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The density asymptotics associated with a stochastic differential equation
Mathl. Comput. Modelling Vol. 20, No. 4/5, pp. 167-171, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177194~$7.00 + 0.00
08957177(94)00128-6
The Density Asymptotics Associated with a Stochastic Differential Equation I. M. DAVIES Department of Mathematics, University College of Swansea Singleton Park, Swansea, SA2 8PP, UK Abstract-We present an initial foray into the use of Laplace Asymptotic Expansions to study the development of the density associated with a stochastic differential equation. We only consider the simplest interesting case herein and by using a variety of viewpoints, we obtain the behaviour of the density for small values of the diffusion constant and time. We demonstrate that the leading order term is indeed the standard one.
1. INTRODUCTION In previous related
work, we have studied
to quantum
stochastic
mechanics.
mechanics
In doing
and Laplace
structure
of the drift in stochastic
derivation
of the most commonly
the study of stochastic view to good stochastic
effect.
differential
problems
in quantum
so, we have touched
asymptotic mechanics
expansions.
or in areas mathematically
upon
semi-classical
Most recently,
equations
Lagrangian
and classical
mechanical
as it were, by concentrating
and obtain the nature of density in terms of b and u without reference to any background our study, a potential
which, to leading order, behaves as classical mechanics In the second section, we move from the stochastic in terms of an expectation.
the expectation
THE
The stochastic
expects.
differential equation to an expression coefficient
fi
required
for the
to rewrite
and gives its structure,
while
on the nature of the leading order term.
STOCHASTIC
differential
structure.
function can be constructed
The third section presents the theorem
in terms of powers of the diffusion
the fourth section concentrates 2.
point of
solely on the
equation
What we find is that, while the drift b dominates
density
the
and refined the
[2,3]. In this, we have approached
from a quantum
Herein, we start from scratch,
mechanics,
we have studied
[l] for a wide class of Hamiltonians
used stochastic
differential
mechanics
DIFFERENTIAL
EQUATION
equation
dXt = b(X,) dt + fidBt, where Bt is Brownian
motion
on lR with the standard
filtration,
(cf. [4]). Th e d ensity, if it exists, satisfies the partial differential
has a solution
for b Lipschitz
equation
Typeset 167
by d&-w
I. M. DAVIES
168
and the invariant density, if it exists, is simply
In our discussion, we only require the finiteness of
fo =exp
i
)
bjb(u)du .
One may quickly show that fc satisfies the second order differential equation
u2 d2fo -7yp+v.f0=0, where V(x) = i(b” + vb’), b’ denoting 2,
we see that f<‘p
(cf. [5]). By noting that
satisfies a(&%) at
The Feynman-Kac formula [6] now gives us
p(z,t) =.fo(z)E
[w{-tl
V(x + 42%)
ds}(f&o)
(x + J;&)] ,
where PO(Z) = p(z,O), lE d enoting expectation when the r.h.s. of the above is well defined. Let us now rewrite the above in some detail to observe the dependence on u. x+J;Bt ;hZ(r
+
ds - ; /
fill,)
exp
-{
b(u) du
I 1 2
J b’(z+d%)ds t o
>
PI++ 4-t)
1.
This may be written in the form p(z,t)
= E [exp
where F(z) = - & ib2(z + z(s)) ds - Jz+z(t) b(u) du. We can expand this expression for p(z, t) in powers of u by use of a Laplace asymptotic expansion. 3.
THE
LAPLACE
ASYMPTOTIC
EXPANSION
There is a large body of work on Laplace asymptotic expansions notably [7-lo]. We will not discuss the relative merits of the different approaches to the problem herein. We will use a result due to [ll]. There is a little introductory notation required namely: Cc[O, t] is the space of continuous functions z : [0, t] -+ R with z(0) = 0, CG[O,t] is that subset of Cc[O, t] comprising of differentiable z, D denotes Frechet differentiation.
Density Asymptotics THEOREM.
(1) F(z)
169
Let F(z) and G(z) b e real valued functionals on Co[O, t] satisfying
- $ J; (z’(s))~ ds h as a unique maximum over Ct[O, t] at x and this maximum is less
than or equal to zero.
