A pathwise construction of an h-transform

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Let R0 be a random probability measure on the finite state space of thetypes E={1,2, . . . , K⁰} for K⁰ ≥2. AssumeR₀ is independent of N. Conditionally on (R₀, N), we define the look-down particle system ξ= (ξ_t(n), t≥0, n∈N):

– The initial types (ξ0(n), n∈N) form an exchangeable sequence valued in E with de Finetti’s measureR₀: Conditionally onR₀, (ξ₀(n), n∈N) is a sequence of independent random variables with law R₀.

– At each atom (t, π) ofN, we associate areproduction event as follows: let j1 < j2 < . . .be the elements of the unique block of the partitionπ which is not a singleton (either it is a doubleton or an infinite set). The individualsj₁ < j₂ < . . . at timetare declared to be the children of the individual j1 at timet−, and receive the type of the parentj1, whereas the types of all the other individuals are shifted upwards accordingly, keeping the order they had before the birth event: for each integer `,ξ<sub>t</sub>(j<sub>`</sub>) =ξt−(j₁) and for eachk /∈ {j<sub>`</sub>, `∈N}, ξ_t(k) =ξt−(k−#J_k) with #Jk the cardinality of the set Jk:={` >1, j`≤k}.

– For each n∈N, the typeξt(n) of the particle at leveln do not evolve between reproduction events which affect level n.

Remark 3.2.1. The integrability condition ^R_(0,1]x²ν(dx)<∞ ensures that finitely many repro-duction events change the type of a particle at a given level in a finite time interval.

For a fixed t≥0, the sequence (ξt(n), n∈N) is exchangeable according to Proposition 3.1 of [35]. This allows to define the random probability measureRt onE as the de Finetti measure of the sequence (ξ_t(n), n∈N):

Rt(dx) = lim

N→∞

1 N

N

X

n=1

δ_ξt(n)(dx) (3.1)

The processR= (Rt, t≥0) takes values in the spaceM_f(E) of finite measures on E (in fact, in the space of probability measures), and we endowM_f(E) with the topology of weak convergence.

We shall work with the càdlàg version of the process R (such a version exists according to Theorem 3.2 of [35]). The processR is called theΛ-Fleming-Viot process without mutation. We stress that, conditionally givenRt, the random variables (ξt(n), n∈N) onE are independent and identically distributed according to the probability measure R_tthanks to de Finetti’s Theorem.

This key fact will be used several times in the following.

We will denote by Pthe law of ξ. We now introduce the relevant filtrations we will work with:

– (F_t=σ{(ξ_s(n), n∈N),0≤s≤t}) corresponds to the filtration of the particle system.

– (G_t=σ{R_s,0≤s≤t}) corresponds to the filtration of the measure-valued processR.

Notice thatξ is a Markov process with respect to the filtrationF, and thatRis a Markov process with respect to the filtrationG.

3.2.2 A pathwise construction of an h-transform Results

The proofs of the results enounced here may be found in the next Subsection. Fix 1≤K ≤K⁰. We assume from now on and until the end of Section 3.2 that:

E(

K

Y

i=1

R₀{i})>0, (3.2)

to avoid empty definitions in the following. Recall the definition of the particle systemξ associated withR. We define fromξ a new particle system ξ^∞ as follows:

(i) The finite sequence (ξ₀^∞(j),1≤j ≤K) is a uniform permutation of{1, . . . , K}, and, inde-pendently, the sequence (ξ₀^∞(j), j≥K+ 1) is exchangeable with asymptotic frequencies R^H₀ , where R^H₀ is the random probability measure with law:

P(R^H₀ ∈A) =E 1_A(R₀) QK

i=1R₀{i}

E(^QK_i=1R₀{i})

! .

(ii) The reproduction events are given by the restriction of the Poisson point measure N (defined as in Subsection 3.2.1) to V :=ⁿ(s, π), π|[K]={{1},{2}, . . . ,{K}}^o, where π|[K] is the restriction of the partition π of N to {1, . . . , K}, that is the atoms of N for which the reproductions events do not involve more than one of the firstK levels.

Remark 3.2.1 ensures that this definition of the particle system ξ^∞ makes sense.

Remark 3.2.2. Note that the particle system (ξ₀^∞(j), j≥1) is no more exchangeable due to the constraint on the K initial levels. Nevertheless, the particle system (ξ₀^∞(j), j > K) is still exchangeable, and we shall view the firstK levels asK independent sources of immigration. This approach will be used in Section 3.2.4.

We also need the definition of the first levelL(t) at which the firstK types appear:

L(t) = inf{i≥K,{1, . . . , K} ⊂ {ξ_t(1), . . . , ξ_t(i)}}, (3.3) with the convention that inf{∅} = ∞. The random variable L(0) is finite if and only if QK

i=1R₀{i} > 0, P-a.s., thanks to de Finetti’s Theorem. The process L(t) is F_t measurable, but not G_t measurable. Notice the random variable L(t) is an instance of the coupon collector problem, based here on a random probability measure R: how many levels do we need to check for seeing the first K types? We define, fori≥1, thepushing rates r_i at leveli:

r_i = i(i−1)

2 c+

Z

(0,1]

ν(dx)1−(1−x)ⁱ−ix(1−x)ⁱ⁻¹.

