Statistics and Probability Letters 80 (2010) 1121–1127

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Diffusion approximation of birth–death processes: Comparison in terms of large deviations and exit points Khashayar Pakdaman a , Michèle Thieullen b , Gilles Wainrib a,b,c,∗ a

Institut Jacques Monod UMR 7592 CNRS, Univ. Paris VII, Paris VI, France

b

Laboratoire de Probabilités et Modèles Aléatoires UMR7599 Univ. Paris VI, Paris VII - CNRS, France Centre de Recherche en Epistémologie Appliquée UMR 7656 Ecole Polytechnique - CNRS, France

c

article

info

Article history: Received 19 June 2009 Received in revised form 9 March 2010 Accepted 10 March 2010 Available online 18 March 2010 Keywords: Diffusion approximation Large deviations Exit points Jump process Birth and death

abstract We compare the large deviations between jump Markov processes and their diffusion approximations for the problem of escape from a domain. It is known that the escape times are generally asymptotically different, leading to a quantitative error. In this paper we show that the escape point can also be different, which implies a qualitative error. In dimension one, for a general class of birth-and-death processes, we show that there always exists a domain [a, b] such that the jump process asymptotically escapes from a and its approximation from b. © 2010 Elsevier B.V. All rights reserved.

1. Introduction To study the impact of the fluctuations on a system, the choice of a noise model is a crucial modelling step. Beyond the technical aspects, one of the major arguments for the use of Brownian motion, especially in multiscale systems, stems from the central limit theorem (CLT) and its functional versions: as soon as there are a large number of independent enough sources of noise, the normal law appears as a universal property. Therefore, diffusion models are seen as legitimate candidates and are widely used in many areas. Critics of these types of models can be made when the independence assumption is not clearly satisfied, leading to non-gaussian distributions. In this paper, we are considering another critical aspect concerning the application of small noise diffusion models, arising often in finite-size effect modelling, especially if one is interested in the large deviations (rare events) of such models. The CLT provides only a second order result so that the fluctuations of upper order statistics, involving large time windows, are not well captured by diffusion models. In order to investigate this issue, we consider a general birth-and-death process on R in the fluid limit with jumps of size +n−1 and −n−1 with rates nr (x) and nl(x) respectively. For instance, one can consider a population model with n independent agents following a birth-anddeath process, whose states can be either 0 or 1, with transition rates α, β > 0, and we are interested in the proportion of the individuals in state 1. We describe a classical way to build a diffusion model, or Langevin approximation, with small noise from a microscopic description of the fluctuation sources: Kurtz (1971) provides a law of large numbers and a CLT leading to the Langevin approximation, which arises for instance in chemical reaction modelling (Gardiner, 1985), in stochastic ion channel kinetics (Fox and Lu, 1994), in population biology (Pollett, 2001) or in queuing theory (Shwartz and Weiss, 1995) in order to capture finite-size effects. See also Pakdaman et al. (in press) for a recent application to neural modelling. Our aim is to show that the qualitative behavior of the Langevin approximation can be radically different from the microscopic model,

∗

Corresponding author at: Institut Jacques Monod UMR 7592 CNRS, Univ. Paris VII, Paris VI, France. E-mail address: [email protected] (G. Wainrib).

