Splitting trees with neutral Poissonian mutations I: Small families.

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1 Splitting trees with neutral Poissonian mutations I: Small families. Nicolas Champagnat,AmauryLambert 2 August 7, 2 Abstract We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations which are not necessarily exponentially distributed and give birth singly at constant rate b. Such a genealogical tree is usually called asplittingtree[9],andthepopulationcountingprocessn t ; t is a homogeneous, binary Crump Mode Jagers process. We assume that individuals independently experience mutations at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e.,the number At of distinct alleles represented in the population at time t, and more specifically, the numbers Ak, t ofallelesrepresentedbyk individuals at time t, k =, 2,...,N t. We mainly use two classes of tools: coalescent point processes, as defined in [5], and branching processes counted by random characteristics, as defined in [, 2]. We provide explicit formulae for the expectation of Ak, t conditional on population size in a coalescent point process,which apply to the special case of splitting trees. Weseparatelyderivethea.s.limitsofAk, t/n t and of At/N t thanks to random characteristics, in the same vein as in [9]. Last, we separately compute the expected homozygosity by applying a method introduced in [4], characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations. MSC 2 subject classifications: Primary 6J8; secondary 92D, 6J85, 6G5, 6G55, 6J, 6K5. Key words and phrases. branching process coalescent point process splitting tree Crump Mode Jagers process linear birth death process allelic partition infinite alleles model Poisson point process Lévy process scale function regenerative set random characteristic. TOSCA project-team, INRIA Nancy Grand Est, IECN UMR 752, Nancy-Université, Campus scientifique, B.P. 7239, 5456 Vandœuvre-lès-Nancy Cedex, France, Nicolas.Champagnat@inria.fr 2 Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 CNRS and UPMC Univ Paris 6, Case courrier 88, 4PlaceJussieu,F-75252ParisCedex5,France, amaury.lambert@upmc.fr

2 Introduction We consider a general branching population, where individuals reproduce independently of each other, have i.i.d. lifetime durations with arbitrary distribution, and give birth at constant rate during their lifetime. We also assume that each birth gives rise to a single newborn. The genealogical tree associated with this construction is known as a splitting tree [8, 9, 5]. The process N t ; t counting the population size is a non-markovian birth death process belonging to the class of general branching processes, or Crump Mode Jagers CMJ processes. Since births arrive singly and at constant rate, these processes are sometimes called homogeneous, binary CMJ processes. Next, individuals are given a type, called allele or haplotype. They inherit their type at birth from their mother, and their germ line change type throughout their lifetime, at the points of independent Poisson point processes with rate θ, conditionalonlifetimesneutralmutations. Thetypeconferred by a mutation is each time an entirely new type, an assumption known as the infinitely-many alleles model. We are interested in the so-called allelic partition partition into types of the population alive at time t. Aconvenientwayofdescribingthispartitionwithoutlabellingtypesistodefinethenumber A θ k, t oftypescarriedbyk individuals at time t. The sequence A θ k, t; k is called the frequency spectrum of the allelic partition. We also denote by A θ t thetotalnumberofdistinct types at time t. ThemostcelebratedmathematicalresultinthissettingisEwens samplingformula, which yields the distribution of the frequency spectrum for the Kingman coalescent tree with neutral Poissonian mutations [7]. Credit is due to G. Yule [2] for the first study of a branching tree with mutations, but the interest for the infinitely-many alleles model applied to branching trees has started with the work of R.C. Griffiths and A.G. Pakes [], where the tree under focus is a Galton Watson tree and each individual, with a fixed probability, is independently declared mutant at birth. A fascinating monography dedicated to general branching processes also undergoing mutations only at birth times is due to Z. Taïb [9]. An extensive use is done there of a.s. limit theorems for branching processes counted by random characteristics, due to P. Jagers and O. Nerman [, 2, 3, 6]. More recently, in a series of three companion papers, J. Bertoin [2, 3, 4] has set up a very general framework for Galton Watson processes with mutations, where he has considered the allelic partition of the whole population from origination to extinction, and studied various scaling limits for large initial population sizes and low mutation probabilities. Branching processes have also been used in the study of multistage carcinogenesis. In this setting, the emphasis is put on the waiting time until atargetmutationoccurs,see[6,8]andthereferencestherein. In this paper, we study the part of the frequency spectrum corresponding to families with a fixed number k of carriers, k, that we call small families. We use three techniques: coalescent point processes, branching processes counted by random characteristics, and Kolomogorov-type equations as a function of the origination time of the tree. In a companion paper [5], we will discuss the part of the frequency spectrum corresponding to the largest or/and oldest families the age of a family being that of their original mutation. 2

3 2 Model and statement of main results 2. Model In this work, we consider genealogical trees satisfying the branching property and called splitting trees [8, 9]. Splitting trees are those random trees where individuals lifetime durations are i.i.d. with an arbitrary distribution, but where birth events occur at Poisson times during each individual s lifetime. We call b this constant birth rate and we denote by V ar.v.distributedasthelifetime duration. Then set Λdr := bpv dr a finite measure on, ] with total mass b called the lifespan measure. We will always assume that a splitting tree is started with one unique progenitor born at time. The process N t ; t counting the number of alive individuals at time t is a homogeneous, binary Crump Mode Jagers process, whichisnotmarkovianunlessλhasanexponentialdensityor is the Dirac mass at {+ } Figure : A coalescent point process for 6 individuals, hence 5 branches. In [5], it is shown that the genealogy of a splitting tree conditioned to be extant at a fixed d time t is given by a coalescent point process, thatis,asequenceofi.i.d.randomvariablesh i = H, i, killed at its first value greater than t. In particular, conditional on N t,n t follows a geometric ditribution with parameter PH < t. More specifically, for any i N t, the coalescence time between the i-th individual alive at time t and the j-th individual alive at time t i.e., the time elapsed since the common lineage to both individuals splits into two distinct lineages is the maximum of H i+,...,h j.thegraphicalrepresentationonfigureisstraightforward.the common law of these so-called branch lengths is given by PH >s= W s, 2. where the nondecreasing function W is such that W = and is characterized by its Laplace transform. More specifically, these branch lengths are the depths of the excursions of the jump 3

