Transcript Document

On some integral estimates for solutions
of SDEs driven by symmetric stable processes
Vladimir P. KURENOK
Department of Electrical and Systems Engineering,
Washington University in St. Louis,
One Brookings Drive, St. Louis, MO 63130-4899, USA
e-mail: [email protected]
Workshop on Stochastic Analysis - Jena 2015
Introduction
We assume Z to be a one-dimensional symmetric stable process of
index  ∈(0, 2] and X to be of the form
𝑋𝑡 = 𝑥0 +
𝑡
𝑏
0
𝑋𝑠− 𝑑𝑍𝑠 +
𝑡
𝑎
0
𝑋𝑠 𝑑𝑠 , 𝑥0 ∈ ℝ.
(A)
For a measurable function 𝑓 ∶ ℝ → 0, ∞ , define
𝑚
1
𝑝
𝑓 𝑝,𝑚 =
( 𝑓 𝑦 𝑑𝑦) 𝑝
−𝑚
to be its 𝐿𝑝 -norm on the interval [-m, m], m ∈ ℕ , p  1 , and let
𝜏𝑚 (X) = inf 𝑡 ≥ 0 ∶ 𝑋𝑡 ≥ 𝑚 .
The estimates of the form
𝑡∧𝜏𝑚 (𝑋)
𝑬
𝑒 −𝜙𝑠 𝜓𝑠 𝑓 𝑋𝑠 𝑑𝑠 ≤ 𝑁 𝑓
𝑝,𝑚
(1)
0
where (𝜙𝑡 ) and (𝜓𝑡 ) are nonnegative processes are called (local) Krylov's estimates.
 N. V. Krylov (Controlled Diffusion Processes, Springer, New York, 1980) proved
them first for diffusion processes, that is when Z is a Brownian motion
process W (α = 2).
 Krylov's estimates for diffusion processes with jumps (W ≠ 0):
- S. Anulova and H. Pragarauskas: Liet. Math. Rinkinys, XVII (1977), 5-26;
- J. P. Lepeltier and B. Marchal: Annales IHP, Vol. 12, No. 1 (1976), 43-103;
- A.V. Melnikov: Stochastics and Stoch. Rep., 10 (1983), 81-102.
 Krylov's estimates for purely discontinuous processes:
- H. Pragarauskas, In: "Probab. Theory and Math. Statist.", B.Grigelionis et al.
(eds.), 579-588, VSP, Utrecht/TEV, Vilnius, 1999 (1 < α < 2 and a = 0);
- V. P. Kurenok: Transactions AMS, Vol. 360, No. 2, 925-938, 2008 (1 < α < 2);
- Xicheng Zhang: AIHP, Vol. 49, no.4, 1057-1079, 2013 (b = 1 and
multidimensional drift a).
 Our goal here is to discuss some versions of Krylov's
estimates for equation (A) in 𝑳𝟐 -norm and their use to prove
the existence of weak solutions for corresponding SDEs.
We recall:
If Z is a symmetric stable process with index 𝛼 ∈ (0, 2 , then its characteristic
function is given by
𝑬 𝑖𝜉𝑍𝑡 = exp −𝑡 𝜉
𝛼
, 𝑡 > 0, 𝜉 ∈ ℝ.
For α ∈ 0,2 , Z is a purely discontinuous Markov process that can be
described by its infinitisimal generator given by
ℒ𝑔 𝑥 =
𝑔 𝑥+𝑧 −𝑔 𝑥 −1
where 𝑐𝛼 is a constant.
∞
𝑧 <1
𝑔′
𝑥 𝑧
𝑐𝛼
𝑧
1+𝛼
𝑑𝑧
For 𝑔 ∈ 𝐿1 ℝ , define
𝑒 𝑖𝑧𝑥 𝑔 𝓏 𝑑𝓏
𝐹𝑔(𝑥) ≔
−∞
to be the Fourier transform of 𝑔. For any function 𝑔 ∈ 𝐶 2 ℝ such that ℒ𝑔 ∈ 𝐿1 , it
holds then
𝐹 ℒ𝑔 𝑥 = − 𝑥 𝛼 𝐹𝑔 𝑥
and
𝐹𝑔′ 𝑥 = 𝑖𝑥𝐹𝑔 𝑥 .