(2) F(x) - ; J; (x’(s))” ds = b. (3) F(z) is measurable. a 1most everywhere with L1, L2 E II%+and Lz < t/4. (4) F(z) I Ll +L2))42 and upper semicontinuous elsewhere. (5) F(z) iscontinuousfor ]]z]] 5 (IL1-b+ll/lL2-3t1)1’2 F(z) has n > 3 continuous fiechet derivatives in a 6 uniform neighborhood of x. (6) is uniformly bounded for r] in a 6 uniform neighborhood lE [exp{(l + e)D2F(x + n)B2/2}] (7) of zero, for some E > 0. (8) ]G(z)] 2 KreK211’11*almost everywhere with Kr,K2 E R+.
(9) G(z)
is measurable. G(z) is continuous at x in the supremum norm 1111. (10) G(z) has n - 2 continuous Frechet derivatives in a 6 uniform neighborhood of x. (11) (12)x’(s) is of bounded variation. Then as X + 0
exp { -
bXv2}E[exp { x-2F(xB)}~(~~)]
= r0 + xrr +
. . . + 0(x”-2).
The proof of this theorem is quite straightforward if one cares to use the proof of a similar result due to [9] as a guide. One could, no doubt, refine the statement and proof if one wished. The important point is the structure of the terms of the series. Defining gj(O)Bj by n-3
G(XB + x) = c
gj(0)(XB)’
+ 0 ( I]XB]]n-2)
j=O
and fj(O)Bj,
similarly, we have
exp {~IX-~}
lE[ exp {X-2F(XB)}
G(XB)]
The only obstacle appears to be our particular form of F but note that F(z) - ;
0t (z’(s))2 ds = -;
J
I’
(z’(s) + b(x + z(s)))2 ds,
which has a maximum of zero at the solution of dz ds-
- -b(x
+ z(s)),
with z(0) = 0. The remaining conditions can be checked; in fact most of the work is contained in [7]. There will be a minor restriction on po but this is not too great an imposition.
THE
4.
LEADING
ORDER
TERM
From the preceeding section, we have the leading order term in the form
exp { -5
’
of b/(x + z(s)) ds} po(x + z(t))E [w { f2(0)B2}] J
170
I. M. DAVIES
where
~[exp{f2(OP2}] =lE
[ev{-il
((by2+ bb”)(Z + Z(S))B,2dS - +J’(z +z(t))B:}] .
We can evaluate this expectation by using an analogue of a result of Montroll [12]. Montroll deals with a Brownian motion slightly different from ours. We have
=($$) 1’2 (2n~~)-~‘~ JrnF(u) -cc
du,
exp { -$}
where
and D satisfying D”(s) + p(s)D(s)
D(t) = 1,
= 0,
D’(t) = 0.
We have
p(s) =
-((q2+bb”)(z+z(s)),
g(s) = St(s), F(u) = exp
-ib’(l:
+ i(t))~~},
and so, for sufficiently small t,
lE[exp { f2(0)B2}]
= ($$)
1’2 (1 + b’(5 + *(t))02)-1’2.
In order to simplify this, we need to re-express (1 + b’(z + z(t))a2).
We begin by noting that
-$g
b’(2 + z(t)) = and that z’(s) satisfies (z’(s))”
+p(s)
z’(s) = 0.
We can then write z’(s) = aD(s)
+ PO(s) j
and evaluate cy and p by use of z’(0) = -O(r)
D-2(u)
and z”(O) = b(z)b’(s).
z’(O)
a=D(0)’
P = b(s) (D(W’(4 and thus,
z’(t) = a + p
t J
=
p.
+ D’(O)),
D-2(s)
0
z”(t)
du
ds,
One finds
DensityAsymptotics
171
This gives us z’(O) l+ b’a2 = z/(t)D(())’ We now have the more appealing result lE [exp { f2(0)B2}]
= (z)
1’2 ,
and substituting this into our original expression for the leading term yields
P(?t) =
(z’(t)>Pob+ 4t)) + qdq. z’(o)
5. CONCLUSION What we have presented in this paper is not in itself novel. What we have done, however, is follow through an initial study of a familiar problem from a different starting point. This is a useful task in many areas as it often leads to a better knowledge of the essentials of a problem. An asymptotic series has been obtained for the density associated with the solution of the stochastic differential equation of interest. The structure of the series is managable and the first order term should be approachable. The classical mechanics expected to appear did so and the leading order term was obtained in a simple and possibly familiar form. It should be straightforward to study drifts of the form b(z, t) instead of just b(z) and diffusions in higher dimension.