4 contour process, say Y t,ofthesplittingtreetruncatedbelowlevelt. They are i.i.d. because Y t is a Markov process. Indeed, it is shown in [5] that Y t has the law of a Lévy process, say Y, without negative jumps, reflected below t and killed upon hitting. The function W is called the scale function of Y,andisdefinedfromtheLaplaceexponentψ of Y : ψx =x e rx Λdr x R ,+ ] Let α denote the largest root of ψ. In the supercritical case i.e., ] rλdr >, and in this case only, α is positive and called the Malthusian parameter, because the population size grows exponentially at rate α on the survival event. Then the function W is characterized by e xr W r dr = ψx x>α. Actually, it is possible to show by path decompositions of the process Y that x W x =exp b dt PJ >t, where J is the maximum of the path of Y killed upon hitting and started from a random initial value, distributed as V. Note that since Y is also the contour process of a splitting tree, J has the law of the extinction time of the CMJ process N =N t ; t started from one individual. In the next section, we consider coalescent point processes without reference to a splitting tree. The law of such a process is merely characterized by a random number N of i.i.d. r.v. H i independent of N, bothwitharbitrarydistributions.inthissetting,2.converselyservesasadefinition of W, which is now an arbitrary nondecreasing function, whereas it was previously seen to be differentiable in the special case of splitting trees. The population size N can be fixed possibly infinite or truly random, e.g. following a geometric distribution. It will be written N t when the law of H is supported by [,t]. In this latter case, any result obtained under the assumption that N follows a geometric distribution can be applied to the case of splitting trees. Throughout this work, we assume that individuals independently experience mutations at Poisson times during their lifetime, that each new mutation event confers a brand new type called haplotype, or allele to the individual, and that a newborn holds the same type as her mother at birth time. The mutation rate is denoted by θ. 2.2 Outline and statement of main results The main technique we use relies on the previously described representation of the genealogy of asplittingtreebyasequenceofi.i.d.r.v.h i i,calledthecoalescentpointprocess. Thisidea was first exploited by Aldous and Popovic [] and Popovic [7] and then it was further developed by Lambert [5]. The common distribution of H,H 2,...is related to the scale function W.Wewillalso use the scale function W θ associated with the lifetime of clonal families standard lifetime truncated at its first mutation event. Section 3 is dedicated to some fine computations in the general framework of coalescent point processes. For example, for a coalescent point process H,H,...,H X ofaget, 4

5 where X is an independent geometric r.v., Theorem 3.3 gives the expectation of A θ k, tu X.Various corollaries are stated, giving the expectation, sometimes conditional on the population size, of specific quantities of biological interest at the fixed time t. Those statements extend results of [4] given under a doubly asymptotic regime t, n. For example, Corollary 3.4 gives the expectation of the number of distinct alleles and of homozygosity probability of drawing two individuals carrying the same allele and Corollary 3. gives the expectation of the number Z y; n amongthen first individuals who carry the ancestral type of lineage y units of time in the past E Z y; n =e θy n PH y k, see Remark 3. for a simple interpretation of this formula. In Section 4, some of the previous results are specified to the case of splitting trees. In particular, Proposition 4. yields the expectation of A θ k, tu Nt,aswellasofZ tu Nt,whereZ t denotesthe number of alive individuals at time t carrying the ancestral allele. The result for A θ k, t canevenbe detailed to the case of haplotypes of a given age. As previously, various corollaries are provided for some quantities such as the homozygosity. Ruling out the information on the population size i.e., taking u =andontheageofthemutation,corollary4.3reads and E t A θ k, t =W t P t Z t =k =W t k= dx θ e θx W θ x 2 k, W θ x e θt W θ t 2 k, W θ t where P t is the conditional probability on survival up until time t. Note also that Subsection 4.2 provides the reader with a more explanatory proof of the previous formulae. The theory of random characteristics [, 2, 3, 6, 9], which is the second main technique we use, is displayed in Section 5. There, the random characteristic of individual i, say, canbefor example the number χ k i t ofmutationsthati has experienced during her lifetime and which are carried by k alive individuals, t units of time after her birth χ i t =ift<. Then the total number of haplotypes carried by k individuals at time t except possibly the ancestral type is the sum over all individuals i dead or alive of χ i t σ i, where σ i is the birth time of individual i. Now according to limit theorems by P. Jagers and O. Nerman [, 2, 3, 6], these sums converge a.s. on the survival event in the supercritical case. Exploiting those limit theorems, we are able to deduce the following a.s. convergences in the supercritical case see Proposition 5., where the limits are computed independently from the results obtained earlier. On the survival event, A θ k, t lim = U k t A θ t U a.s. and A θ t lim = U a.s., t N t 5

6 where and U k := dx θ e θx W θ x 2 k, W θ x U := k U k = dx θ e θx W θ x. In the final section Section 6, we consider G θ t :=Z tz t /2+ k kk A θk, t/2, that we term absolute homozygosity, in reference to standard homozygosity, which is defined as Ḡ θ t =2G θ t/n t N t. Homozygosity is a well-known measure of diversity, that can be seen as the probability that two randomly sampled distinct individuals or sequences share the same allele. In the spirit of backward Kolmogorov equations, we derive the dynamics of the expectation of G θ tu Nt as the origination time of the tree moves back in time. Then the expected standard and absolute homozygosity can be computed. In passing, we recover formulae obtained in Section 4 by totally different methods. Specifically, we get E t G θ t =W tw 2θ t. 3 Expected haplotype frequencies for coalescent point processes In this section, unless otherwise specified, we assume that the lineage of individual, sometimes called lineage, is infinite, and that all other branch lengths are i.i.d., distributed as some r.v. H. To each H i corresponds an individual, that we call individual i. We also assume that mutations occur according to a Poisson point process on edge lengths with parameter θ. 3. The next branch with no extra mutation We let E θ denote the set of individuals who carry no more mutations but possibly less than individual some of and at most exactly the mutations carried by, but no other mutation. We call such individuals, -type individuals same type as some point on lineage at some time in the past. Set K θ := and for i, define Kθ i as the label of the i-th individual in E θ.inaddition,set and H θ i := max{h j : K θ i <j Kθ i+ } B θ i := Kθ i Kθ i. See Figure 2 for a graphical representation of these quantities on a typical coalescent point process with mutations. We write B θ,h θ inlieuofb θ,hθ andwedefinew θx; γ by We will also need the following notation W θ x; γ := E x,γ, ]. γ Bθ,H θ x W x; γ := γph x x,γ, ]. 6

