A remark on conditions that a diffusion in the natural scale is a martingale

0.1 MB·

by  Yuuki Shimizu

A remark on conditions that a diffusion in the
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1 natural scale is a martingale
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2

n
a Yuuki Shimizu and Fumihiko Nakano ∗
J

1
January 24, 2017
2

]
R
P
. Abstract
h
t
a Weconsideradiffusionprocesses{X }onanintervalinthenatural
t
m
scale. Some results are known under which {X } is a martingale, and
t
[
we give simple and analytic proofs for them.

1
v
Mathematics Subject Classification (2010): 60J60, 60G44
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9
5 1 Introduction
0
.
1
Let −∞ ≤ l < l ≤ ∞ and let m be a Borel measure with
0 − +
7 supp m = (l ,l ). We denote by {X } ,{P } the mini-
1 − + t t≥0 x x∈(l−,l+)
: mal diffusion process on (l−,l+) with thne speed measure m aond the scale
v
function s(x) = x. It is well known that a local martingale {X } is a
i t
X
martingale if and only if {X : T is a stopping time with T ≤ t} is
T
r
a uniformly integrable for any t ≥ 0. Here our aim is to have more explicit
condition for the one-dimensional diffusions in the natural scale. If |l | < ∞,
±
{X } is bounded so that it is a martingale. If l = −∞, l < ∞, this can
t − +
be reduced to the case of l < ∞, l = ∞ by replacing X by −X . Hence
− + t t
it suffices to consider the following two cases.
∗
DepartmentofMathematics,GakushuinUniversity,1-5-1,Mejiro,Toshima-ku,Tokyo,
171-8588,Japan.
1
Case I : −∞ < l , l = +∞, Case II : l = −∞, l = +∞.
− + − +
LetP(l ,l )bethesetofBorelmeasureson(l ,l ), andforµ ∈ P(l ,l )
− + − + − +
let P (·) := P (·)µ(dx). According to Lemma 4.1 ([1], Lemma 2),
µ (l−,l+) x
{Xτ} is a P -martingale for some µ ∈ P(l ,l ) with |x|µ(dx) < ∞ if
t µR − + (l−,l+)
and only if {Xτ} is P -martingale for any x ∈ (l ,l ). We further set
t x − + R
τ := inf{t ≥ 0|X = a}, τ := lim τ , τ := τ ∧τ
a t ± a + −
a→l±
Xτ := X .
t t∧τ
Kotani [1] showed the following theorem.
Theorem 1.1 [1] {Xτ} is a P -martingale for any x ∈ (l ,l ) if and only
t x − +
if
Case I :
xm(dx) = ∞, r ∈ (l ,∞)
−
Z[r,l+)
Case II :
xm(dx) = ∞ and |x|m(dx) = ∞, r ∈ (−∞,∞).
Z[r,l+) Z(l−,r]
By Feller’s criterion, P (τ = ∞) = 1 if |l | = ∞, ♯ = ±∞. Thus Theorem
x ♯ ♯
1.1 implies that {Xτ} is a martingale if and only if the boundaries at infinity
t
are natural. Hulley, Platen [2] derived another condition. Let
d2
Lf := f
dmdx
be the generator of {X } and for λ > 0 let f (resp. f ) be the positive
t − +
increasing (resp. positive decreasing) solution to the equation Lf = λf,
which are unique up to constants unless the boundary is regular.
Theorem 1.2 [2] {Xτ} is a P -martingale for any x ∈ (l ,l ) if and only
t x − +
if
Case I :
lim f′ (z) = ∞
z→∞ −
Case II :
lim f′ (z) = ∞ and lim f′ (z) = −∞.
z→∞ − z→−∞ +
2
Gushchin, Urusov, and Zervos [3] derived a condition that {Xτ} is a sub-
t
martingale or a supermartingale.
Theorem 1.3 [3] {Xτ} is a P -submartingale if and only if ∞xm(dx) =
t x r
∞, r ∈ (l ,l ).
− + R
By [2] Proposition 3.16, 3.17, this condition is equivalent to lim f′ (t) =
t→∞ −
∞. Together with Theorem 1.3 we thus have
Theorem 1.4 {Xτ} is a P -submartingale if and only if lim f′ (t) = ∞.
t x t→∞ −
Moreover in [3], they further derived a condition in Case I such that {Xτ}
t
is a strict P supermartingale, that is, {Xτ} is a P -supermartingale but is
x t x
