Learning Session on Genealogies of Interacting Particle Systems

Transcription

1 Learning Session on Genealogies of Interacting Particle Systems A.Depperschmidt, A.Greven University Erlangen-Nuremberg Singapore, 31 July - 4 August 2017

2 Tree-valued Markov processes

3 Contents 1 Introduction 2 Discrete population models 3 Large population limits 4 Flows of bridges and coalescents 5 Stochastic equations

4 1 Introduction 2 Discrete population models 3 Large population limits 4 Flows of bridges and coalescents 5 Stochastic equations

5 Goals We present here from today s perspective the essentials of a series of papers from on tree-valued Markov processes with particular focus on some specific stochastic population models and the flow of bridges coding.

6 The description of processes via flow of bridges has been created in three papers of Bertoin and Le Gall 2003, 2005, 2006: [ 1 ] Bertoin, Jean; Le Gall, Jean-François Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003), no. 2, [ 2 ] Bertoin, Jean; Le Gall, Jean-François Stochastic flows associated to coalescent processes. II. Stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 3, [ 3 ] Bertoin, Jean; Le Gall, Jean-Francois Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 (2006), no. 1-4, This approach has been generalized and unified using stochastic equations by Dawson and Li 2012 [ 4 ] Dawson, Donald A.; Li, Zenghu Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 (2012), no. 2,

7 and subsequently a lot of applications where developed: [ 5 ] Labbé, Cyril Genealogy of flows of continuous-state branching processes via flows of partitions and the Eve property. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 3, [ 6 ] Foucart, Clément Generalized Fleming-Viot processes with immigration via stochastic flows of partitions. ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012), no. 2, In particular there is some work which gives a connection with the session on Lookdown constructions. [ 7 ] Gufler, Stephan Pathwise construction of tree-valued Fleming-Viot processes arxiv: v3

8 Models Two populations processes are central: Fleming-Viot + Generalized Fleming-Viot with immigration Continuous state branching processes with immigration and a process appearing as a dual to the first above: Kingman and Λ-coalescents. The first two are related via a rescaling theorem and the first and the third via duality.

9 Coding There is a specific coding of their evolving states: States: Bridges Evolving States: Flows of Bridges This coding is reminiscent of the ordering in look-down constructions. This allows to obtain representations on one probability space which solve particular stochastic equations which induce weak solutions to some specific martingale problem. Also some genealogical information is nicely represented.

10 1 Introduction 2 Discrete population models 3 Large population limits 4 Flows of bridges and coalescents 5 Stochastic equations

11 Cannings model Consider population of fixed size N identified with {1,..., N}. ξ i,n = #children of individual i {1,..., N} in generation n Z Cannings model: For each n, the family (ξ i,n, i {1,..., N}) is exchangeable N 0 -valued subject to N ξ i,n = N. i=1

12 Consistent labelling and discrete bridges Consistent labelling: For 1 i j N the offspring of individual i have smaller labels than that of j. For each n, Y n : {0,..., N} {0,..., N} defined by Y n (k) := k ξ i,n = #offspring of first k individuals of n-th generation. i=1 Y n is non-decreasing, with Y n (0) = 0, Y n (N) = N, i.e. a discrete bridge (Bertoin, Le Gall, 2003).

13 Cannings model and flows of bridges n + 2 Y n Y n Y n+1 Y n n n Consistent labelling means that lineages do not cross-over. Interpretation: Increments of Y n are ξ i,n, i.e. offspring numbers of individuals in generation n. In general: Increments of Y n+k 1 Y n are offspring numbers of individuals in generation n in generation n + k. Note: Normalising leads to bridges from [0, 1] to [0, 1], i.e. to distribution functions with exchangeable jump sizes at deterministic jump times.