7 e c 2 c 3 c 4 b 5 b 6 d 7 d 8 a 9 f a f e c b d H θ a B θ Figure 2: On this coalescent point process, the 8-th individual is the first one whose type is the same as some point on lineage anywhere in the past, so that 8 E θ and B θ =8. ThemaximumHθ of the first B θ branch lengths is shown. Also note that Eθ and B θ 2 =2. Theorem 3. The bivariate sequence Bi θ,hθ i ; i is a sequence of i.i.d. random pairs. In addition, the following formula holds for all x and γ, ] W θ x; γ =e θx W x; γ+θ x W y; γ e θy dy. Remark 3.2 Differentiating both sides of the previous equation w.r.t. the first variable yields dw θ x; γ =e θx dw x; γ. Also, the formula in the previous statement was shown in [4] in the special case γ =. Proof. First observe that the pair K θ,hθ doesnotdependonthehaplotypeofindividual,and that the i-th, -type individual is also the next individual after Ki θ with no mutation other than those carried by individual Ki θ.thisensuresthatkθ i Kθ i,hθ i hasthesamelawaskθ,hθ, and the independence between Ki θ Kθ i,hθ i andpreviouspairsisduetotheindependenceof branch lengths and the fact that new mutations can only occur on branches with labels strictly greater than Ki θ. As for the formula relating W θ and W,weconsidertherenewalprocessS defined by S =and S n = n i= Bθ i.next,foranyintegerk, let F k denote the event F k := { n :S n = k, M n x}, where M n := max{hi θ : i n}. LetT k denote the time elapsed since the lineages of individual andindividualk have split up, that is, T k =max{h i : i k}. NoticethatbydefinitionofHi θ, 7

8 T k = M n on the event {S n = k}, sothat F k = { n :S n = k, T k x}. So F k is the event that the lineage of individual k has had no mutation between time T k and present time i.e., no mutation on the part of its lineage not common with individual, and T k x. By standard properties of Poisson processes, we get PF k =E e θt k,t k x = PH x k e θx + θ x PH y k e θy dy. 3. Note that the r.h.s. of this equation is obtained using the integration by parts formula for càdlàg functions i.e. functions continuous on the right and admitting left limits at each points of the space, like PH x: if f is continuously differentiable and g is càdlàg with bounded variation, x fxgx =fg + f ygydy + fydgy. 3.2,x] Equation 3. yields On the other hand, γ k PF k =e θx W x; γ+θ k x W y; γ e θy dy. PS n = k, M n x γ k PF k = γ k k k n = E γ Sn,M n x n = E γ P n i= Bθ i,h θ x,..., Hn θ x n = n n E γ Bθ,H θ x = E γ Bθ,H θ x, which yields the desired result. 3.2 Expected haplotype frequencies for geometrically distributed population sizes Let X denote some independent geometric random variable with parameter γ,thatis,px n =γ n for any n. In the infinite-allele model, each haplotype is characterized by its most recent mutation. We denote by A θ k, y; γ thenumberofhaplotypeswhosemostrecentmutationoccurredbetweentime y and present time and which are carried by k individuals among {,,...,X}. 8

9 Theorem 3.3 For all k, y>, γ, ], u [, ], E u X A θ k, y; γ = γ uγ 2 dx θ e θx k W θ x; uγ 2. W θ x; uγ Let I θ y; γresp. I θ y; n denote the number of individuals among {,,...,X} resp. {,,...,n} whose most recent mutation appeared between time y and present time. Let A θ y; γresp. A θ y; n denote the number of distinct haplotypes represented in {,,...,X} resp. {,,...,n} whosemostrecentmutationappearedbetweentime y and present time. Let Ḡθy; n denotetheconditional probability that two distinct individuals randomly drawn conditional probability here refers to this sampling, given the coalescent point process and the mutation times on branches from {,,,n} share the same haplotype and that the most recent mutation of this common haplotype appeared between time y and present time. Corollary 3.4 For any integer n, E I θ y; n = n exp θy, E A θ y; n = n dx θ e θx PH θ >x+ and in the case where the law of H has no atom dx θ e θx E B θ n, H θ x. 3.3 n E Ḡθy; kn k y n = 2 PH dx PH x k e θx e θx e θy. nn k= Remark 3.5 The first expectation can readily be deduced from some exchangeability argument, since each individual carries a mutation with age smaller than y with probability exp θy there is no edge effect since the ancestral lineage is infinite. Remark 3.6 In [4], a pathwise result was shown for the number A θ,n of distinct haplotypes represented in {,,,n}, namely lim n n A θ,n= dx θ e θx PH θ >x a.s. Remark 3.7 In the case where the law of H admits atoms, the computation of EḠθy; n can be done following the same line as in the proof below, using the fact that dw x; γ has an atomic part. The computation gives n E Ḡθy; n = 2 k= { n k y k nn + x [,y] µ n.a. H dx PH xk e θx e θx e θy PH x k PH <x k e θx e θx e θy, where µ n.a. H is the non-atomic part of the law of H. 9

10 Proof of Corollary 3.4. For the first expectation, taking u = in the theorem, E I θ y; γ =E k ka θ k, y; γ = e θy γ, using repeatedly Fubini Tonelli theorem and k kxk = x 2 for any x [,. The result then follows from the inversion of the generating function using γ = n n + γγn. For the second expectation, E A θ y; γ =E k A θ k, y; γ = γ dx θ e θx W θ x; γ = dx θ e θx E γ Bθ,H θ x. γ Next invert the generating function as follows γ E γ Bθ,H θ x = n + γγ n PB θ = j, H θ xγ j n j = n γγ n n + kpb θ = k, H θ x n k= = n γγ n E n + B θ,b θ n, H θ x, which entails E A θ y; n = = = dx θ e θx n + E n + B θ,b θ n, H θ x dx θ e θx n +PH θ >x+e n + n + B θ {B θ n},h θ x dx θ e θx n +PH θ >x+e n + {B θ >n} + B θ {B θ n},h θ x, which yields the result. For the third expectation, we use the fact that the expected number of unordered pairs of individuals sharing the same haplotype younger than y equals where n nn + γγ Ḡ θ y; n =E[Ḡθy; γ], 2 n Ḡ θ y; γ := k 2 Now since k 2 kk xk =2x x 3,weget EḠθy; γ = γ = γ = γ kk A θ k, y; γ. 2 dx θ e θx W θ x; γ x dx θ e θx e θz dw z; γ dw z; γ e θz e θz e θy,