PART I: Estimates in the case of 1<α<2
We assume that:
 𝐾 > 0 is a constant;
 𝑓
is a nonnegative, measurable function such that 𝑓 ∈ 𝐶0 ∞ ℝ where
𝐶0 ∞ ℝ denotes the class of all infinitely differentiable real valued functions
with compact support defined on ℝ;
 𝑍 is a symmetric stable process of index α ∈ 1,2 adapted to a filtration 𝔽;
 D denotes the class of all 𝔽 -predictable processes (𝛾𝑡 ) such that 𝛾𝑡 ≤ 𝐾.
For a process 𝑋 𝛾 (controlled process) of the form
𝑑𝑋𝑡
𝛾
= 𝑑𝑍𝑡 + 𝛾𝑡 𝑑𝑡
and any 𝜆 > 0, define the value function 𝑣 𝑥 , 𝑥 ∈ ℝ, by
∞
𝑒 −𝜆𝑡 𝑓 𝑥 + 𝑋𝑡
𝑣 𝑥 = sup 𝑬
𝛾∈𝐷
0
𝛾
𝑑𝑡.
 By standard arguments (N.V. Krylov and H. Pragarauskas: "Traditional
derivation of Bellman equation for general controlled stochastic processes", Lit.
Math. Rink., 21, 146-152, 1982), the function v will satisfy the following Bellman
equation (  is deterministic!)
𝑠𝑢𝑝 ℒ𝑣 𝑥 − 𝜆𝑣 𝑥 + 𝛾𝑣 ′ 𝑥 + 𝑓 𝑥
=0
𝛾≤𝐾
which holds a.e. in ℝ.
 The above equation is then equivalent to
ℒ𝑣 − 𝜆𝑣 + 𝐾 𝑣 ′ + 𝑓 = 0.
(2)
Lemma 1.
𝑣 𝑥 ≤𝑁 𝑓
For all 𝑥 ∈ ℝ, it holds
∞
2
1
2
(3)
𝑓 2 𝑦 𝑑𝑦
≔𝑁
−∞
where the constant N depends on K and α only.
 The proof uses Fourier transform technique combined with the Parseval's
identity. As the result, it holds for all 𝑦 ∈ ℝ and 𝜆 ≥ 𝜇
∞
𝑣 2 (𝑦) ≤ 𝑁
𝑓 2 (𝑥)𝑑𝑥
−∞
where
1
𝑁= 2
𝜋
and µ > 0 is such that
for all 𝑥 ∈ ℝ.
∞
𝑥
𝛼
+𝜆
−2 𝑑𝑥
𝑥
𝛼
+𝜇
2
<∞
−∞
≥ 4𝐾 2 𝑥
2
Theorem 1.
𝑎 𝑥
Suppose X is a solution of the equation (A) with 𝛼 ∈ 1,2 and
≤𝐾𝑏 𝑥
𝛼
(4)
for all 𝑥 ∈ ℝ.
Then, for any 𝑥 ∈ ℝ, 𝜆 ≥ 𝜇 , and any measurable function 𝑓 ∶ ℝ → 0, ∞ , it holds
∞
𝐸
0
where 𝜓𝑡 =
𝑡
0
𝑒 −𝜆𝜓𝑢 𝑏 𝑋𝑢
𝑏 𝑋𝑠
𝛼 𝑑𝑠
𝛼𝑓
𝑥 + 𝑋𝑢 𝑑𝑢 ≤ 𝑁 𝑓
2
(5)
and the constant N depends on K and α only.
 Accordingly, a local version holds as well:
𝑡∧𝜏𝑚 𝑋
𝐸
𝑏 𝛼 𝑓 𝑋𝑢 𝑑𝑢 ≤
0
𝑁 𝑓
where N depends then on K, α, m, and t.
 The proof uses Ito's formula and Lemma 1.
2,𝑚
(6)
Application of the estimates in case 1 < α < 2:
Theorem 2. Assume that Z is a symmetric stable process of index 1 < α < 2
and there exist positive constants δ1 and δ2 such that
𝑏 𝑥
≥ 𝛿1 ,
𝑎 𝑥 + |𝑏 𝑥 |≤ 𝛿2
for all 𝑥 ∈ ℝ.
Then, for any 𝑥0 ∈ ℝ, there exists a solution of the equation (A).
 In particular, if 𝑎 𝑥
≤ 𝛿2 for all 𝑥 ∈ ℝ, then the equation (A) with b = 1 has a
solution for any 𝑥0 ∈ ℝ.