14 Genealogies via partition processes For m N 0 : i, j {1,..., N} are in the same partition element at time m if the time to most recent common ancestor of individuals i and j from generation 0 is at most m. time 0 { {1},{2},{3},{4},{5} } { {1},{2},{3},{4, 5} } { {1, 2, 3},{4, 5} } { {1, 2, 3},{4, 5} } time { {1, 2, 3, 4, 5} }

15 Genealogies via flows of bridges For m < n set B m,n = Y m+1 Y n, B m,m = Id. Increments of B 0,n partition the set of individuals of the present (time 0) population according to their ancestral relation n generations ago. time 0 B 0,0 = (0, 1, 2, 3, 4, 5) B 0,1 = (0, 0, 1, 2, 3, 5) B 0,2 = (0, 0, 3, 5, 5, 5) B 0,3 = (0, 3, 3, 5, 5, 5) time B 0,4 = (0, 0, 5, 5, 5, 5)

16 Properties of flows of (discrete) bridges (i) Cocycle property: For every l m n B l,m B m,n = B l,n (ii) Law of B m,n depends only on n m and for all n 1 n 2 n k are independent. B n1,n 2,..., B nk 1,n k

17 Branching models and bridges Similar objects can be defined for discrete time branching models: bridges have to be replaced by discrete subordinators, i.e. non-decreasing random walks that map N 0 to N 0.

18 1 Introduction 2 Discrete population models 3 Large population limits 4 Flows of bridges and coalescents 5 Stochastic equations

19 Genealogies of large populations Under suitable conditions for large population sizes the genealogies of Cannings models can be approximated by Kingman coalescent and more generally by Λ-coalescents and Ξ-coalescents. Möhle and Sagitov [MS01] give full classification of limits. Flows of (normalized) bridges are in one-to-one correspondence to such coalescents and to mass coalescents [BLG03]. Mass coalescent in discrete case: Take instead of sizes of partition elements of {1,..., N} their relative sizes normalised by N.

20 Bridges Definition 1 (Bertoin, Le Gall, 2003). A bridge is a [0, 1]-valued random process B = (B(r), r [0, 1]) with (i) B(0) = 0, B(1) = 1, paths nondecreasing and right-continuous. (ii) B has exchangeable increments. Theorem 2 (General characterization of bridges. Kallenberg, 1973). B is a bridge iff there is sequence of rv s β = (β i, i N) with β 1 β 2 0 and i=1 βi 1 and an i.i.d. sequence U 1, U 2,... of uniform on [0, 1] rv s (indep. of β) so that B(r) = ( 1 β i) r + i=1 i=1 β i 1 {U i, r [0, 1] r} Iterpretation: U i and β i are jump times resp. jump sizes. Between the jumps B increases linearly with drift (1 i=1 βi ).

21 Flows of bridges Definition 3. A flow of bridges is a family (B s,t : < s t < ) of bridges so that: (i) For every s < t < u, B s,u = B s,t B t,u a.s. (ii) The law of B s,t only depends on t s, and for s 1 < s 2 < < s n, the bridges B s1,s 2, B s2,s 3,..., B sn 1,s n are independent. (iii) B 0,0 = Id and B 0,t Id in probability as t 0, in the sense of Skorohod s topology.

22 Large population limit of Cannings models Enrich the neutral Cannings model with types in set E = [0, 1]. Define X t (i) := type of individual i in generation t, and the empirical measure Z N t := 1 N N i=1 δ Xt (i). For time scale choose the probability of picking two siblings at random c N := E[ξ 1,N(ξ 1,N 1)] N 1 [ N = E i=1 ξ i,n N ] (ξ i,n 1). (N 1) Theorem 4. Assume Z N 0 µ M 1 (E) and further conditions, then (Z N t/c N ) converges weakly to Fleming-Viot process (Z t ).

23 Further conditions : Example Assume c N 0, and E[ξ 1,N (ξ 1,N 1)(ξ 1,N 2)] N 2 c N 0, as N. Then (Z t ) is Fleming-Viot with genealogy given by Kingman s coalescent. Kingman s n-coalescent is a process (π n t ) t 0 on partitions of {1,..., n} starting with π n 0 = {{1},..., {n}} and when there are k partition elements are present, then each pair coalesces at rate 1, i.e. the total coalescence rate is ( k 2). Kingman s coalescent is the extension to partitions of N.