11 where differentiation of W is understood w.r.t. the first variable. Then we use the fact, when the law of H has no atom, dw z; γ = γph dz γph z 2 = PH dz nγ n PH z n. n The proof ends writing the product series between the last entire series and γ 2 = n n + γ n. Before proving the theorem, we insert a paragraph in which we state and prove a preliminary key result A key lemma We denote by l i the time elapsed since the i-th most recent mutation on the lineage of individual, also called lineage. Let N i y; γ denotethenumberof, -type individuals in {,...,X} whose most recent mutation time in its haplotype is l i if l i y, andn i y; γ =otherwise. We also define,y-type individuals as those individuals that have the same type as the point at time y on lineage. In other words, an individual is of,y-type if the most recent mutation of its haplotype is l i for the unique i such that l i y<l i,withtheconventionthatl :=. In the same vein,, [, y]-type individuals are those individuals that have the same type as some point on lineage at any time between time y and present time. We denote by Z y; γ thenumberof,y-type individuals of {,,...,X}. NotethatZ y; γ = N i ; γ wherei is such that l i y<l i.alsoseti y; γ thenumberof, [,y]-type individuals of {,,...,X} and I y; γ thenumberof, -type individuals of {,,...,X} whose most recent mutation appeared between time y and present time. Otherwise said, I y; γ =I y; γ+z y; γ and I y; γ = i N i y; γ Lemma 3.8 For all k, y>, γ, ], u [, ], i E u X,N i y,γ =k = γ dz θ e θz uγ W z; uγ W θ z; uγ 2 k W θ z; uγ and E u X,Z y; γ =k = γ W y; uγ k e θy uγ W θ y; uγ 2. W θ y; uγ Corollary 3.9 For all y>, γ, ], u [, ], E u X I y; γ = γ uγ dz θ e θz W z; uγ and E u X Z y; γ = γ uγ e θy W y; uγ. Note that, in the case γ = u =,onemustreplace γ/ uγ byintheseresults.

12 Proof. Use the formulae in Lemma 3.8 and Fubini Tonelli theorem repeatedly, in particular to see that E I y; γ u X = E u X N i y; γ = ke u X N i y;γ=k = k E u X N i y;γ=k. i i k k i The proof ends using k kxk = x 2 for all x [,. Let n be a non-negative integer. In the next corollary, Z y; n denotesthenumberof,y-type individuals of {,,...,n} and I y; n thenumberof, -type individuals of {,,...,n} whose most recent mutation appeared between time y and present time. Corollary 3. For all y> and n, E I y; n = θz PH zn+ dz θ e PH >z and θy PH yn+ E Z y; n =e. PH >y Proof. We use γ = k γk along with W z; γ = γph z = n γ n PH z n. Plugging these equalities into the first formula of the first corollary evaluated at u =yields E I y; γ = dz θ e θz y W z; γ = dz θ e θz γ n γ n n PH z k. Inverting the generating function yields the expression proposed for E I y; n. The very same line of reasoning can be applied to get E Z y; n. Remark 3. Keeping the expression in the proof of the theorem under the shape of a sum is more informative. Indeed, differentiating each side of the equality, we then get n E I dy; n =dy θ e θy k= PH y k, where I dy; n denotes the number of, -type individuals of {,,...,n} whose most recent mutation is of age in y,y + dy. Theinterpretationofthisnewexpressiongoesasfollows.Thetermθdy is the probability that a mutation occurred on lineage in the time interval y,y + dy backwards in time; the term PH y k is the probability that the lineage of individual k split off lineage more recently than y; theterme θy is the probability that the lineage of individual k has undergone no mutation in the last y units of time. k= 2

13 Proof of Lemma 3.8. Set D := and for i 2, D i := min{j :H θ j >l i }. Also recall the renewal process S n = n i= Bθ i.thenwehaveforalli N i y; γ = li y i= + D i+ the indicator function of i = beingduetothecountofindividualinthatcase. First, we work conditionally on the values v i of the ages l i of mutations of lineage. Using repeatedly the lack-of-memory property of X, wegetforalli 2andk E u X,N i y; γ =k l j = v j,j = v i y E u S D i,x S Di E u Bθ,B θ X, H θ v i H θ >v i k E u Bθ,B θ X, H θ v i E u X,B θ >X + E u X,B θ X, H θ >v i, where the last multiplicative term equals E u X,B θ >X + E u X,B θ X, H θ >v i Similarly for i =andk, j=d i S j X, = E u X E u X,B θ X, H θ v i u Bθ,B θ X, H θ v i = E u X E = γ E uγ Bθ,H θ v i uγ γ = uγw θ v i ; uγ. E u X,N y; γ k = k l j = v j,j = v y E u Bθ,B θ X, H θ v E u X E u Bθ,B θ X, H θ v k γ = v y E uγ Bθ,H θ v uγw θ v ; uγ. Now elementary probabilistic reasoning shows that for i 2 E u S D i,x S Di l j = v j,j = k = k k PH θ v i PH θ >v i E u Bθ,B θ X H θ v i PH θ >v i E uγ Bθ,H θ v i = PH θ >v i W θ v i ; uγ. 3

14 As a consequence, for all i 2, E u X,N i y; γ =k l j = v j,j = v i y γ uγ W θ v i ; uγ W θ v i ; uγ k E uγ Bθ,H θ v i E uγ Bθ,v i <H θ v i, whereas E u X,N y; γ =k l j = v j,j = v y γ uγ k E uγ Bθ,H θ v. W θ v ; uγ It is well-known that for the Poisson point process of mutations, so that i 2 where Since we get i Pl i dx, l i dz = θi x i 2 E u X,N i y; γ =k = γ uγ i 2 i 2! e θz dx dz <x<z,i 2, z dz dx θi x i 2 k i 2! e θz F θ x, z; uγ, W θ z; uγ W θ z; uγ F θ x, z; uγ :=W θ x; uγe uγ Bθ,x<H θ z. 3.4 E u X,N y; γ =k = γ uγ E u X,N i y; γ =k = γ uγ Now observe that F θ x, z; uγ =W θ x; uγ E = W θ x; uγ = W θx; uγ W θ z; uγ, so that the integration by parts formula 3.2 yields z +θ dx e θx F θ x, z; uγ =+ dz θ e θz k, W θ z; uγ W θ z; uγ dz θ e θz k [ z ] +θ dx e θx F θ x, z; uγ. W θ z; uγ W θ z; uγ uγ Bθ,H θ z E W θ x; uγ W θ z; uγ [e θx W θx; uγ W θ z; uγ ] z uγ Bθ,H θ x z e θx + W θ z; uγ dw θx; uγ, where differentiation of W is understood w.r.t. the first variable. Since by Theorem 3., dw θ x; uγ = e θx dw x; uγ, we get z +θ dx e θx W z; uγ F θ x, z; uγ = W θ z; uγ, 3.5 4