0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2010.03.006

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even for such a simple example. The problem of the exit from a domain, which is very important in many applications, has been investigated (Kushner, 1982; Hanggi et al., 1984; Matkowsky et al., 1991; Doering et al., 2004) in a similar setting showing that the exit times are asymptotically different for the microscopic Markov model and its Langevin approximation. These previous studies focused on the differences in the mean exit time from a domain, and not the exit point. Here we address the question of the exit point and show that, in a simple and general class of models, it can be dramatically different in the sense that it can lead to very different qualitative outcome. Namely, we consider the exit from an interval [a, b] and we show that there exist some a < b for which the exit points are different for the original Markov jump process and its diffusion approximation. As the chosen example is simple and general, arising directly from modelling (not an artificial and complicated counter-example), the same effect should be observable and have an impact in many other models. In a metastable system, our result shows that the succession of states can be very different. So, in a modelling perspective, the discrete nature of the underlying microscopic objects may not be reducible to a diffusion approximation. Thus, the CLT and the universality of the gaussian law should not be perceived as sufficient justification for the use of a diffusion model, especially in the perspective of large deviations. In terms of the Kramers–Moyal expansion (Risken, 1989), neglecting terms of order strictly higher than 2, even if they are going to zero faster than the term of order 2, may lead to very different behavior. Note that, as pointed out in Pawula (1967) Theorem, if the Kramers–Moyal expansion does not stop after the second term, then it must contain an infinite number of terms. When the level of noise is asymptotically small, the residual components can still decide radically the outcome with a qualitative impact. With an example taken from epidemiology we show that such a qualitative difference can change completely the interpretation of the result. Thus, the choice of a noise model should be made very carefully, and arguments based on a functional CLT or asymptotic expansions of the master equation (Van Kampen, 1981) to derive a small diffusion term from a microscopic description might not be sufficient to obtain consistent results. In Section 2, we describe the general birth-and-death model and our main result with some examples. In Section 3 we give the proof of our main result. For the paper to be self-contained, we recall the results from Kurtz (1971) in Appendix A and the Freidlin–Wentzell theory (Freidlin and Wentzell, 1998) related to the problem of exit from a domain in Appendix B. 2. Model and main result Let r and l two positive C 3 real functions. For n ∈ N, we consider:

• a pure jump Markov process (Xn (t ))t ≥0 with jumps of size 1/n with rate nr (x) and jumps of size −1/n with rate nl(x) • its Langevin approximation (xn (t ))t ≥0 , i.e. the diffusion process: 1 p dxn (t ) = [r (xn (t )) − l(xn (t ))]dt + √ r (xn (t )) + l(xn (t ))dWt . n

Formally, the term r (xn (t )) − l(xn (t )) has a gain-loss interpretation and the term (r (xn (t )) + l(xn (t )))/n corresponds to the variance. In Appendix A we recall how to derive this diffusion or Langevin approximation using Kurtz (1971). We assume that there exists x0 ∈ R such that r (x0 ) − l(x0 ) = 0 and r 0 (x0 ) − l0 (x0 ) < 0. The point x0 can be seen as a stable fixed point for the deterministic limit dynamical system x0 = r (x) − l(x). We assume that Xn (0) = xn (0) = x0 . We consider the problem of exit from an interval [a, b] with a < x0 < b for both the jump Markov process and its Langevin approximation. More precisely, we define τnM (a) = inf{t ≤ 0, Xn (t ) < a}, τnM (b) = inf{t ≤ 0, Xn (t ) > b} and τnL (a) = inf{t ≤ 0, xn (t ) < a}, τnL (b) = inf{t ≤ 0, xn (t ) > b}, where M and L stand respectively for Markov and Langevin. For n ∈ N, we say that the exit point for a realization of the jump Markov process (resp. Langevin approximation) is a if τnM (a) < τnM (b) (resp. τnL (a) < τnL (b)). We say that b is the exit point if the previous inequalities are inverted. Thus we are interested in comparing the following limits of probabilities: PaM = lim P τnM (a) < τnM (b)





n→∞

PaL = lim P τnL (a) < τnL (b) .





n→∞

It means that we ask the question: in the limit n → ∞, what is the difference in the exit point for the jump Markov process and its Langevin approximation? Our main result is the following: Theorem. We define ξ = r (x0 ) = l(x0 ) and assume ξ 6= 0. We further assume the following sufficient conditions: 2

2

(H1 ) ξ (l00 (x0 ) − r 00 (x0 )) 6= (l0 (x0 ) − r 0 (x0 )) 1

(H2 ) (l00 (x0 )(l00 (x0 ) − l0 (x0 )) − r 00 (x0 )(r 00 (x0 ) − r 0 (x0 ))) + ξ −1 (l0 (x0 ) − r 0 (x0 ))2 6= 0. 2