24 Bridges and Kingman mass coalescent If π = (A 1, A 2,..., ) is an exchangeable partition of N define the asymptotic frequency of a block A in the partition by A {1,..., n} f A = lim n n (exists by de Finetti!). Consider Kingman s coalescent (π t ) t 0 starting from partition of N into singletons. Conditioned on π t = k for some t > 0, let β 1 t,..., β k t be the asymptotic frequencies of blocks of π t, U 1,..., U k i.i.d. uniform on [0, 1]. The partition π t can be represented by the bridge k βt1 i {U i r} i=1 Recall: β t = (β 1 t,..., β k t ) is unif. distr. on {x [0, 1] k : k i=1 x i = 1}.

25 Λ-coalescent P := set of partitions of N, Λ a finite measure on [0, 1]. We assume Λ({0}) = 0. Λ-coalescent is a P-valued process (Π t ) t 0. Its restriction to {1,..., n}, the n-λ-coalescent, is a process on partitions of {1,..., n}. Jump rates: When there are b blocks (2 b n) then for any k (2 k b) blocks the rate of merging into one block is given by λ b,k = 1 0 x k (1 x) b k 1 x 2 Λ(dx). Interpretation: Setting Λ(dx) = x 2 Λ(dx) mark at a Λ-P.P.P. the partitions with probability x and coalesce the marked ones. This process is dual to the Λ-Fleming-Viot.

26 Λ-Fleming-Viot particle approximation Define an N-individuals continuous time Λ-Cannings model: Take a Λ-Poisson point process on [0, 1] [0, ) with intensity d Λ dt. Transitions: At a point (x, t) mark all N individuals independently with 1 with probability x; with 0 with probability 1 x. Choose among the 1-individuals one individual at random. All 1 s are killed and replaced by the descendants of the chosen one. The expected number of marked 1 pairs during time [0, T ] is finite: Λ(dr) N(N 1) T r 2 r 2 N(N 1) = T Λ([0, 1]), (1) 2 2 hence only finitely many jumps occur.

27 1 Introduction 2 Discrete population models 3 Large population limits 4 Flows of bridges and coalescents 5 Stochastic equations

28 Some basic ideas Next we present some details on the ideas coding the evolution of Λ-Fleming-Viot via flow of bridges: Generalized Λ-Fleming-Viot processes, which model the evolution of a continuous population with fixed mass 1, have appeared in articles by Donnelly and Kurtz [DK99a], [DK99b], and were studied later in [BLG03], [BLG05],...

29 Fleming-Viot processes as flows of bridges It is convenient to view a generalized Fleming-Viot process as a stochastic flow (F t, t 0) on [0, 1], such that for each t 0, F t : [0, 1] [0, 1] is a (random) right-continuous non-decreasing map with F t (0) = 0 and F t (1) = 1. We should think of the unit interval as a population, and then of F t as the distribution function of a (random) probability measure df t (x) on [0, 1]. The transitions of the flow are Markovian, and more precisely, for every s, t 0, we have F t+s = F s F t, (2) where F s is a copy of F s independent of (F r, 0 r t). (Recall the cocycle and indep. increments properties of discrete bridges!)

30 Partitioning the population via bridges For every 0 r 1 < r 2 1, the interval (F t (r 1 ), F t (r 2 )] represents the sub-population at time t which consists of descendants of the sub-population (r 1, r 2 ] at the initial time. This means the two-point motions (F t (r 1 ), F t (r 2 )) play a role for the genealogy. For example we can ask for the eventual ancestor of the whole population.