15 which ends the proof for the first formula. Let us turn to Z y; γ. The same kind of reasoning as previously shows that E u X,Z y; γ =k l j = v j,j = v i <y<v i E u S D i,x S Di E u Bθ,B θ X, H θ y H θ >v i i i 2 + i= k E u Bθ,B θ X, H θ y E u X,B θ >X + E u X,B θ X, H θ >y. Referring to the calculations above, we easily get E u X,Z y; γ =k l j = v j,j = γ uγ i W θ y; uγ v i <y<v i W θ y; uγ k [ i= + i 2W θ v i ; uγe uγ Bθ,v i <H θ y Integrating over the law of the Poisson point process of mutations yields E u X,Z y; γ =k = γ uγ e θy W θ y; uγ + γ uγ i 2 y dz W θ y; uγ dx θi x i 2 i 2! e θz W θ y; uγ where F θ was defined in 3.4. Thanks to equation 3.5, we get E u X,Z y; γ =k = γ uγ e θy W θ y; uγ which is the desired formula Proof of Theorem 3.3 = γ W y; uγ e θy uγ W θ y; uγ 2 W θ y; uγ W θ y; uγ k ]. k F θ x, y; uγ, W θ y; uγ k [ +θ k, ] dx e θx F x, y; uγ Let M n k, y; γ denotethenumberofhaplotypeswhosemostrecentmutationoccurredbetweentime y and present time on the n-th branch with i.i.d. lengths H n,excepth =+, and which are carried by k individuals among {,,..., X} hence among {n,n +,..., X}. In particular, A θ k, y; γ = n M n k, y; γ. First, M k, y; γ = i N i y,γ=k, so thanks to Lemma 3.8, E u X M k, y; γ = dz F k, z; uγ, 5

16 where we have used the following definition F k, z; uγ := γ W z; uγ k θe θz uγ W θ z; uγ 2. W θ z; uγ Second, for all n, by the lack-of-memory property of the geometric variable X, E u X M n k, y; γ [ = u n PX n PH n dxe u X M k, x; γ + PH n ye u X M k, y; γ ] [ x ] =uγ n PH dx dz F k, z; uγ+ph y dz F k, z; uγ =uγ n dz F k, z; uγ PH z. Now since A θ k, y; γ = n M nk, y; γ, we get E u X A θ k, y; γ = = = hence the result, recalling the definition of F. dz F k, z; uγ+ uγ n dz F k, z; uγ PH z n [ dz F k, z; uγ + uγ ] PH z uγ dz F k, z; uγ[ uγw z; uγ], 4 Splitting trees: Expected haplotype frequencies at fixed time 4. Joint expected haplotype frequencies with population size distribution In this subsection, we apply the results of the previous section to a splitting tree started at time t from one single individual and conditioned to be extant at present time. Then the population at present time is {,,...,N t }, wheren t is the population size and N t followsthegeometric distribution with parameter γ t := PH t t>, that is, P t N t n =γt n for any integer n, where P t denotes the probability conditional on the population being extant at time, that is, t units of time after foundation. We recall that, in the case of splitting trees, the law of the branch lengths H is always absolutely continuous w.r.t. Lebesgue s measure. The difference with the previous section is that the lengths of branches are still i.i.d. but distributed as H conditional on H t. Asaconsequence,everythingwehavedoneintheprevioussectionholds for the standing population of a splitting tree founded t units of time ago and conditioned upon survival up to t, replacingγ with γ t and W with from Theorem 3. W t x; α := αph x H t x [,t],α, ]. 6

17 In particular we now use W t θ instead of W θ,with W t θ x; α =e θx W t x; α+θ x dy W t y; α e θy. Noticing that W t x; uγ t =Wx; u, we also have W t θ x; uγ t=w θ x; u, where we stick to the notation from the previous section, namely, and W x; u = uph x W θ x; u =e θx W x; u+θ x x,u, ], dy W y; u e θy. We call a derived haplotype ahaplotypewhichisdifferentfromtheancestralhaplotype. Thenthe following statement stems readily from Theorem 3.3 and Lemma 3.8. Recall that W x =W x; and that W θ x =W θ x;. Proposition 4. Let A θ k, t denote the number of derived haplotypes represented by k individuals in the standing population of a splitting tree founded t units of time ago and Z t the number of individuals in the standing population carrying the ancestral haplotype. Then for all t and u, ], and E t u N t A θ k, t = W t; u2 W t E t u N t,z t =k = dx θ e θx k W θ x; u 2. W θ x; u W t; u2 W t e θt k W θ t; u 2. W θ t; u Remark 4.2 Not to overload with notation, we have not considered the alleles of age less than y. If A θ k, y, t denotes the number of derived haplotypes of age less than y, representedbyk individuals in the standing population of a splitting tree founded t units of time ago, then we get the same formula as in the previous statement, but where the upper bound of the integral has changed E t u N t A θ k, y, t = W t; u2 W t t dx θ e θx W θ x; u 2 W θ x; u k. The following corollary is obtained by taking u =inthelaststatement. Amoreexplanatoryproof is given in the next subsection. Corollary 4.3 We have and E t A θ k, t =W t P t Z t =k =W t dx θ e θx W θ x 2 k W θ x e θt W θ t 2 k. W θ t 7

18 The same kinds of calculations as those done for the corollaries of the previous section yield the following statement, where the first equation could readily be deduced by exchangeability arguments. Corollary 4.4 Recall that Z t is the number of individuals in the standing population carrying the ancestral type and set A θ t the number of derived haplotypes represented in the standing population. Then for any positive real number t and positive integer n, and EA θ t N t = n =n EZ t N t = n =n exp θt dx θ e θx E PH t Bθ {H θ x} + dx θ e θx E B θ n PH t Bθ,H θ x. Proof. The firstresult is clear letting y go to + in Corollary 3.. In view of 3.3 in Corollary 3.4, in order to prove the second result, we only need to check that PH θ >x=e PH t Bθ {H θ x} and ẼB θ n, H θ x =E B θ n PH t Bθ,H θ x, where P is the law of the coalescent point process when the r.v. H i arei.i.d.withcommonlaw PH H t. Now, PH θ x =PH θ x i B θ, H i t = PB θ = k, H θ xph t k k = E PH t Bθ {H θ x}. The second equality, very similar, is left to the reader. Recall that G θ t denotestheabsolutehomozygosityinthestandingpopulation,thatis, then we easily get G θ t = Z tz t 2 Proposition 4.5 For all t and u, ], E t u N t G θ t = + k 2 kk A θ k, t, 2 W t; u2 W 2θ t; u. W t Note that explicit formulas can also be obtained for the expectation of the standard homozygosity Ḡ θ t =2G θ t/n t N t, which is the probability that two randomly sampled individuals in the population at time t have the same haplotype. Formulas are given in Section 6, where they are obtained thanks to an alternative proof based on moment generating function computations. 8