not a P -martingale.
x
Theorem 1.5 [3] Let −∞ < l , l = ∞. Then {Xτ−} is a strict P -
− + t x
supermartingale if and only if
lim E [X ] = l
t→∞ x t∧τ− −
for any x ∈ (l ,l ).
− +
We believe that Theorem 1.5 is also true for l = −∞. The goal of this
−
paper is :
(1) To give a simple analytic proof of Theorem 1.4 without using the results
in [2]. We note that the proofs of Proposition 3.16, 3.17 in [2] is more or less
probabilistic using Tanaka’s formula.
(2) To give a simple analytic proof of Theorem 1.5 ; the original proof of
that in [3] is done by embedding {X } into the geometric Brownian motion
t
on the torus.
The rest of this paper is organized as follows. In Section 2(resp. Section
3), we give a proof of Theorem 1.4 (resp. Theorem 1.5). In Appendix, we
prepare some tools for these proofs.
2 A proof of Theorem 1.4
In Case I, the statement follows from Theorem 1.2, for {Xτ−} is always a
t
P -supermartingale being bounded frombelow. Henceforth we consider Case
x
II.
3
Suppose {X } is a P -submartingale and let z < x. Then {Xτz} is
t x t
bounded from blow so that it is a P -martingale. For λ > 0, let fz (resp. fz)
x − +
be the positive increasing (resp. positive decreasing) solution to the equation
Lf = λf such that fz(z) = 0. Then we have
−
f (z)
fz(x) = f (x)− − f (x), fz(x) = f (x).
− − f (z) + + +
+
Since f′ is increasing, we have
+
f (z) f (z)
f′ (x) = fz′(x)+ − f′ (x) ≥ fz′(x)+ − f′ (z), x ∈ (z,∞).
− − f (z) + − f (z) +
+ +
Applying Theorem 1.2 to {Xτz} yields lim fz′(t) = ∞ and thus
t t→∞ −
lim f′ (t) = ∞.
t→∞ −
Conversely, suppose lim f′ (t) = ∞ and let z < x. Then
t→∞ −
∞ z z f (x)
lim z e−λtP (τ < t)dt = lim E [e−λτz] = lim −
x z x
z→∞ Z0 z→∞ λ z→∞ λf−(z)
f (x) 1
−
= lim = 0
z→∞ λ f′ (z)
−
where we used Lemma 4.3 and l’Hospital’s rule. By Fatou’s lemma,
∞
e−λtliminfzP (τ < t)dt = 0.
x z
Z0 z→∞
Hence liminf zP (τ < t) = 0 so that we can find a sequence {z } ⊂
z→∞ x z n
(x,∞) with lim z = ∞ such that
n→∞ n
lim z P (τ < t) = 0.
n→∞ n x zn
On the other hand {Xτzn} is a P -submartingale being bounded from above
t x
and
x ≤ E [X ] = z P (τ < t)+E [X ;τ ≥ t].
x t∧τzn n x zn x t zn
Since lim P (τ ≥ t) = 1, x ≤ E [X ]. Markov property implies {X } is
n→∞ x zn x t t
a P -submartingale.
x
4
3 A proof of Theorem 1.5
Without losing generality, we may suppose l < 0. For λ > 0, let f (resp.
− −
f ) be the positive increasing (resp. positive decreasing) solution to the
+
equation Lf = λf such that f (l ) = 0. Let G be Green’s function of L :
− −
1f (y)f (x) (y < x)
G(x,y,λ) := h − +
1f (x)f (y) (x ≤ y)
( h − +
h := f (x)f′ (x)−f (x)f′ (x).
+ − − +
Then we have
∞ ∞
G(x,y,λ)(y−l )m(dy) = E e−λt(X −l )dt . (3.1)
− x t∧τ− −
Zl− hZ0 i
Let α := lim f (t). Then f′ ∈ L1(a,∞) for a ∈ (l ,∞) and
+ t→∞ + + −
∞
f (x) = α − f′ (y)dy.
+ + +
x
Z
Therefore lim f′ (x) = 0. The equation Lf = λf yields
x→∞ + + +
∞
f′ (x) = −λ f (y)m(dy)
+ +
x
Z
∞
f (x) = α +λ (y −x)f (y)m(dy)
+ + +
x
Z
so that we have
∞
λ yf (y)m(dy) = f (x)−α −xf′ (x).
+ + + +
x
Z
Similarly,
y
f′ (y) = f′ (l )+λ f (z)m(dz)
− − − −
Zl−
x
f (x) = f′ (l )(x−l )+λ (x−y)f (y)m(dy)
− − − − −
Zl−
x x
λ yf (y)m(dy) = f′ (l )(x−l )−f (x)+λx f (y)m(dy).
− − − − − −
Zl− Zl−
Substituting them into (3.1) yields
∞ x−l α f (x)
e−λtE [X −l ]dt = − − + − . (3.2)
x t∧τ− − λ λh
Z0
5
We note that (3.2) and Lemma 4.1 also proves Theorem 1.1 in Case I.
Suppose {Xτ−} is a strict P -supermartingale. The discussion above implies
t x