31 Primitive Eve Define the primitive Eve, e as follows. e := inf { y [0, 1] : lim t F t(y) = 1 } = sup { y [0, 1] : lim t F t(y) = 0 } The process is a Markov process on [0, 1] with (3) ϱ t ({e}) = F t (e) F t (e ), t 0 (4) ϱ t ({e}) = F t (e) F t (e ) t 1. (5) Interpretation: ϱ t ({y}) = F t (y) F t (y ) corresponds to the mass of the progeny at time t of the individual y from initial population. Thus primitive Eve is the common ancestor at the initial generation of most of the individuals at time t when t is sufficiently large.

32 Construction of flow: simple case A simple construction can be given in the case 1 0 x 2 Λ(dx) <. Existence and uniqueness of flows in -case via stochastic equations will be explained in the next section. Let ((T i, U i, ξ i ), i N) denote the sequence of atoms of a Poisson random measure on [0, ) [0, 1] [0, 1] with intensity dt du Λ(dx), (6) ranked in the increasing order of the first coordinate.

33 Process (F t, t 0) starts from F 0 = Id, remains constant on [T i 1, T i ) (with the usual convention T 0 = 0), and for every i N F Ti = i F Ti 1 (7) where i (r) = ξ i 1 {Ui r} + r(1 ξ i ), r [0, 1]. (8) At each time T i, an individual in the population at time T i 1 is picked uniformly at random and gives birth to a sub-population of size ξ i. Simultaneously, the rest of the population shrinks by factor 1 ξ i.

34 Dual picture In [BLG03] it is shown that Λ-coalescent started from the partition of N into singletons is in one-to-one correspondence to the flows of bridges (B s,t, < s t < ). Consequence of the preceding construction: the mass frequency coalescent at time t has the same distribution as the ranked sequence of jump sizes of F t. Define the dual flow ( ˆB s,t, < s t < ) by Then we have the duality relation: ˆB s,t = B t, s. ϱ([0, y]) = F t (y) = ˆB 0,t (y).

35 A similar picture of flow of bridges can be formulated for branching processes. These processes appear in asymptotic description of the left corner of the bridge.

36 Branching and the asymptotics of Λ-Fleming-Viot Goal: One wants to study the behaviour of the low ranks in Λ-Fleming-Viot. Assumptions: For each integer k 1 let Λ k (dx) be a finite measure on [0, 1] with Λ k ({0}) = 0 and let {X k (t, v) : t 0, v [0, 1]} be defined by (12) from a Poisson random measure {M k (ds, dz, du)} on (0, ) (0, 1] 2 with intensity ds z 2 Λ k (dz) du. Suppose that z 2 (z z 2 )Λ k (k 1 dz) converges weakly as k to a finite measure on (0, ) denoted by z 2 (z z 2 )Λ (dz).

37 The rescaled p-point motion {(kx k (kt, r 1 /k),..., kx k (kt, r p /k)) : t 0} (9) converges in distribution to that of the weak solution flow of the stochastic equation (branching) t Y t (v) = v x1 {u Ys (v)}ñ(ds, dx, du), (10) where Ñ(ds, dx, du) is a compensated Poisson random measure on [0, ) (0, ) 2 with intensity ds z 2 Λ (dz) du.

38 In order to get such asymptotics, [DL12] apply the machinery of limit theorems for semimartingales [JS03] to the so-called generalized Fleming-Viot processes, which were shown in [BLG03] to be duals to the coalescents with multiple collisions.

39 1 Introduction 2 Discrete population models 3 Large population limits 4 Flows of bridges and coalescents 5 Stochastic equations

40 The basic stochastic equations A key point is to write now out stochastic equations which generate the flow of bridges for the specified models. We execute this now following [DL12] starting with branching, which is technically the simplest case, because of the branching property.

41 Branching Suppose that σ 0 and b are constants, v γ(v) is a nonnegative and nondecreasing continuous function on [0, ) and (z z 2 )m(dz) is a finite measures on (0, ). Let {W (ds, du)} be a white noise on (0, ) 2 based on the Lebesgue measure ds du. Let {N(ds, dz, du)} be a Poisson random measure on (0, ) 3 with intensity ds m(dz)du. Let {Ñ(ds, dz, du)} be the compensated measure of {N(ds, dz, du)}.