 To compare: N. I. Portenko (Random Oper. and Stoch. Equ., Vol. 2, No. 3, 1994)
proved the existence of solutions for the equation (A) with b = 1 if there is 𝑝 >
𝛼−1
−1
such that 𝑎 ∈ 𝐿𝑝 ℝ .
 Sufficient conditions in the Theorem 2 are different from those of Portenko and the
proof method used by Portenko is a different one from using the Krylov's estimates.
PART II:
Integral estimates in the case of 0 < α < 2
We assume:
 Z is a symmetric stable process of index 0 < α < 2;
 𝑓 is a nonnegative, measurable function such that 𝑓 ∈ 𝐶0 ∞ ℝ ;
 𝛾 is a fixed constant.
For  > 0, we consider the linear fractional ODE:
ℒ𝑣 𝑥 − 𝜆𝑣 𝑥 + 𝛾𝑣 ′ 𝑥 + 𝑓 𝑥 = 0.
Lemma 2.
(7)
Let 𝛾 ≠ 0. Then, for all 𝑥 ∈ ℝ and 0 < α < 2, it holds
𝑣 𝑥 ≤𝑁 𝑓
2,
where the constant 𝑁 depends on α and 𝛾.
If 𝛾 = 0, then the estimate holds only for all 1/2 < α < 2.
 The proof uses again Fourier transform technique combined with the Parseval
identity. As the result, one has for all 𝑥 ∈ ℝ and > 0:
∞
𝑣 2 (𝑥)
where
𝑁
≤ 2
4𝜋
∞
𝑁≔
−∞
𝑓 2 𝑧 𝑑𝑧
−∞
𝑑𝜉
𝜉 𝛼 + 𝜆 2 + 𝛾2𝜉2
One sees that N < ∞ for 𝛾 ≠ 0 and 0 < α < 2 while N < ∞ in the case of
𝛾 = 0 and 1/2 < α < 2.
 Addition of the drift term 𝛾𝑡 (𝛾 ≠ 0) to the process Z allows to expand the
analytical estimate of the sup-norm of the function 𝑣 𝑥 through the 𝐿2 -norm of
the function 𝑓 𝑥 from the range 1/2 < α < 2 to the entire range 0 < α < 2 of the
index. In this sense the addition of the drift plays a "regularizing effect" on the
estimates.
Assume now that X is of the form
𝑑𝑋𝑡 = 𝑏 𝑋𝑡− 𝑑𝑍𝑡 + 𝛾 𝑏
𝛼
𝑋𝑡 𝑑𝑡,
(B)
where 𝑡 ≥ 0 and 𝑋0 = 𝑥0 ∈ ℝ .
One has then the following (local) Krylov's estimate:
Theorem 3. Let 𝛾 ≠ 0 and X be of the form (B). Then, for all 0 < α < 2, 𝑡 ≥ 0,
m ∈ ℕ, and any measurable function f: ℝ → 0, ∞ , one has
𝐄
𝑡∧𝜏𝑚 𝑋
0
𝑏 𝛼 (𝑋𝑠 )𝑓 𝑋𝑠 𝑑𝑠 ≤ 𝑁 𝑓
2,𝑚
where the constant N depends on α, , m, and t.
If 𝛾 = 0 , then the estimate is true for 1/2 < α < 2 only.
(8)
Application of the estimates in case 0 < α < 2:
Theorem 4. Let 0 < α < 2 and assume that there exists a constant
𝐾 > 0 such that 𝑏 𝑥
≤ 𝐾 for all 𝑥 ∈ ℝ and 𝑏
−2𝛼
∈ 𝐿𝑙𝑜𝑐 ℝ . Then,
for any 𝑥0 ∈ ℝ, there exists a solution of the equation (B) for any 𝛾 ≠
0 and 𝛾 = 0.
 𝐿𝑙𝑜𝑐 ℝ denotes the class of locally integrable functions on ℝ.
Proof idea:
For any fixed 𝜸 ≠ 𝟎 :
 One chooses a sequence of Lipshitz continuous and uniformly bounded
functions 𝑏𝑛 , 𝑛 ≥ 1, converging to b so that, for any fixed n and the given
process Z, there is a solution of the equation
𝑡
𝑋𝑡 = 𝑥0 +
0
𝑡
𝑏𝑛 𝑋𝑠− 𝑑𝑍𝑠 + 𝛾
0
𝑏𝑛
𝛼
𝑋𝑠 𝑑𝑠,
𝑡 ≥ 0.
