CONTRATO DE TRABAJO N° 001

The line of this procedure is as follows: We start by setting up the transition probabili- ties PN of the stochastic process in which we count the numbers of individual objects for

fixed system sizeN. The state space of this process is discrete, and the evolution of the

transition probabilities is described by the master equation. We then consider a sequence of discrete state space processes in which the state variables denote the intensive variables, i. e. the fractions of different classes of objects. For the system size tending to infinity, this sequence converges to a process with state variables changing continuously in space. The limit of the according sequence of master equations is approximated by a forward diffusion equation which is taken as a description for the limiting process. The limit is obtained by replacing difference quotients by the respective derivatives.

This technique has been used e. g. by Goel and Richter-Dyn (1974) for the approximation of the univariate birth and death process. Gillespie (1980) in a way reverses the method by considering an approximation legitimate only if its discretised version reduces to the master equation; to that end, derivatives are replaced by difference quotients.

The approximation method introduced in this section may seem obvious; however, it apparently has not been formulated in generality in the literature before. The following hence presents a new procedure. Due to space constraints, only main results are shown here. Some newly proved statements which enable this proceeding have been moved to SectionB.1 in the appendix.

Assuming that at most one event can occur during a small time interval of length ∆t,

we can establish an equation for PN(t+ ∆t,X) by summing over all possible nonzero

jumps ∆X 6=0 to arrive at state X ∈ DN at timet+ ∆t:

PN(t+ ∆t,X) = X ∆X PN(t,X −∆X, t+ ∆t,X)PN(t,X−∆X) + 1−X ∆X PN(t,X, t+ ∆t,X+ ∆X) PN(t,X).

The probability PN(·,X1,·,X2) is assumed zero here for all X1,X2 6∈ DN. The first line

collects all possibilities for transitions to the desired state at the desired time. The second line is the probability that the process has already been in stateX at timet and remained

there during the considered time interval. That is why the master equation, which results out of this equation, is also called a gain-loss equation. Subtract PN(t,X) on both sides,

divide by ∆t and let ∆t→0. We then obtain

∂PN(t,X) ∂t = X ∆X WN(t,X−∆X,∆X)PN(t,X−∆X)−WN(t,X,∆X)PN(t,X)

with transition rates

WN(t,X,∆X) = lim ∆t↓0

1

∆tPN(t,X, t+ ∆t,X+ ∆X)

as a description for the continuous time process with discrete state space. This is the forward master equation (4.8). For an uncountable set of possible jumps, the sum could easily be replaced by an integral. The functional form of WN is determined by the jump ∆X. For

an alternative notation, one can assign to each possible jump an index ifrom a set I and

write WN,i(t,X) = WN(t,X,∆i) for the corresponding jump ∆i, resulting in

∂PN(t,X) ∂t = X i∈I WN,i(t,X−∆i)PN(t,X−∆i)−WN,i(t,X)PN(t,X) . (4.15)

Instead of the extensive variableXwe now regard the intensive variablex= X/N. Consider

a sequence of processes with (still discrete) state spaces CN =N−1DN corresponding to a

sequence of numbers N which tends to infinity. The master equation for each process is ∂pN(t,x) ∂t = X i∈I wN,i(t,x−ε∆i)pN(t,x−ε∆i)−wN,i(t,x)pN(t,x) (4.16) with pN(τ,x, t,y) = PN(τ, Nx, t, Ny), wN,i(t,x) = WN,i(t, Nx) and ε = N−1. In order

to approximate the jump process by a diffusion process, this master equation should be approximated by a Kolmogorov equation. That again means that the difference terms in (4.16) should be replaced by derivatives with respect to the components of x. The single

summands in (4.16) are not of the form of difference quotients though, so this step is not immediately admissible. However, it is always possible to express each of these summands by a collection of difference quotients of some order. This is proven in Lemma B.3 in SectionB.1 in the appendix. Then, the master equation becomes

∂pN(t,x) ∂t = X i∈I X k∈Ii D_k|k|(wN,i·pN)(t,x) = X i∈I X k∈Ii ε|k|D |k| k (wN,i·pN)(t,x) ε|k| , (4.17)

where the notationD_k|k|stands for difference operators as introduced in DefinitionB.1, theIi

are appropriate sets of vectorsk= (0, k1, . . . , kn)0 as used in LemmaB.3, and|k|= Pnj=1kj.