42 Stochastic equation for Branching We shall see that for any v 0 there is a pathwise unique nonnegative solution of the stochastic equation t Y t (v) = v + σ 0 Ys (v) 0 W (ds, du) + t + 0 t 0 Ys (v) 0 0 [γ(v) by s (v)]ds zñ(ds, dz, du). (11) It is not hard to show each solution Y (v) = Y t (v) : t 0 is a continuous state branching process with immigration (CBI-process).

43 Theorem 5. There is a unique nonnegative strong solution to (11).

44 Then it is natural to call the two-parameter process {Y t (v) : t 0, v 0} a flow of CBI-processes. We prove that the flow has a version with the following properties: (i) for each v 0, t Y t (v) is a càdlàg process on [0, ) and solves (11). (ii) for each t 0, v Y t (v) is a nonnegative and nondecreasing càdlàg process on [0, ).

45 Λ-Fleming-Viot Suppose that σ 0 and b 0 are constants v γ(v) is a nondecreasing continuous on [0, 1] with 0 γ(v) 1 z 2 ν(dz) is a finite measure on (0, 1] {B(ds, du)} is white noise on (0, ) (0, 1] based on ds du, {M(ds, dz, du)} is a Poisson random measure on (0, ) (0, 1] 2 with intensity ds ν(dz)du. We show that for any v [0, 1] there is a pathwise unique solution X (v) = {X t (v) : t 0} to the equation t X t (v) = v + σ + b + 0 t [ ] 1 {u Xs (v)} X s (v) B(ds, du) [γ(v) X s (v)] ds 0 t z [ ] 1 {u Xs (v)} X s (v) M(ds, dz, du). (12)

46 We show that there is a version of the random field {X t (v) : t 0, 0 v 1} with the following properties: (i) for each v [0, 1], t X t (v) is càdlàg on [0, ) and solves (12); (ii) for each t 0, v X t (v) is nondecreasing and càdlàg on [0, 1] with X t (0) 0 and X t (1) 1.

47 More on generalized Fleming-Viot Theorem 6. The path-valued Markov process {X (v) : 0 v 1} has a ϱ-càdlàg modification. Consequently, there is a version of the solution flow {X t (v) : t 0, 0 v 1} of (12) with the following properties: (i) for each v [0, 1], t X t (v) is càdlàg on [0, ) and solves (12); (ii) for each t 0, v X t (v) is nondecreasing and càdlàg on [0, 1] with X t (0) 0 and X t (1) 1. We call the solution flow {X t (v) : t 0, v [0, 1]} of (12) specified above a generalized Fleming-Viot flow. The law of the flow is determined by the parameters (σ, b, γ, ν).

48 The p-point motion For any 0 r 1 < < r p 1 the p-point motion is equivalent to {(B t,0 (r 1 ),..., B t,0 (r p )) : t 0} (13) {(X t (r 1 ),..., X t (r p )) : t 0}. (14) Therefore, the solutions of (12) give a realization of the flow of bridges associated with the Λ-coalescent process. In Kingman case they showed that the p-point motion {(B t,0 (r 1 ),..., B t,0 (r p )) : t 0} is a diffusion process in D p := {x = (x 1,..., x p ) R p : 0 x 1 x p 1} (15) with generator A 0 defined by A 0 f (x) = 1 2 p i,j=1 x i j (1 x i j ) 2 f x i x j (x). (16)

49 The p-point motion of inverses Given a Λ-coalescent flow {B s,t : < s t < }, we define the flow of inverses by B 1 s,t (v) = inf{u [0, 1] : B s,t (u) > v}, v [0, 1), (17) and B 1 s,t (1) = B 1 s,t (1 ). Kingman case: p-point motion {(B 1 0,t (r 1 ),..., B 1 0,t (r p )) : t 0} is a diffusion process in D p with generator A 1 given by A 1 f (x) = A 0 f (x) + p i=1 ( 1 2 x i ) f x i (x), (18)

50 More on CBI-processes Suppose that σ 0 and b are constants, and (u u 2 )m(du) is a finite measure on (0, ). Let ϕ be a function given by φ(z) = bz σ2 z (e zu 1 + zu)m(du), z 0. (19) A Markov process with state space R + := [0, ) is called a CB-process with branching mechanism φ if it has transition semigroup (p t ) t 0 given by R + e λy p t (x, dy) = e xv t(λ), λ 0, (20) where (t, λ) v t (λ) is the unique locally bounded nonnegative solution of d dt v t(λ) = φ(v t (λ)), v 0 (λ) = λ, t 0.