 One proves then that the sequence of processes 𝑋 𝑛 , 𝑍 , 𝑛 = 1,2, … is tight and
converges in weak sense so that there is a limiting process 𝑋, 𝑍 with 𝑍 being a
symmetric stable process of the same index α.
 One shows then - with help of Krylov's estimates – that the process 𝑋, 𝑍
satisfies the equation (B).
For 𝜸 = 𝟎:
 One chooses a sequence of real numbers 𝛾𝑛 ≠ 0, 𝑛 = 1, 2, … such that lim 𝛾𝑛 = 0.
𝑛→∞
For any fixed 𝑛 = 1, 2, … . , there is a solution 𝑋 𝑛 , 𝑍 𝑛 of the equation
𝑑𝑋𝑡 = 𝑏 𝑋𝑡− 𝑑𝑍𝑡 + 𝛾𝑛 𝑏
𝛼
𝑋𝑡 𝑑𝑡,
𝑋 = 𝑥0 ∈ ℝ (9)
where 𝑍 𝑛 is a symmetric stable process of index 0 < α < 2.
 One proves as before that the sequence of processes 𝑋 𝑛 , 𝑍 𝑛 , 𝑛 = 1,2, … is tight
and converges in weak sense so that there is a limiting process 𝑋, 𝑍 with 𝑍
being a symmetric stable process of the same index α.
 One shows then that the limiting process 𝑋, 𝑍 satisfies the equation
𝑑𝑋𝑡 = 𝑏 𝑋𝑡− 𝑑𝑍𝑡 ,
(𝐶)
where 𝑡 ≥ 0 and 𝑋0 = 𝑥0 ∈ ℝ .
 In the last step, one uses the Skorokhod's lemma for the convergence of stable
integrals instead of Krylov's estimates since the coefficient b in (9) does not
depend on n.
Discussion of the existence result in case 𝜸 = 𝟎 :
 𝛼 = 2 (H.J. Engelbert and W. Schmidt): equation (C) has a non-trivial
solution for any 𝑥0 ∈ ℝ if and only if 𝑏 −2 ∈ 𝐿𝑙𝑜𝑐 ;
 𝛼 ∈ 1,2 (P.A. Zanzotto, 2002): the same result as for 𝛼 = 2 where the
condition 𝑏 −2 ∈ 𝐿𝑙𝑜𝑐 was replaced by the condition 𝑏
−𝛼 ∈
𝐿𝑙𝑜𝑐 ;
 0 < α ≤ 1: there are only sufficient conditions, for example:
1) There exists a number 𝛿 > 1 such that
𝑏
−𝛿 ∈
𝐿𝑙𝑜𝑐 ;
2) There exists a number 𝑈 > 0 with 𝑙 𝐵𝑈 < ∞ such that
𝐵𝑈 = 𝑦 ∈ ℝ: 𝑏 𝑦
>𝑈
where l is the Lebesgue measure onℝ (P.A. Zanzotto, 2002).
 The existence conditions of Theorem 4 are different from those of Zanzotto.
Moreover, consider the function
𝑏 𝑥 =
−2, 𝑥 < 1
𝑥, 𝑥 ∈ −1,1
2, 𝑥 > 1
in the case of 0 < α < 1/2 .
 It does not satisfy the existence conditions of Zanzotto but does satisfy the
assumptions of Theorem 4 so that there is a solution of the equation (C).
 Besides, Theorem 4 also improves the local integrability condition on the
coefficient b found by Zanzotto for the case of 0 < α < 1/2 in general.
 Contrary to the condition required by Zanzotto, Theorem 4 assumes only
𝑏
−𝛿 ∈
𝐿𝑙𝑜𝑐 where 𝛿 = 2𝛼 < 1 for 0 < α < 1/2 .
Summary
 One starts with an integro-differential equation, for example:
ℒ𝑣 − 𝜆𝑣 + 𝐾 𝑣 ′ + 𝑓 = 0
where f is a given function.
 One proves that
𝑠𝑢𝑝 𝑣 𝑥 ≤ 𝑁 𝑓
 One derives a Krylov type estimate
𝑡
𝑬 𝑒 −𝜑𝑠 𝜓𝑠 𝑓 𝑠, 𝑋𝑠
0
𝑝
𝑑𝑠 ≤ 𝑁 𝑓
𝑝
where process X is related to the integro-differential equation.
 The estimates can be used to prove the existence of solutions of related SDEs
with measurable coefficients.