Then there exist intervals [a, b] with a < x0 < b, such that, either PaM = 1 and PaL = 0, or PaM = 0 and PaL = 1. A detailed proof is given in Section 3. We give here in a few lines an overview of the proof. The main tool is the so-called quasipotential, defined by Freidlin and Wentzell (see Appendix B for a more detailed presentation), since it is directly related to the problem of escape from a domain. It is shown in Freidlin and Wentzell (1998) that the exit point from a domain D is

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Fig. 1. Quasipotential for Example 1 in the Markov (crosses) and in the Langevin (circles) cases; ordinates: quasipotentials UM (x) and UL (x) (without unit); abscissa: x (without unit).

the point z, if unique, that minimizes the quasipotential on the boundary of D. It means that the probability to exit from D in a small neighborhood of z tends to 1 as the noise goes to zero. So, if we denote by UM and UL the quasipotentials respectively for the jump Markov process and for its Langevin approximation, we want to show that there exists an interval [a, b] with a < x0 < b, such that:

• either UM (a) < UM (b) and UL (a) > UL (b): the exit point is a for the jump Markov process and b for its Langevin approximation

• or UM (b) < UM (a) and UL (b) > UL (a): the exit point is b for the jump Markov process and a for its Langevin approximation. To this end we need some asymmetry which is provided by the conditions (H1 ), (H2 ) of the Theorem. These two conditions are only sufficient, but they are not very restrictive. We now give some numerical examples where the exit points are different for a good choice of parameter values. These values were found by computing numerically the quasipotentials. Remark. The assumptions (H1 )(H2 ) of the Theorem are generically satisfied because they involve only first and second order derivatives evaluated at a single point. Moreover, the result is robust in the sense that the set of pairs (a, b) such that, either PaM = 1 and PaL = 0, or PaM = 0 and PaL = 1, contains a product [a0 , a1 ] × [b0 , b1 ] with a0 < a1 and b0 < b1 . Example 1. We consider the example given in the introduction, which appears naturally in many models, with r (x) = (1 − x)α and l(x) = xβ . In this setting, the hypotheses of the Theorem (H1 ) and (H2 ) reduce to α 6= β . In the following numerical example, we exhibit a case where the exit point is different in the Langevin and in the Markov case: α = 0.9; β = 0.5; a = x0 − 0.35; b = x0 + 0.3. We then compute the following numerical approximations (SCILAB precision = 2.22 · 10−16 ): UM (a) ≈ 0.2527918 < UM (b) ≈ 0.2563880 UL (a) ≈ 0.2334383 > UL (b) ≈ 0.2244103. Thus, the jump Markov process will escape from a, whereas its Langevin approximation will escape from b (Fig. 1).

Example 2. In the Susceptible-Infected-Susceptible model of epidemiology (Pollett, 2001), which is widely used, the same effect can be observed for some combination of the model parameter and the chosen boundaries. In this model, r (x) = Λx(1 − x) and l(x) = x, where Λ corresponds to the infection rate. If Λ > 1, x0 = 1 − Λ1 is a stable equilibrium point. In this setting, the hypothesis (H1 ) reduces to Λ 6= 0 and Λ 6= 1 and (H2 ) reduces to Λ 6= 34 , so they are satisfied with Λ > 1. With Λ = 2, x0 = 0.5, a = 0.1 b = 0.815, we obtain numerically (SCILAB precision  = 2.22 · 10−16 ): UM (a) ≈ 0.129008 < UM (b) ≈ 0.1310633 UL (a) ≈ 0.1270555 > UL (b) ≈ 0.1266729.