51 Given any β 0 we can also define a transition semigroup (q t ) t 0 on R + by { t } e λy q t (x, dy) = exp xv t (λ) βv s (λ)ds. (21) 0 R + A nonnegative real-valued Markov process with transition semigroup (q t ) t 0 is called a CBI-process with branching mechanism φ and immigration rate β. It is easy to see that both (p t ) t 0 and (q t ) t 0 are Feller semigroups.

52 Theorem 7. There is a unique nonnegative strong solution of the stochastic equation t Y t =Y 0 + σ t + 0 Ys 0 0 Ys 0 0 t W (ds, du) + (β by s )ds 0 zñ(ds, dz, du). Moreover, the solution {Y t : t 0} is a CBI-process with branching mechanism φ and immigration rate β.

53 Using Ito s formula one can see that {Y t (v) : t 0} solves the martingale problem associated with the generator L defined by Lf (x) = 1 2 σ2 xf (x) + (β bx)f (x) + x 0 D z f (x)m(dz). (22) Then it is a CBI-process with branching mechanism φ and immigration rate β. Here D Z =: z f (x) f (x)z (23) z f (x) = f (x + z) f (x) (24)

54 Key results of solution theory of Dawson and Li Setup Let {W (ds, du)} be a white noise on (0, ) E with intensity ds π(dz). Let {N 0 (ds, du)} and {N 1 (ds, du)} be Poisson random measures on (0, ) U 0 and (0, ) U 1 with intensities ds µ 0 (du) and ds µ 1 (du), respectively. The s-component represents time. Suppose that {W (ds, du)}, {N 0 (ds, du)} and {N 1 (ds, du)} are defined on some complete probability space (Ω, F, P) and are independent of each other. Let {Ñ0(ds, du)} denote the compensated measure of {N 0 (ds, du)}.

55 Gaussian white noise Let (E R +, E B(R + ), ν(dx)dt) be a σ-finite measure space. A Gaussian white noise based on ν(dx) dt is a random set function, W, on E B(R + ) such that (a) W (B [a, b]) is a Gaussian random variable with mean zero and variance ν(b) b a. (b) if (A [a 1, a 2 ]) (B [b 1, b 2 ]) =, then W (A [a 1, a 2 ] (B [b 1, b 2 ])) = W (A [a 1, a 2 ])+W (B [b 1, b 2 ]) and W (A [a 1, a 2 ]) and W (B [b 1, b 2 ]) are independent.

56 A nonnegative càdlàg process {x(t) : t 0} is called a solution of t x(t) =x(0) + t t 0 0 E b(x(s ))ds + σ(x(s ), u)w (ds, du) t 0 U 1 g 1 (x(s ), u)n 1 (ds, du), U 0 g 0 (x(s ), u)ñ0(ds, du) (25) if it satisfies the stochastic equation almost surely for every t 0. We say {x(t) : t 0} is a strong solution if, in addition, it is adapted to the augmented natural filtration generated by {W (ds, du)}, {N 0 (ds, du)} and {N 1 (ds, du)}.

57 Theorem 8. Suppose that (σ, b, g 0, g 1 ) are admissible parameters satisfying conditions (a) (d). Then the pathwise uniqueness of solutions holds for (25). Theorem 9. Suppose that (σ, b, g 0, g 1 ) are admissible parameters satisfying conditions (a), (c), (e). Then there is a unique strong solution to (25).