α > 0. We shall show below that
+
α f (x)
lim x−l − + − = 0. (3.3)
−
λ→0 h
(cid:16) (cid:17)
Let φ, ψ be the solution to Lf = λf with the initial condition
φ(0) = 1, φ′(0) = 0
ψ(0) = 0, ψ′(0) = 1.
Then f satisfy
±
φ(x) φ(x)
f (x) = φ(x)− lim ψ(x), f (x) = φ(x)− lim ψ(x).
+ x→∞ ψ(x)! − x→l− ψ(x)!
ψ, ψ can be composed by the method of successive approximation :
∞ x
φ(x) = 1+ λnφ (x), φ (x) = 1, φ (x) = (x−y)φ (y)m(dy)
n 0 n n−1
n=1 Z0
X
∞ x
ψ(x) = x+ λnψ (x), ψ (x) = x, ψ (x) = (x−y)ψ (x)m(dy)
n 0 n n−1
n=1 Z0
X
which is convergent locally uniformly w.r.t. λ [4] which yields
limφ(x) = 1, limφ′(x) = 0, limψ(x) = x, limψ′(x) = 1.
λ→0 λ→0 λ→0 λ→0
Moreover
ψ(x) 0 1 0
lim − lim = lim dx = dx = −l
−
λ→0 x→l− φ(x)! λ→0 Zl− (φ(x))2 ! Zl−
implies
x 1
limf (x) = 1− , limf′ (x) = − .
λ→0 − l− λ→0 − l−
On the other hand, by α > 0 and by Lemma 4.2, we have ∞xm(dx) < ∞,
+ r
r ∈ (l ,∞) so that we can find g satisfying
− R
∞
g(x) = 1+λ (y −x)g(y)m(dy)
x
Z
6
bysuccessive approximation. Using α > 0, lim f′ (t) = 0, lim g(t) =
+ t→∞ + t→∞
1 and lim g′(t) = 0, we have
t→∞
f (x)g′(x)−f′ (x)g(x) = 0
+ +
which implies f (x) = Cg(x) for some positive constant C. Because
+
lim g(x) = 1, lim g′(x) = 0,
λ→0 λ→0
limf (x) = C, limf′ (x) = 0.
+ +
λ→0 λ→0
Therefore
α f (x) C 1− x
lim x−l − + − = x−l − l− = 0
λ→0 − h ! − C · −1 (cid:16)−0· 1(cid:17)− x
l− l−
(cid:16) (cid:17) (cid:16) (cid:17)
proving (3.3). Since X is a supermartingale, f(t) := E [X − l ] ∈
t∧τ− x t∧τ− −
C1[0,∞) is monotone decreasing which shows that lim f(t) exists and
t→∞
f′ ∈ L1(0,∞). Thus by (3.2) and Lemma 4.4
lim E [X −l ] = 0.
t→∞ x t∧τ− −
Conversely, suppose that lim E [X −l ] = 0. Then
x t∧τ− −
t→∞
∞
limλ e−λtE [X −l ]dt = 0
x t∧τ− −
λ→0 Z0
which implies α > 0 since otherwise it would contradict to (3.2), (3.3).
+
Therefore {Xτ−} is not a martingale.
t
4 Appendix
Lemma 4.1 (Lemma 2 in [1])
Suppose {X } is a P -martingale for some µ ∈ P(l ,∞). Then for any
t∧τ− µ −
t ≥ 0, x ∈ (l ,∞),
−
E [X ] = x. (4.1)
x t∧τ−
Conversely, if (4.1) is valid, then {X } is a P -martingale for any µ ∈
t∧τ− µ
P(l ,∞) with ∞|x|µ(dx) < ∞.
− l−
R
7
Lemma 4.2 Let λ > 0 and let f be the positive decreasing solution to
+
Lf = λf with α := lim f (x). Then the following three conditions are
+ +
x→∞
equivalent.
(1) α = 0
+
∞
(2) ym(dy) = ∞
a
Z
∞
(3) λ (y −x)f (y)m(dy) = f (x).
+ +
x
Z
Lemma 4.3 Let f be the ones defined in the proof of Theorem 1.5. Then
±
f (x)
E [e−λτa] = + , a < x
x
f (a)
+
f (x)
E [e−λτb : τ < τ ] = − , −∞ ≤ l < x < b.
x b − −
f (b)
−
Lemma 4.4 Suppose f ∈ C1[0,∞) and f′ ∈ L1(0,∞). Then
(1) lim f(t) exists, and
t→∞
(2) lim f(t) = λlim ∞e−λtf(t)dt.
t→∞ λ↓0 0
R
References
[1] S. Kotani, : On a condition that one-dimensional diffusion processes are
martingales, In memoriam Paul-Andr’e Meyer: S’eminaire de Probabilit’es
XXXIX, Lecture Notes in Mathematics, 1874(2006), pp.149 - 156, Springer.
[2] H. Hulley, E. Platen, : A visual classification of local martingales, Quan-
titative Finance Research Centre, Research Paper 238(2008), University of
Technology, Sydney
[3] A. Gushchin, M. Urusov, and M. Zervos, : On the submartingale / super-
martingale property of diffusions in natural scale, Proceedings of the Steklov
Institute of Mathematics Vol. 287(2014), pp 122-132.
[4] K. Ito, : Essentials of stochastic process, Translations of mathematical mono-
graphs, vol. 231, AMS.
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