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Here again, the jump Markov process will escape from a, whereas its Langevin approximation will escape from b. This can be interpreted as follows: for this population the stable state is half infected, and the probability of a rare escape above 81.5% infected before a rare escape below 10% infected will be asymptotically 0 for the jump Markov model and 1 for its Langevin approximation. 3. Proof of the main result As explained in the overview of the proof, we want to compute the quasipotentials for both processes and compare them. Then, using Freidlin and Wentzell (1998), we relate this analysis to the exit point problem. We refer the reader to Appendix B for a basic remainder of Freidlin–Wentzell theory. Computation of the quasipotentials. To compute the quasipotentials we need to compute first the hamiltonians for our processes. This can be seen as a description of their exponential moments. By Theorem 4.3 Chapter 5 p. 159 of Freidlin and Wentzell (1998), we have a way to compute the quasipotential: find a function U, vanishing at x0 , continuously differentiable and satisfying H (x, U 0 (x)) = 0 for x 6= x0 and such that U 0 (x) 6= 0 for x 6= x0 where:

• in the jump Markov case: HM (x, α) = (eα − 1)r (x) + (e−α − 1)l(x) • in the Langevin approximation: HL (x, α) = (r (x) − l(x))α + (r (x) + l(x))α 2 . 0 Here we note that HL is the second order expansion of HM in α . Actually, solving HL (x, UL0 (x)) = 0 and HM (x, UM (x)) = 0, we can find UL and UM explicitly:

• in the jump Markov case: Z x UM (x) = ln(l(u)/r (u))du. x0

• in the Langevin approximation: Z x r (u) − l(u) UL (x) = −2 du. x0 r (u) + l(u) Comparison of the quasipotentials. We recall that the exit point is given by the point of the boundary at which the minimum of the quasipotential is attained, if unique (see Theorem 3 in Appendix B). We are looking for an interval [a, b] with a < x0 < b, such that:

• either UM (a) < UM (b) and UL (a) > UL (b): the exit point is a for the jump Markov process and b for its Langevin approximation

• or UM (b) < UM (a) and UL (b) > UL (a): the exit point is b for the jump Markov process and a for its Langevin approximation. Those two functions UM and UL are close to one another when x is close to x0 , up to order 3. More precisely, ∀0 ≤ k ≤ (k) (k) 3, UM (x0 ) = UL (x0 ). To simplify the notation in what follows, we make the assumption that x0 = 0; this will obviously not affect the result. Remember that we are assuming ξ = l(x0 ) = r (x0 ) 6= 0. For a < 0 small, we have the following Taylor expansions: K3 3 L4 a2 + a + a4 + o(a4 ) 2! 3! 4! K2 2 K3 3 M4 4 UM (a) = a + a + a + o(a4 ) 2! 3! 4! UL (a) =

K2

where K2 = ξ −1 (l0 (x0 ) − r 0 (x0 )), K3 = ξ −1 (l00 (x0 ) − r 00 (x0 )) − ξ −2 (l0 (x0 ) − r 0 (x0 )) and where we assume that M4 6= L4 , because of assumption (H2 ). Since we made the assumption that r 0 (x0 ) − l0 (x0 ) < 0, thus K2 > 0. We choose a small enough such that there exist zaL > 0 and zaM > 0 with UL (a) = UL (zaL ) and UM (a) = UM (zaM ). We remark that zaL 6= zaM . Indeed, if zaM = zaL = z, then subtracting UM and UL at a and z, we obtain: (M4 − L4 )a4 = (M4 − L4 )z 4 + o(a4 ), which implies z = −a + o(a). But one cannot have UL (a) = UL (−a + o(a)) unless K3 = 0. Assumption (H1 ) of the Theorem ensures that K3 6= 0, so that it is not possible to have zaM = zaL (Fig. 2). We then choose b between zaM and zaL . By the monotonicity of UL and UM after x0 we obtain a relevant interval [a, b] to conclude the proof. Indeed, if zaM < b < zaL , then 2

UM (a) = UM (zaM ) < UM (b) and

2

UL (a) = UM (zaL ) > UL (b).