The first component of kis zero becauset is fixed on the right hand side of Equation (4.17),

4.3 Approximation Methods 59

It seems feasible now to approximate the difference quotients D_k|k|/ε|k|_{by proper derivatives} as εgoes to zero. However, the consideration of ε tending to zero, i. e.N tending to infinity,

involves two limiting procedures: First, convergence of the difference quotients, and second, convergence of the functionspN andwN,i. Accurate mathematical treatment of this limit is

elaborate and beyond the purpose of this chapter. However, in many examples the scaled function wi =N−1wN,i does not depend on N anymore. We assume that this is the case

here (at least asymptotically), so that Equation (4.16) equals (if necessary, asymptotically)

∂pN(t,x) ∂t = X i∈I wi(t,x−ε∆i)pN(t,x−ε∆i)−wi(t,x)pN(t,x) ε . (4.18)

Furthermore, it seems plausible that Equation (4.17) remains true if pN is replaced by its

limit function p(which is assumed to exist), so Equation (4.17) turns into ∂p(t,x) ∂t = X i∈I X k∈Ii ε|k|−1D |k| k (wi·p)(t,x) ε|k| .

Provided that p and the wi are sufficiently often differentiable, it follows that — regarding

the limits of the difference quotients as ε tends to zero — the master equation becomes ∂p(t,x) ∂t = X k=(k1,...,kn)0 ε|k|−1 ∂ |k|_f k(t,x)p(t,x) ∂xk1 1 · · ·∂xknn ! (4.19) for some finite set of differentiable functions fk, k ∈ Nn0. Assume that the derivatives

are bounded. After restriction to terms up to order O(ε), i. e. ignoring smaller terms

with |k| ≥3, Equation (4.19) can then be rewritten as

∂p(t,x) ∂t =− n X j=1 ∂hµj(x, t)p(t,x) i ∂xj + 1 2N n X j,k=1 ∂2h_Σ jk(x, t)p(t,x) i ∂xj∂xk , (4.20)

where µj and Σjk with j, k = 1, . . . , n are the components of a vectorµ and a matrix Σ.

These can be determined according to Algorithm B.1and Example B.1 in the appendix. In some special cases, there are also explicit formulas for µ and Σ — see for instance

Example B.2.

Heuristically, the space-continuous limit of the initial jump process is described by Equa- tion (4.20). That is the forward diffusion equation (4.7) if Σ is positive definite. This

equation corresponds to a diffusion process with drift vectorµ and diffusion matrix Σ/N,

i. e. the intensive Markov jump process can be approximated by a diffusion satisfying the SDE

dxt=µ(xt, t)dt+√1

N σ(xt, t)dBt , xt0 =x0,

where σ is a square root of Σ, i. e. Σ=σσ0. The matrix σ is not necessarily unique as

Strictly speaking, since the Kolmogorov equation has been obtained using heuristic argu- ments, the Lipschitz continuity of µ and σ needs to be checked at this point in order to

ensure the existence of a solution to the above SDE (cf. Section 3.2.3). Note that such a solution is an approximation and not a limit as the system size parameterN is still part of

the diffusion matrix.

The expansion of the backward master equation can be performed similarly and is a special case of the approximation of the infinitesimal generator considered in the next section.

Example 4.2. Recall the SI model from Example 4.1 on page 51. The stochastic process

counting the absolute number S of susceptibles in a population of size N is described by

the master equation (4.9). Now consider the fraction s =S/N of susceptible individuals

and define α= λN. Let pN(t, s) denote the transition probability of the according intensive

process,wN,1(t, s) =WN(t, N s,−1) =N αs(1−s)the transition rate andw1(t, s) =αs(1−s)

the scaled transition rate, i. e. wN =N w1. Then the master equation of the intensive jump

process reads

∂pN(t, s)

∂t =

w1(t, s+ε)pN(t, s+ε)−w1(t, s)pN(t, s)

ε ,

where ε = N−1_{. This corresponds to Equation (4.18) above. The right hand side of the}

master equation is already of the form of a difference quotient with respect to a fixed vector (·, ε)0, ∂pN(t, s) ∂t = D1 (0,1)0_,(·,ε)0(w₁·p_N)(t, s) ε .

The dot in the vector (·, ε)0 of small parameters means that it is needless to fix its first

component as no derivative with respect to the first argument of w1 ·pN is considered.

According to ExampleB.1, the above quotient should not be approximated by (∂/∂s)(w1 ·

p)(t, s) but by Formula (B.7), i. e. ∂p(t, s) ∂t = ∂(w1·p)(t, s) ∂s + ε 2 ∂2(_w 1 ·p)(t, s) ∂s2 , (4.21)

where pN has been replaced by its limit function p. This is the Kolmogorov forward equation

that has already been stated by Equation (4.13) for the extensive process. The Kolmogorov equation (4.21)corresponds to a diffusion process with driftµ(s, t) =−w1(t, s) =−αs(1−s)

and diffusion N−1Σ(_{s, t}) = _εw

1(t, s) =αs(1−s)/N, i. e. to the solution of the SDE

dst=−αst(1−st)dt+ 1 √ N q αst(1−st)dBt (4.22)

with an appropriate initial value. For N → ∞, (4.21) and (4.22) reduce to

∂p(t, s) ∂t = ∂(w1·p)(t, s) ∂s and dst=−αst(1−st)dt.

4.3 Approximation Methods 61