REFERENCES 1. I.M. Davies, On the drift in stochastic mechanics, J. Phys. A 22, 3195-3198, (1989). 2. I.M. Davies, On the stochastic Lagrangian and a new derivation of the Schrijdinger equation, J. Phys. A 22, 3199-3203, (1989). 3. I.M. Davies, On the stochastic Lagrangian in manifolds, In Stochastics and Quantum Mechanics, (Edited by A. Truman and I. M. Davies), pp. 69-75, World Scientific Press, Singapore, (1992). It6 Calculus, John Wiley 4. L.C.G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales: and Sons, Chichester, (1987). 5. A. IPruman and D. Williams, A general&d Arc-Sine law and Nelson’s stochastic mechanics of one-dimensional time-homogeneous diffusions, In Diflusion Processes and Related Problems, (Edited by M. Pinsky), Birkhiluser, Boston, (1991). 6. B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, (1979). 7. M. Schilder, Some asymptotic formulas for Wiener integrals, nans. Amer. Math. Sot. 125, 63-85, (1966). 8. M. Pincus, Gaussian processes and Hammerstein integral equations, Tmns. Amer. Math. Sot. 134, 193-214, (1968). 9. I.M. Davies and A. Truman, Laplace asymptotic expansions of conditional Wiener integrals and generalised Mehler kernel formulas, J. Math. Phys. 23, 2059-2070, (1982). 10. R.S. Ellis and J. Rosen, Asymptotic expansions of Gaussian integrals, Bull. Amer. Math. Sot. 3, 705-709, (1980). 11. I.M. Davies and A. Truman, (unpublished result), (1980). 12. E.W. Montroll, Markhoff chains, Wiener integrals and Quantum theory, Commun. Pure and Appl. Math. 5, 415-453, (1952).
08957177(94)00128-6
The Density Asymptotics Associated with a Stochastic Differential Equation I. M. DAVIES Department of Mathematics, University College of Swansea Singleton Park, Swansea, SA2 8PP, UK Abstract-We present an initial foray into the use of Laplace Asymptotic Expansions to study the development of the density associated with a stochastic differential equation. We only consider the simplest interesting case herein and by using a variety of viewpoints, we obtain the behaviour of the density for small values of the diffusion constant and time. We demonstrate that the leading order term is indeed the standard one.
1. INTRODUCTION In previous related
work, we have studied
to quantum
stochastic
mechanics.
mechanics
In doing
and Laplace
structure
of the drift in stochastic
derivation
of the most commonly
the study of stochastic view to good stochastic
effect.
differential
problems
in quantum
so, we have touched
asymptotic mechanics
expansions.
or in areas mathematically
upon
semi-classical
Most recently,
equations
Lagrangian
and classical
mechanical
as it were, by concentrating
and obtain the nature of density in terms of b and u without reference to any background our study, a potential
which, to leading order, behaves as classical mechanics In the second section, we move from the stochastic in terms of an expectation.
the expectation
THE
The stochastic
expects.
differential equation to an expression coefficient
fi
required
for the
to rewrite
and gives its structure,
while
on the nature of the leading order term.
STOCHASTIC
differential
structure.
function can be constructed
The third section presents the theorem
in terms of powers of the diffusion
the fourth section concentrates 2.
point of
solely on the
equation
What we find is that, while the drift b dominates
density
the
and refined the
[2,3]. In this, we have approached
from a quantum
Herein, we start from scratch,
mechanics,
we have studied
[l] for a wide class of Hamiltonians
used stochastic
differential
mechanics
DIFFERENTIAL
EQUATION
equation
dXt = b(X,) dt + fidBt, where Bt is Brownian
motion
on lR with the standard
filtration,
(cf. [4]). Th e d ensity, if it exists, satisfies the partial differential
has a solution
for b Lipschitz
equation
Typeset 167
by d&-w
I. M. DAVIES
168
and the invariant density, if it exists, is simply
In our discussion, we only require the finiteness of
fo =exp
i
)
bjb(u)du .