58 Let E, U 0 and U 1 be Polish spaces and suppose that π(dz), µ 0 (du) and µ 1 (du) are σ-finite Borel measures on E, U 0 and U 1, respectively. We say the parameters (σ, b, g 0, g 1 ) are admissible if: x b(x) is a continuous function on R + satisfying b(0) 0; (x, u) σ(x, u) is a Borel function on R + E satisfying σ(0, u) = 0 for u E; (x, u) g 0 (x, u) is a Borel function on R + U 0 satisfying g 0 (0, u) = 0 and g 0 (x, u) + x 0 for x > 0 and u U 0 ; (x, u) g 1 (x, u) is a Borel function on R + U 1 satisfying g 1 (x, u) + x 0 for x 0 and u U 1.

59 Let us formulate the following conditions: (a) There is a constant K 0 so that b(x) + g 1 (x, u) µ 1 (du) K(1 + x) (26) U 1 for every x 0. (b) There is a non decreasing function x L(x) on R + and a Borel function (x, u) ḡ 0 (x, u) on R + U 0 so that for every x 0: and E sup g 0 (y, u) ḡ 0 (x, u) 0 y x σ(x, u) 2 π(du) + [ḡ 0 (x, u) ḡ 0 (x, u) 2 ]µ 0 (du) L(x). U 0 (27)

60 (c) For each m 1 there is a non decreasing concave function z r m (z) on R + so that 0+ r m(z) 1 dz = and b 1 (x) b 1 (y) + g 1 (x, u) g 1 (y, u) µ 1 (du) r m ( x y ) (28) U 1 for every 0 x, y m. (d) For each m 1 there is a nonnegative non decreasing function z ϱ m (z) on R + so that 0+ ϱ m(z) 2 dz =, E σ(x, u) σ(y, u) 2 π(du) ϱ m ( x y ) 2 (29) and, setting l 0 (x, y, u) := g 0 (x, u) g 0 (y, u), U 0 µ 0 (du) 1 0 l 0 (x, y, u) 2 (1 t)1 { l0 (x,y,u) n} ϱ m ( (x y) + tl 0 (x, y, u) ) 2 dt c(m, n) (30) for every n 1 and 0 x, y m. Here c(m, n) 0 is a constant.

61 (e) For each u U 0 the function x g 0 (x, u) is non decreasing, and for each m 1 there is a nonnegative and non decreasing function z ϱ m (z) on R + so that 0+ ϱ m(z) 2 dz = and E σ(x, u) σ(y, u) 2 π(du) + l 0 (x, y, u) l 0 (x, y, u) 2 µ 0 (du) ϱ m ( x y ) 2 (31) U 0 for all 0 x, y m, where l 0 (x, y, u) = g 0 (x, u) g 0 (y, u).

62 The key technical tool in comparison We say the comparison property of solutions holds for a stochastic equation if for any two solutions {x 1 (t) : t 0} and {x 2 (t) : t 0} satisfying x 1 (0) x 2 (0) we have P{x 1 (t) x 2 (t) for all t 0} = 1. Theorem 10. Let (σ, b, g 0, g 1 ) be admissible parameters satisfying conditions (a) (d). In addition, assume that for every u U 1 the function x x + g 1 (x, u) is non decreasing. Then the comparison property holds for the solutions of (25).

63 Theorem 11. Let (σ, b, g 0, g 1) and (σ, b, g 0, g 1 ) be two sets of admissible parameters satisfying conditions (a) (d). In addition, assume that: (i) for every u U 1, x x + g 1(x, u) or x x + g 1 (x, u) is non decreasing; (ii) b (x) b (x) and g 1(x, u) g 1 (x, u) for every x 0 and u U 1. Suppose that {x (t) : t 0} is a solution of (25) with (b, g 1 ) = (b, g 1), and {x (t) : t 0} is a solution of the equation with (b, g 1 ) = (b, g 1 ). If x (0) x (0), then P{x (t) x (t), for all t 0} = 1.

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