN t, Pt,x
\bigr\}
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
1252 N. V. KRYLOV
such that the process
X1 =
\bigl\{
(\sanst t, xt),\infty ,\scrN t+, Pt,x
\bigr\}
is Markov and for any (t, x) \in \BbbR d+1 there exists a d-dimensional Wiener process wt, t \geq 0, which
is a Wiener process relative to \=\scrN t, where \=\scrN t is the completion of \scrN t with respect to Pt,x, and such
that with Pt,x-probability one, for all s \geq 0, \sanst s = t+ s and
xs = x+
s\int
0
\sigma (t+ u, xu) dwu +
s\int
0
b(t+ u, xu) du. (6.1)
Proof. Define a = \sigma 2,
Lu(t, x) = \partial tu(t, x) +
1
2
aijDiju(t, x) + biDiu(t, x)
and introduce \Pi t,x as the set of probability measures on (\Omega ,\scrN ) such that P
\bigl(
(\sanst 0, x0) = (t, x)
\bigr)
= 1,
E
T\int
0
| b(\sanst t, xt)| dt <\infty \forall T <\infty , (6.2)
and the process
\eta t(u) = u(\sanst t, xt) -
t\int
0
Lu(\sanst s, xs) ds
is a martingal relative to \{ \scrN t\} for all u \in C\infty
0 (\BbbR d+1).
According to Lemma 4.3, if Pt,x \in \Pi t,x, then the assertion of the theorem regarding (6.1) holds
and (6.2) is true. Therefore, by Theorem 2 of [4] to prove the present theorem, it suffices to show
that \Pi t,x \not = \varnothing and \{ \Pi t,x\} is a Markov system relative to (\sansT ,\scrN \sanst ) and ([0,\infty ),\scrN t+).
That \Pi t,x \not = \varnothing follows from Theorem 3.1(i). Let us prove that \{ \Pi t,x\} is a B-system. To achieve
this, as it follows from [4], it suffices to show that if (tn, xn) \rightarrow (t, x) and Pn \in \Pi tn,xn , then there
exists a subsequence n(k) \rightarrow \infty and P 0 \in \Pi t,x such that for any f \in C\infty
0 (\BbbR d+2)
En(k) \mathrm{e}\mathrm{x}\mathrm{p}
\left( \infty \int
0
e - tf(t, \sanst t, xt) dt
\right) \rightarrow E0 \mathrm{e}\mathrm{x}\mathrm{p}
\left( \infty \int
0
e - tf(t, \sanst t, xt) dt
\right) ,
where En(k), E0 are the expectation signs with respect to Pn(k), P 0, respectively. The reader will
easily derive this property from Theorem 3.1 (ii) by using Taylor’s series and observing that
E
\left( \infty \int
0
e - tf(t, \sanst t, xt) dt
\right) n =
= E
\infty \int
0
. . .
\infty \int
0
e - t1f(t1, \sanst t1 , xt1) . . . e
- tnf(tn, \sanst tn , xtn) dt1 . . . dtn.
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ON TIME INHOMOGENEOUS STOCHASTIC ITÔ EQUATIONS WITH DRIFT IN Ld+1 1253
What remains is to prove that for (\sansT ,\scrN t) and ([0,\infty ),\scrN t+) the conditions 2 and 3 are satisfied
of the definition of Markov system in [4]. This is done by almost literally repeating the corresponding
part of the proof of Theorem 3 of [4]. One need only replace there xt with (\sanst t, xt).
The theorem is proved.
Acknowledgment. The author is sincerely grateful to A. I. Nazarov, who pointed out an error
in the first version of the article, to Hongjie Dong and Doyoon Kim for spotting several misprints
bordering with errors, and to Xicheng Zhang whose comment allowed the author to avoid an incorrect
statement.
References
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Received 20.08.20
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 9
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| id | umjimathkievua-article-6280 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| resource_txt_mv | umjimathkievua/19/411233bb830ccc8d6e086effdf599c19.pdf |
| spelling | umjimathkievua-article-62802022-03-26T11:02:07Z On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$ О неоднородных по времени стохастических уравнениях Ито со сносом в $L_{d+1}$ On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$ Krylov, N. V. Krylov, N. V. Krylov, N. V. Itˆo’s equations with singular drift Markov diffusion processes Itˆo’s equations with singular drift Markov diffusion processes UDC 519.21 We prove the solvability of Itô stochastic equations with uniformly nondegenerate bounded measurable diffusion and drift in $L_{d+1}(R^{d+1}).$Actually, the powers of summability of the drift in $x$ and $t$ could be different. Our results seem to be new even if the diffusion is constant. The method of proving the solvability belongs to A. V. Skorokhod.Weak uniqueness of solutions is an open problem even if the diffusion is constant. Мы доказываем разрешимость стохастических уравнений Ито с равномерно невырожденной и ограниченной матрицей диффузии и со сносом в L_{d+1}(R^{d+1}). На самом деле, степени суммируемости сноса по x и t могут быть различными. Этот результат является новым даже если диффузия постоянна. Метод, который мы используем, принадлежит А.В. Скороходу. Вопрос о слабой единственности решений открыт даже если диффузия постоянна. УДК 519.21 Про неоднорiднi за часом стохастичнi рiвняння Іто з переносом в $L_{d+1}$ Доведено розв'язність стохастичних рівнянь Іто з рівномірно невиродженою та обмеженою матрицею дифузії і з переносом в $L_{d+1}(R^{d+1}).$ Справді, показники інтегровності по $x$ і $t$можуть відрізнятися. Цей результат є новим навіть коли дифузія стала. Метод, який ми використовуємо, належить А. В. Скороходу. Питання про слабку єдиність є відкритим навіть коли дифузія стала. Institute of Mathematics, NAS of Ukraine 2020-09-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6280 10.37863/umzh.v72i9.6280 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 9 (2020); 1232-1253 Український математичний журнал; Том 72 № 9 (2020); 1232-1253 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/6280/8753 Copyright (c) 2020 Микола Володимирович Крилов |
| spellingShingle | Krylov, N. V. Krylov, N. V. Krylov, N. V. On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$ |
| title | On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$ |
| title_alt | О неоднородных по времени стохастических уравнениях Ито со сносом в $L_{d+1}$ On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$ |
| title_full | On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$ |
| title_fullStr | On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$ |
| title_full_unstemmed | On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$ |
| title_short | On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$ |
| title_sort | on time inhomogeneous stochastic itô equations with drift in $l_{d+1}$ |
| topic_facet | Itˆo’s equations with singular drift Markov diffusion processes Itˆo’s equations with singular drift Markov diffusion processes |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6280 |
| work_keys_str_mv | AT krylovnv ontimeinhomogeneousstochasticitoequationswithdriftinld1 AT krylovnv ontimeinhomogeneousstochasticitoequationswithdriftinld1 AT krylovnv ontimeinhomogeneousstochasticitoequationswithdriftinld1 AT krylovnv oneodnorodnyhpovremenistohastičeskihuravneniâhitososnosomvld1 AT krylovnv oneodnorodnyhpovremenistohastičeskihuravneniâhitososnosomvld1 AT krylovnv oneodnorodnyhpovremenistohastičeskihuravneniâhitososnosomvld1 |