19 Proof. We use Proposition 4. and the fact that k 2 kk xk 2 =2/ x 3.Anintegration by parts yields E t u N t G θ t W t; u2 = e θt W t; u2 W θ t; u + dx θ e θx W θ x; u W t W t W t; u2 = e θt W t; u2 [ ] t W θ t; u + e θx W θ x; u W t W t + dx e θx W θ x; u, where differentiation is understood w.r.t. the first variable. Recalling that W θ x; u =e θx W x; u provides the announced formula. 4.2 An explanatory proof of Corollary 4.3 Consider the standing population at time t conditioned on being nonempty probability measure P t. For any real number y,t, for any non-negative integer i, letc i y; dy, D i y ande i y denote the following events C i y; dy :={i N t, the i-th branch length has size H i y and carries a mutation with age in y,y + dy} D i y :={the type carried by the lineage of the i-th individual at time t y has at least one alive representative} E i k, y :={the type carried by the lineage of the i-th individual at time t y has k alive representatives} Then define A θ k, t, y; dy asthenumberofhaplotypesofageintheintervaly,y+dy representedby exactly k alive individuals at time t. Hereafter,wecomputetheexpectationunderP t of A θ k, t, y; dy. The result will follow from the equality Now it is readily seen that A θ k, t = A θ k, t, y; dy. A θ k, t, y; dy = i C i y;dy E i k,y so that E t A θ k, t, y; dy = i P t C i y; dy E i k, y. Next observe that E i k, y D i y, so that P t C i y; dy E i k, y = P t C i y; dyp t D i y C i y; dyp t E i k, y D i y C i y; dy = P t C i y; dyp t D yp t E k, y D y. 9

20 Thus, we record that E t A θ k, t, y; dy =P t D yp t E y D y i P t C i y; dy. 4. We will now prove the three following equalities i P t C i y; dy = θdy W t W y, 4.2 W y e θy P t D y =, W θ y 4.3 P t E k, y D y = k. W θ y W θ y 4.4 These three equalities, along with 4., yield the expected expression E t A θ k, t, y; dy =θdywt e θy W θ y 2 k, 4.5 W θ y which now sheds light on the meaning of each of the terms in the formula given in Corollary 4.3. Let us now prove equations 4.2, 4.3 and 4.4. First, P t C i y; dy = P t N t i θdy i= + = i θdy W t [ = θdy i= + i W t i= + i i PH y H<t W y W t W t i W y W t so we get 4.2. Second, let L denote an independent exponential r.v. with parameter θ, sothaty L + is the age of the oldest mutation on lineage with age smaller than y, withtheconventionthatthisage is zero when there is no such mutation. Then either L y, andd y isrealizedbecauselineage has carried the same type since time t y, orl<yand D y isrealizediffthenextbranchwith no extra mutation than for which the maximum of past branch lengths exceeds t L satisfies that this maximum does not exceed y see Subsection 3.. Conditional on L = x, thislasteventoccurs with probability PH θ y H θ >y x. As a consequence, we get P t D y = e θy + dx θ e θx W θy x W θ y = dx θ e θx W θ y x W θ y = e θy W θ y du θ e θu W θ u, and an integration by parts using the relationship between W and W θ see Remark 3.2 yields 4.3. Finally, 4.4 stems from the definition of W θ see again Subsection ],

21 5 Splitting trees: A.s. convergence of haplotype frequencies In this section, we rely on the theory of random characteristics introduced in the seminal papers [, 6] and further developed in [2, 3] and especially in [9], where the emphasis, as here, is on branching populations experiencing mutations but there the mutation scheme is different, since mutation events occur simultaneously with births. We will assume that the splitting tree starts at time with one individual. Then recall from the last subsection that N t denotes the number of individuals alive at time t, A θ t denotesthenumber of derived haplotypes carried by alive individuals at time t, A θ k, t denotesthenumberofderived haplotypes carried by k alive individuals at time t, andz t denotesthenumberofaliveindividuals at time t carrying the ancestral haplotype. For any individual i, inthepopulation,weletχ i t resp.χ k i t be the number of mutations that i has experienced during her lifetime that are carried by alive individuals resp. by k alive individuals t units of time after her birth χ i t =ift<. Then χ and the χ k are random characteristics, in the sense given in the previously cited papers. In particular, A θ t = i χ i t σ i, and A θ k, t = i χ k i t σ i, where σ i denotes the birth time of i and the sum is taken over all individuals, dead or alive at time t, in the population. This allows us to make use of limit theorems for individuals counted by random characteristics proved in [, 2, 3, 6], using the formulation of [9, Appendix A]. Different limit theorems hold depending whether the random characteristic is individual, inthesensethatitonly depends on the life history of the focal individual, or general, inthesensethatitmayalsodepend on the life history of the whole descendance of the focal individual, which is for example the case of χ and χ k. Recall that b is the birth rate of our homogeneous Crump Mode Jagers process, that V denotes arandomlifetimeduration,andthatα denotes the Malthusian parameter, which satisfies ψα =, where ψ is defined in 2.2. Let us restate the results in [9, Appendix A] in our setting. Set β := ue αu dµu,, ] where the last integral is a Stieltjes integral w.r.t. the nondecreasing function µt =E# offspring born on,t] = bet V = r tλdr.,+ ] Also for any random characteristic, say χ, define χα asitslaplacetransformatα χα := dt e αt χt,,+ 2

22 where it is implicit that χ is the characteristic of the progenitor born at time. Hereafter, we apply Theorems and 5 of [9, Appendix A], which apply to general random characteristics. These theorems need some technical assumptions to hold, which we verify at the end of the proof of the next statement. These theorems ensure first that and second that, on the survival event, lim t e αt EA θ k, t = E χ k α β A θ k, t lim = E χ k α t A θ t E χα a.s. In addition to verifying the validity of the aforementioned technical assumptions, it remains to compute the quantities β, E χα ande χ k α. With the following definitions, U k := dx θ e θx W θ x 2 k, W θ x and U := k U k = dx θ e θx W θ x, we have β = ψ α/α, E χ k α =U k /b and of course E χα =U/b. This can be recorded in the following proposition. Proposition 5. In the supercritical case, lim t e αt EA θ k, t = αu k bψ α 5. and And on the survival event, lim t e αt EA θ t = A θ k, t lim = U k t A θ t U αu bψ α. 5.2 a.s. Remark 5.2 Note that it can be shown similarly that and that, for example, lim t e αt EN t = α bψ α, A θ t lim = U a.s. t N t This is reminiscent of Theorem 3.2 in [4] where the same limit is obtained after conditioning on the population size to equal n and letting n.thisa.s.convergenceismadepossiblebyembedding all populations of fixed size on the same space thanks to an infinite coalescent point process: the population of size n is that generated by the first n values of the coalescent point process. 22