One may quickly show that fc satisfies the second order differential equation
u2 d2fo -7yp+v.f0=0, where V(x) = i(b” + vb’), b’ denoting 2,
we see that f<‘p
(cf. [5]). By noting that
satisfies a(&%) at
The Feynman-Kac formula [6] now gives us
p(z,t) =.fo(z)E
[w{-tl
V(x + 42%)
ds}(f&o)
(x + J;&)] ,
where PO(Z) = p(z,O), lE d enoting expectation when the r.h.s. of the above is well defined. Let us now rewrite the above in some detail to observe the dependence on u. x+J;Bt ;hZ(r
+
ds - ; /
fill,)
exp
-{
b(u) du
I 1 2
J b’(z+d%)ds t o
>
PI++ 4-t)
1.
This may be written in the form p(z,t)
= E [exp
where F(z) = - & ib2(z + z(s)) ds - Jz+z(t) b(u) du. We can expand this expression for p(z, t) in powers of u by use of a Laplace asymptotic expansion. 3.
THE
LAPLACE
ASYMPTOTIC
EXPANSION
There is a large body of work on Laplace asymptotic expansions notably [7-lo]. We will not discuss the relative merits of the different approaches to the problem herein. We will use a result due to [ll]. There is a little introductory notation required namely: Cc[O, t] is the space of continuous functions z : [0, t] -+ R with z(0) = 0, CG[O,t] is that subset of Cc[O, t] comprising of differentiable z, D denotes Frechet differentiation.
Density Asymptotics THEOREM.
(1) F(z)
169
Let F(z) and G(z) b e real valued functionals on Co[O, t] satisfying
- $ J; (z’(s))~ ds h as a unique maximum over Ct[O, t] at x and this maximum is less
than or equal to zero.
(2) F(x) - ; J; (x’(s))” ds = b. (3) F(z) is measurable. a 1most everywhere with L1, L2 E II%+and Lz < t/4. (4) F(z) I Ll +L2))42 and upper semicontinuous elsewhere. (5) F(z) iscontinuousfor ]]z]] 5 (IL1-b+ll/lL2-3t1)1’2 F(z) has n > 3 continuous fiechet derivatives in a 6 uniform neighborhood of x. (6) is uniformly bounded for r] in a 6 uniform neighborhood lE [exp{(l + e)D2F(x + n)B2/2}] (7) of zero, for some E > 0. (8) ]G(z)] 2 KreK211’11*almost everywhere with Kr,K2 E R+.
(9) G(z)
is measurable. G(z) is continuous at x in the supremum norm 1111. (10) G(z) has n - 2 continuous Frechet derivatives in a 6 uniform neighborhood of x. (11) (12)x’(s) is of bounded variation. Then as X + 0
exp { -
bXv2}E[exp { x-2F(xB)}~(~~)]
= r0 + xrr +
. . . + 0(x”-2).
The proof of this theorem is quite straightforward if one cares to use the proof of a similar result due to [9] as a guide. One could, no doubt, refine the statement and proof if one wished. The important point is the structure of the terms of the series. Defining gj(O)Bj by n-3
G(XB + x) = c
gj(0)(XB)’
+ 0 ( I]XB]]n-2)
j=O
and fj(O)Bj,
similarly, we have
exp {~IX-~}
lE[ exp {X-2F(XB)}
G(XB)]
The only obstacle appears to be our particular form of F but note that F(z) - ;
0t (z’(s))2 ds = -;
J
I’
(z’(s) + b(x + z(s)))2 ds,
which has a maximum of zero at the solution of dz ds-
- -b(x
+ z(s)),
with z(0) = 0. The remaining conditions can be checked; in fact most of the work is contained in [7]. There will be a minor restriction on po but this is not too great an imposition.
THE
4.