Notice that r₁ = 0 and that r_i is finite for every i ≥ 1 since ^R_(0,1]x² ν(dx) < ∞. From the construction of the look-down particle system, these pushing rates ri may be understood as the rate at which a type at leveliis pushed up to higher levels (not necessarilyi+ 1) by reproduction events at lower levels. Let us define a process Q= (Q_t, t≥0) as follows:

Q_t= 1_{L(t)=K}

P(L(0) =K)e^rKt.

Lemma 3.2.3. The process Q= (Q_t, t≥0)is a non negative F-martingale, and

∀A∈ F_t, P(ξ^∞∈A) =E(1_A(ξ) Qt). (3.4) We need the following definition of the process:

Mt= QK

i=1Rt{i}

E(^QK_i=1R0{i})e^rKt. By projection on the smaller filtration G_t, we deduce Lemma 3.2.4.

3.2 A product typeh-transform 95

Lemma 3.2.4. The process M = (Mt, t≥0)is a non negative G-martingale.

This fact allows to define the processR^H = (R^H_t , t≥0) absolutely continuous with respect to R= (R_t, t≥0) on eachG_t,t≥0, with Radon Nykodim derivative:

∀A∈ G_t, P(R^H ∈A) =E(1_A(R) M_t). (3.5) The processR^H is the product typeh-transform of interest. Intuitively, the ponderation byM favours the paths in which the first K types are present in equal proportion. Also notice that equation (3.5) agrees with the definition ofR^H₀ . We shall deduce from Lemma 3.2.3 and Lemma 3.2.4 the following Theorem, which gives the pathwise construction of the h-transform R^H of R.

Theorem 3.2.5. Let 1≤K ≤K⁰. We have that:

(a) The limit of the empirical measure:

R^∞_t (dx) := lim

N→∞

1 N

N

X

n=1

δ_ξ^∞

t (n)(dx) exists a.s.

(b) The process (R^∞_t , t≥0)is distributed as (R^H_t , t≥0).

Let us comment on these results. The processξ^∞ is constructed by changing the initial condition and forgetting (as soon as K ≥2) specific reproduction events in the look-down particle system of ξ. Lemma 3.2.3 tells us that this procedure selects the configurations ofξ in which the first K levels are filled with the first K types at initial time without any “interaction” between these first K levels at a further time. Theorem 3.2.5 tells us that the processR^∞ constructed in this way is an h-transform of R and Lemma 3.2.4 yields the following simple probabilistic interpretation of the Radon Nikodym derivative in equation (3.5): the numerator is proportional to the probability that the first K levels are occupied by the firstK types at timet, whereas the denominator is proportional to the probability that the firstK levels are occupied by the firstK types at time 0. We shall see in Section 3.2.3 that the processes ξ^∞ and R^∞ also arise by conditioning the processes ξ and R on coexistence of the first K types.

Proofs

Proof of Lemma 3.2.3. From the de Finetti theorem, conditionally onRt, the random variables (ξ_t(i), i∈N) are independent and identically distributed according to R_t. This implies that:

P(L(t) =K|G_t) =K!

K

Y

i=1

Rt{i}. (3.6)

In particular, we have:

P(L(0) =K) =K!E(

K

Y

i=1

R₀{i}), which, together with (3.2), ensures that Q_t is well defined.

Then, let us defineW ={π, π_|[K] ={{1},{2}, . . . ,{K}}}, and Vt ={(s, π),0≤s≤t, π∈W}, and also the set difference W^c=P_∞\W andV_t^c={(s, π),0≤s≤t, π∈W^c}. We observe that:

– From the de Finetti Theorem, the law of ξ^∞₀ , as defined in (i), is that of ξ0 conditioned on {L(0) =K}.

– The law of the restriction of a Poisson point measure on a given subset is that of a Poisson point measure conditioned on having no atoms outside this subset: thus N conditioned on having no atoms in V_t^c (this event has positive probability) is the restriction of N toV_t. Since the two conditionings are independent, we have, for A∈ F_t:

P(ξ^∞∈A) =P(ξ∈A|{L(0) =K} ∩ {N(V_t^c) = 0})

This implies from the construction ofN that:

P(N(V_t^c) = 0) = e^−µ(Wc)t=e^−rKt. (3.8) Notice that

{L(t) =K}={L(0) =K} ∩ {N(V_t^c) = 0}. (3.9) From (3.7), (3.8) and (3.9), we deduce that:

P(ξ^∞∈A) =E Observe now that Aalso belongs to F_s as soon ass≥t, which yields:

P(ξ^∞∈A) =E(1_A(ξ)Q_s).

Comparing the two last equalities ensures that (Qt, t≥0) is aF-martingale. ut Proof of Lemma 3.2.4. We know from Lemma 3.2.3 that (Q_t, t ≥0) is a F-martingale. Since G_t⊂ F_t for everyt≥0, we deduce that (E(Qt|G_t), t≥0) is aG-martingale. But using (3.6) for the second equality, so that (Mt, t≥0) is aG-martingale.

u t Proof of Theorem 3.2.5. From Lemma 3.2.3, ξ^∞ is absolutely continuous with respect toξ on F_t. The existence of the almost sure limit of the empirical measure claimed in point (a) follows from (3.1). We now project on G_t the absolute continuity relationship on F_t given in Lemma 3.2.4. Let A∈ G_t:

P(R^∞∈A) =E(1_A(R)Qt) =E(1_A(R)E(Qt|G_t)) =E(1_A(R)Mt) =P(R^H ∈A),

where we use Lemma 3.2.3 for the first equality and the definition of R^H for the last equality.

This proves point (b). ut

3.2 A product typeh-transform 97