23 Remark 5.3 In [5], it is proved in the supercritical case α > thatthesurvivalprobabilityis α/b and that the scale function W has the following asymptotic behaviour lim W t te αt = ψ α. One could have used these two facts and the monotone convergence theorem to recover 5. and 5.2 from Corollary 4.3. In the following proof, we prefer to show the agreement with Corollary 4.3 by computing directly β, E χα and E χ k α. Proof. Let us first prove that β = ψ α/α. Recallingthedefinitionofβ, weget β = be =,+ ] = α 2 du ue αu {u<v } Λdr,+ ] r du ue αu Λdr e αr αre αr = α 2 α ψα α ψ α = ψ α α. Next let us compute E χ k α. Denote by R a,b t the number of individuals alive at time t descending clonally from the time interval a, b. More specifically, for a progenitor individual alive on the time interval a, b andexperiencingnomutationbetweentimesa and b, R a,b t is the number of individuals alive at t including possibly this progenitor descending from those daughters of the progenitor who were born during the time interval a, b, and that still carry the same type that the progenitor carried at time a. Inparticular,sinceW θ is the scale function associated with the clonal reproduction process P R a,b t = k = PNt a θ = k ζ = b a = PNt a θ ζ = b apnt a θ = k Nt a θ W θ t b k = t>b W θ t a W θ t a W θ t a, 5.3 where N θ is the population size process of a clonal splitting tree and ζ is the lifetime of the progenitor. Now let us start with a progenitor with lifetime distributed as V and denote by l i the time of the 23

24 i-th point of a Poisson point process with intensity θ the i-th mutation of the progenitor. Then E χ k α = E = E = E = E dt e αt i dt e αt i {l i <V t} dz e θz z R l i,v l i+ t = k dy θi+ y i i! z V t dt e αt dzθ e θz dy θ e θy R Vθ dt e αt t dy θ e θy R y,v θ t = k, {y<v t} y,v z t R = k y,v z t = k where V θ denotes the minimum of V and of an independent exponential r.v. with parameter θ. Then E χ k α = = = which, thanks to 5.3, yields E χ k α = where and = dt e αt dt e αt dx θ e θx,, dx θe θx W θ x dx θe θx W θ x F 2 x :=, W θ x W θ x F x :=, Let us compute F and F 2.Set ψ θ x :=x PV θ du PV θ du PV θ du k dy dx u+x x PV θ du, k F x F 2x W θ x, PV θ du, PV θ du u+x x {y<u} θe θy P R y,u t = k u+x x {t x<u} θe θt x P R t x,u t = k dt e θ αt dt e θ αt P R t x,u t = k, u+x x, dt e θ αt dt e θ αt W θ t u t>u W θ x t>uw θ t u. e rx b PV θ dr x. Then [4] ψ θ x =xψx + θ/x + θ, and /ψ θ is the Laplace transform of W θ. Also recall that ψα =,sothatψ θ α θ =. First,ifθ = α, thenf x =, u PV θ du = ψ α+/b = /b. Second,ifθ α, then F x = eθ αx α θ, PV θ du e α θu = eθ αx bα θ α θ ψ θα θ, 24

25 so that whatever the respective values of α and θ, F x = b eθ αx. We use Laplace transforms to compute F 2.Foranyκ>, dx κ e κx F 2 x = = =,,, PV θ du u PV θ due κu dt e θ αt W θ t u u t u dx κ e κx dt e θ α κt W θ t u PV θ due κu e θ α κu ds e θ α κs W θ s = b κ + α θ ψ θκ + α θ α θ ψ θ α θ ψ θ κ + α θ κ = bψ θ κ + α θ b, so that and As a consequence, we get E χ k α = F 2 x = b eθ αx W θ x b, F x F 2x W θ x = bw θ x. dx θe θx bw θ x 2 k, W θ x which is the announced U k /b. Last, let us check the technical assumptions required for Theorems and 5 in [9, Appendix A] to hold. For the first theorem, we have to check the following two requirements sup e αu Eχu < 5.4 n [n,n+] t Eχt isa.e. continuous. 5.5 For the second theorem, we have to check the following two requirements The following equality in distribution is easily seen <η<α, E sup e ηt χt < 5.6 t <η<α, ˆµη <. 5.7 χt = i {T i t V } { P j N jt S j {Ti <S j <T i+ t V } A}, 25

26 where V is distributed as a lifetime, the T i aretherankedatomsofanindependentpoissonpoint process with rate θ mutation times, the S i aretherankedatomsofanindependentpoissonpoint process with rate b birth times, the N i formanindependentsequenceofi.i.d.homogeneous, binary CMJ processes descendances of daughters, and A is taken equal to N, butcanbetaken equal to {k} in the case of the random characteristic χ k.inanycase,χis dominated by a Poisson point process with rate θ, sothateχt θt. This ensures that 5.4 holds. As for 5.5, notice from the last displayed equation that Eχt = i F it, where F i t := u PT i du, T i+ ds [u, PV drp N j t S j {u<sj <s t r} A. j Because T i has a density w.r.t. Lebesgue measure, each F i is everywhere continuous on, say, [,t ]. In addition, for any t [,t ], F i t PT i t PT i t and i PT i t =θt <, sowe get continuity of t Eχt on[,t ]bydominatedconvergence. Becauset is arbitrary, t Eχt is continuous everywhere. Let us treat the last two requirements. The last requirement 5.7 merely stems from the obvious inequality µt bt. Toprove5.6,becauseχ is dominated by a Poisson point process, it suffices to show that for any Poisson point process Y with rate, say, and for any η>, E sup t e ηt Y t <. In fact, setting M c t :=e ηt Y t + c, we claim that for large enough c, Mc 2 is a supermartingale. Then using the inequality Psup t Mc 2 t z c/z, weget Psup t Y t e ηt y Psup t Y t + c e ηt y =Psup t M 2 c t y2 c y 2, so that Esup t Y t e ηt <. TheonlythinglefttoshowisthatM 2 c is a supermartingale. Writing F t forthenaturalfiltrationofy and P s for a Poisson random variable with parameter s independent of Y t,weget EM c t + s 2 F t =e 2ηt+s E Y t + c + P s 2 = e 2ηt+s Y t + c + s 2 + s M c t 2, where the last inequality holds for any s, t ifthereissomepositivec depending only on η such that e 2ηs x + s 2 + s x 2 x c, s. Then we study the function f : s x 2 e 2ηs x + s 2 s. Since f s =4η 2 x 2 e 2ηs 2, f is nondecreasing on [, + assoonasx 2 /2η 2.Ontheotherhand,f = 2ηx 2 2x. Letx be the largest root of x 2ηx 2 2x. Assoonasx x, f. Setting c := max/η 2,x, as soon as x c, f andf is nondecreasing on [,, so that f is nondecreasing on [,. Since f =, we conclude that f is non-negative on [,, so that Mc 2 indeed is a supermartingale. 6 Expected homozygosities through moment generating functions We consider again the coalescent point process of Section 3, constructed from H =+ and the i.i.d. sequence of r.v. H i i,withcommonlawph. Let us recall that, in the case of splitting 26