LEADING
ORDER
TERM
From the preceeding section, we have the leading order term in the form
exp { -5
’
of b/(x + z(s)) ds} po(x + z(t))E [w { f2(0)B2}] J
170
I. M. DAVIES
where
~[exp{f2(OP2}] =lE
[ev{-il
((by2+ bb”)(Z + Z(S))B,2dS - +J’(z +z(t))B:}] .
We can evaluate this expectation by using an analogue of a result of Montroll [12]. Montroll deals with a Brownian motion slightly different from ours. We have
=($$) 1’2 (2n~~)-~‘~ JrnF(u) -cc
du,
exp { -$}
where
and D satisfying D”(s) + p(s)D(s)
D(t) = 1,
= 0,
D’(t) = 0.
We have
p(s) =
-((q2+bb”)(z+z(s)),
g(s) = St(s), F(u) = exp
-ib’(l:
+ i(t))~~},
and so, for sufficiently small t,
lE[exp { f2(0)B2}]
= ($$)
1’2 (1 + b’(5 + *(t))02)-1’2.
In order to simplify this, we need to re-express (1 + b’(z + z(t))a2).
We begin by noting that
-$g
b’(2 + z(t)) = and that z’(s) satisfies (z’(s))”
+p(s)
z’(s) = 0.
We can then write z’(s) = aD(s)
+ PO(s) j
and evaluate cy and p by use of z’(0) = -O(r)
D-2(u)
and z”(O) = b(z)b’(s).
z’(O)
a=D(0)’
P = b(s) (D(W’(4 and thus,
z’(t) = a + p
t J
=
p.
+ D’(O)),
D-2(s)
0
z”(t)
du
ds,
One finds
DensityAsymptotics
171
This gives us z’(O) l+ b’a2 = z/(t)D(())’ We now have the more appealing result lE [exp { f2(0)B2}]
= (z)
1’2 ,
and substituting this into our original expression for the leading term yields
P(?t) =
(z’(t)>Pob+ 4t)) + qdq. z’(o)
5. CONCLUSION What we have presented in this paper is not in itself novel. What we have done, however, is follow through an initial study of a familiar problem from a different starting point. This is a useful task in many areas as it often leads to a better knowledge of the essentials of a problem. An asymptotic series has been obtained for the density associated with the solution of the stochastic differential equation of interest. The structure of the series is managable and the first order term should be approachable. The classical mechanics expected to appear did so and the leading order term was obtained in a simple and possibly familiar form. It should be straightforward to study drifts of the form b(z, t) instead of just b(z) and diffusions in higher dimension.
REFERENCES 1. I.M. Davies, On the drift in stochastic mechanics, J. Phys. A 22, 3195-3198, (1989). 2. I.M. Davies, On the stochastic Lagrangian and a new derivation of the Schrijdinger equation, J. Phys. A 22, 3199-3203, (1989). 3. I.M. Davies, On the stochastic Lagrangian in manifolds, In Stochastics and Quantum Mechanics, (Edited by A. Truman and I. M. Davies), pp. 69-75, World Scientific Press, Singapore, (1992). It6 Calculus, John Wiley 4. L.C.G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales: and Sons, Chichester, (1987). 5. A. IPruman and D. Williams, A general&d Arc-Sine law and Nelson’s stochastic mechanics of one-dimensional time-homogeneous diffusions, In Diflusion Processes and Related Problems, (Edited by M. Pinsky), Birkhiluser, Boston, (1991). 6. B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, (1979). 7. M. Schilder, Some asymptotic formulas for Wiener integrals, nans. Amer. Math. Sot. 125, 63-85, (1966). 8. M. Pincus, Gaussian processes and Hammerstein integral equations, Tmns. Amer. Math. Sot. 134, 193-214, (1968). 9. I.M. Davies and A. Truman, Laplace asymptotic expansions of conditional Wiener integrals and generalised Mehler kernel formulas, J. Math. Phys. 23, 2059-2070, (1982). 10. R.S. Ellis and J. Rosen, Asymptotic expansions of Gaussian integrals, Bull. Amer. Math. Sot. 3, 705-709, (1980). 11. I.M. Davies and A. Truman, (unpublished result), (1980). 12. E.W. Montroll, Markhoff chains, Wiener integrals and Quantum theory, Commun. Pure and Appl. Math. 5, 415-453, (1952).