Remark. Exact non-asymptotic solution of the two-sided boundary escape problem are available using the scale function. If one defines (up to a constant and a multiplicative factor) the scale function of a one-dimensional Markov process X as an increasing real function satisfying: Px (τc < τd ) =

s(x) − s(d) s(c ) − s(d)

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Fig. 2. Schematic representation of the quasipotentials near x0 .

then a consequence of the Markov property is that s(X ) is a local martingale, which provides an equation for s. In the Langevin approximation case, s satisfies:

(l(x) + r (x))s00 (x)/2 + (r (x) − l(x))s0 (x) = 0 so that the probability of escaping through the lower side of the interval [c , d] is: Px [τnL (c ) < τnL (d)] =

s(d) − s(x) s(d) − s(c )

with s being the scale function associated with the Langevin diffusion process, such that:



s0 (x) = exp −2N

Z

x

r (y) − l(y) r (y) + l(y)

As a consequence, Px [τnL (c ) < τnL (d)] = to 1 as n → ∞ and to 0 otherwise.



dy .

Rd x

enUL (y) dy/

Rd c

enUL (y) dy, so that if UL (c ) < UL (d) then that probability converges

Similarly, for the original jump Markov process, s : {0, 1n , . . . , 1} → R satisfies: n (ri (si+1 − si ) + li (si−1 − si )) = 0 where ri = r (i/n), li = l(i/n) and si = s(i/n). Denoting δ si = si+1 − si and solving the above equation, one finds:

δ si =

i Y

! li /ri

δ s0 .

j =1





Pi Taking the logarithm and dividing by n, one defines ULn (i) := n−1 ln j=1 li /ri + ln(δ s0 ) which is nothing but a discretization that converges to the quasipotential UL when n → ∞. We conclude as above that, if cn and dn are integers: Px [τnM (c ) < τnM (d)] =

dn X i=cn

enUL (i) n

X dn

enUM (i) n

i=cn

so that if UM (c ) < UM (d) then that probability converges to 1 as n → ∞ and to 0 otherwise. Acknowledgement The authors thank the anonymous referee for insightful comments. Appendix A. Jump Markov processes in the fluid limit Here we recall results from Kurtz (1971) to show the mathematical basis of the Langevin approximation. k Let (xn )n∈N be a sequence of homogeneous Markov jump R processes with state spaces En ⊂ R , intensities λn (x) and jump laws µn (x, dy). Let the flow be defined as Fn (x) = λn (x) E (z − x)µn (x, dz ). The following theorem states that if the flow n

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admits a limit and if the second order moment of the jump size converges uniformly to zero then the Markov jump process converges to the solution of a deterministic ordinary differential equation (cf. Kurtz, 1971). Theorem 1 (cf. Kurtz, 1971). If the following holds: (1) limn→+∞ supx∈En |Fn (x) − F (x)| = 0 with F : Rk → Rk globally lipschitz, R (2) limn→+∞ |xn (0) − x0 | = 0 a.s., and (3) limn→+∞ supx∈En Kn (x) = 0, with Kn (x) = λn (x) E |z − x|2 µn (x, dz ). Then, n   ∀δ > 0, ∀t > 0, limn→+∞ P sup0≤s≤t |xn (s) − x(s)| > δ = 0, where x is the only solution with x(0) = x0 of the ODE: x0 = F (x).

After the law of large numbers, it is natural to look for a central limit theorem, which will lead to the Langevin approximation. Theorem 2 (cf. Kurtz, 1971). With the same notations as in A.1 Theorem 1, we define: gijn (x) = αn2 λn (x)

Z

(zi − xi )(zj − xj )µn (x, dz ) En

and the (k, k) matrix Gn (x) = gijn (x) , where αn is a normalizing sequence of positive numbers going to infinity. Suppose that (1) Fn and Gn converge uniformly to F and G, where F is bounded, L-lipschitz continuous, and G is bounded uniformly continuous, (3) xn (0) → x0 and (2) there is a sequence n decreasing to zero such that




lim sup α λ (x) 2 n n

n→∞ x∈E

n

Z αn |z −x|>n

|z − x|2 µn (x, dz ) = 0.