27 trees, the law of H has a density w.r.t. Lebesgue s measure. We introduce the derivative of log W t: ptdt = PH t + dt H>t=W tph dt. 6. For any time t, weconsiderthesplittingtreeobtainedfromh,...,h Nt, wheren t := inf{i :H i >t}. WethendefinethestandardhomozygosityḠθt astheprobabilitythattwodistinct randomly sampled individuals in the population at time t share the same haplotype, and the absolute homozygosity G θ t asthenumberofpairsofdistinct individuals in the population at time t that share the same haplotype. Note that both of these quantities are on the event {N t =}, andon the complement event, Ḡ θ t = 2G θt N t N t. 6.2 The notation G θ t coincideswiththatofsubsection4.. WealsorecallthatZ t denotesthe number of individuals sharing the ancestral haplotype, defined here as the haplotype of individual at time t. Our goal in this section is to compute E t G θ t and E t Ḡθt using another method than in Section 3. As in [4], we characterize the joint law of G θ t,n t,z t as time increases in a similar fashion as for branching processes, in order to obtain backward Kolmogorov equations for moment generating functions involving these random variables. The result will then follow by solving these equations. Proposition 6. For all t, the expected absolute homozygosity is given by E t G θ t = W tw 2θ t, whereas the expected standard homozygosity is given by E t Ḡθt = e 2θt W t 2W t +2θ e 2θs W s W t W s 6. Joint dynamics of G θ t, N t and Z t [ log W t log W s W t W s ] ds. W t Consider two splitting trees of age t, withrespectiveabsolutehomozygosity,populationsize,number of ancestral individuals and height processes G θ t, N t, Z t, H i i and G θ t, N t, Z t, H i i. We call merger of these two splitting trees the splitting tree obtained from the sequence of heights H =+,H,...,H Nt,H,H,...,H N t, whereh is obtained from the infinite branch H by cutting the part below t. Inaddition, allthemutationtimesarekeptunchangedoneachbranch of the tree. After this merger event, the new splitting tree has population size N t + N t,thenewnumberof ancestral individuals is Z t+z t andthenewabsolutehomozygosityis,countingfirstthepairs of ancestral individuals Z t+z tz t+z t + G θ t Z tz t 2 2 = G θ t+g θ t+z tz t. + G θ t Z tz t 2 27

28 Now, we have G θ,n,z =,, and, if the law of G θ t,n t,z t is known for some t, then, on the time interval [t, t + dt], either a mutation occurs on the ancestral branch, with probability θdt, and G θ t + dt,n t+dt,z t + dt = G θ t,n t,, or H Nt [t, t + dt], with probability ptdt defined in 6., and G θ t + dt,n t+dt,z t + dt = G θ t+g θ t+z tz t,n t + N t,z t+z t, where G θ t,n t,z t is an i.i.d. copy of G θt,n t,z t, or nothing happens the probability that both previous events occurs is odt. In other words, when the ancestral time t increases, the process G θ t,n t,z t jumps to G θ t,n t, with rate θ and to G θ t+g θ t+z tz t,n t + N t,z t+z t with instantaneous rate pt. Of course, the previous argument is quite informal, but it could easily be made rigorous by considering all the possible events that could occur in the time interval [t, t + s], and letting s. In particular, the Kolmogorov equations of the following subsection can easily be justified this way. 6.2 Moment generating functions computations We define the moment generating functions Lt, u =E t G θ tu Nt Mt, u, v =E t u Nt v Z t, 6.4 for all u, v [, ] and t. Since G θ t =ifn t andthequantitiesinsidetheexpectationsare bounded by N 2 t,thesefunctionshavefinitevalues.ourgoalhereistocomputeexplicitexpressions for these quantities. Note that, for any i.i.d. triples of nonnegative r.v. G θ,n,z andg θ,n,z, EG θ + G θ + Z Z un+n 2 =2EG θ u N 2 Eu N + EZ u N 2. Using this equation and the previous construction of the process, we can write the forward Kolmogorov equation for the moment generating functions L and M: forallu, v [, ] and t, ] t Lt, u = θ + ptlt, u+θlt, u+pt [2 ult, u Mt, u, + v Mt, u, L,u=, and { t Mt, u, v = θ + ptmt, u, v+θmt, u, + pt u Mt, u, v 2 M,u,v=v

29 The explicit computation of the solutions of these equations requires several steps. First, for fixed u and v, thefunctionmt, u, v issolutiontoanodeknownasriccati sequation.inthecase where v =,thefunctionft =Mt, u, is solution to f = pfuf, which is known as Bernoulli s equation. It can be solved by making the change of unknown function f =/f, whichmakestheodelinear.thisyields ft =Mt, u, = u + uexp W t; u psds = W t, 6.7 where we used that p is the derivative of the function log W t. Second, for all u, v [, ], the function Mt, u, is a particular solution of 6.6 with different initial condition. Hence, the function gt =Mt, u, v Mt, u, = Mt, u, v ft solvesthe Bernoulli ODE ġ = θ + p 2upfg + upg 2, for which the previous trick again works. This yields exp t θ + ps 2upsfsds Mt, u, v =ft+ v u t psexp. s ds θ + pτ 2upτfτdτ Since uw s; uph ds isthederivativeoflogw ; u, it follows from 6.7 that Hence, we obtain Mt, u, v = ps 2ufsds =logw t 2logW t; u. 6.8 W t; u W t + e θt W t; u v u. e θs W s; u 2 PH ds Observing that uw s; u 2 PH ds isthederivativeofw ; u, an integration by parts and Theorem 3. finally yield W t; u Mt, u, v = e θt W t; u v W t v + W. θt; u We then compute Mt, u, = W t; u W t = ft and v Mt, u, = W t; u2 e θt W t =: qt. Third, the linear equation 6.5 can be explicitly solved: Lt, u =exp s ps 2ufsds psq 2 sexp pτ 2ufτdτ ds. 29