Then Rn (t ) = αn xn (t ) − xn (0) −

Rt 0

Fn (xn (u))du



= αn Zn (t ) converges in law to the diffusion R(t ) with characteristic

function: 1X

E [exp(iθ R(t ))] = exp

2

θi θj

ij

!

t

Z

gij (x(s))ds . 0

Rt

Remark. This means that the process R(t ) is a driftless diffusion process: R(t ) = 0 σR (x(s))dBt with B a standard kdimensional Brownian motion, x the solution of the ODE: x0 = F (x) and σR the square root matrix of G. Moreover, Theorem 2 leads to consider un solution of the following stochastic differential equation with initial condition un (0) = x0 : dun (t ) = Fn (un (t ))dt + √1 σR (un (t ))dBt which will be called the diffusion or Langevin approximation of the jump Markov N

process xn (t ).

Appendix B. Freidlin–Wentzell theory and the exit problem We present here a quick overview of the main results of the Freidlin–Wentzell (Freidlin and Wentzell, 1998) theory that are important for our problem. We consider a general situation, that allows jumps, where the processes Xth we are interested in is a Markov process on r R with initial distribution Pxh and infinitesimal generator, defined for f ∈ C 2 with compact support, by: A f (x) = h

X i

# Z " X h X i,j 1 0 00 b (x)fi (x) + a (x)fij (x) + f (x + hβ) − f (x) − h βi fi (x) µx (dβ). 2 i,j h Rr i i

0

To account for large fluctuations of this process, we consider exponential moments and Legendre transforms, with analogy with the Cramér theorem for sums of independent variables. Here the role played by those independent variables is played by the independent increments of the process X h . Define: H (x, α) =

X

bi (x)αi +

X

ai,j (x)αi αj +

i ,j

i

Z Rr

[e(α,β) − 1 − (α, β)]µx (dβ).

Then we denote by L(x, β) the Legendre transform of H (x, α): L(x, β) = sup{(α, β) − H (x, α)}. α

For a Rr valued function φt , T1 ≤ t ≤ T2 , we define the following functional: S (φ) = ST1 T2 (φ) =

Z 

T2

 T1 +∞

L(φt , φ˙ t )dt

if φ is abs. continuous and the integral converges otherwise.

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Under some restrictions on H and L, Freidlin and Wentzell prove that h−1 S (φ) is the action functional for the process (Xth , Pth ) as h → 0 and uniformly in the initial point x (cf. Freidlin and Wentzell, 1998 Thm 2.1 p. 146). We now consider the problem of exit from a domain and introduce the quasipotential. Let D be a domain of Rr with a smooth boundary ∂ D. Denote xt the deterministic limit of Xth when h → 0. Let us assume that, for x ∈ ∂ D, (b(x), n(x)) < 0 with n being the exterior normal, then xt does not leave D, but Xth will leave D with probability 1 as h → 0. The exit time and exit point are determined by the quasipotential, which is the infimum of the action functional for functions starting at x ∈ D and ending on the boundary. Suppose that O is an asymptotically stable equilibrium point, and ∀x ∈ D, xt (x) → O as t → ∞ without leaving D (we say D is attracted to O). We define the quasipotential as: V (x, y) = infT {S0T (φ); φT = x, φT = y}. Theorem 3 (cf. Freidlin and Wentzell, 1998). 1. For the mean exit time τ h = inf{t ; Xth 6∈ D}: for all x ∈ D, lim λ(h) ln Ex [τ h ] = inf V (0, y) = V0 .

h→0

y∈∂ D

2. For the exit point: if there exists a unique y0 ∈ ∂ D such that V (0, y0 ) = infy∈∂ D V (0, y) then, ∀δ > 0, ∀x ∈ D, lim Px [|Xτhh − y0 | < δ] = 1.

h→0

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