LogFeller et Ray Knight

Transcription

1 LogFeller et Ray Knight Etienne Pardoux joint work with V. Le and A. Wakolbinger Etienne Pardoux (Marseille) MANEGE, 18/1/1 1 / 16

2 Feller s branching diffusion with logistic growth We consider the diffusion dz t = Z t (θ γz t )dt + 2 Z t dw t, t, (1) with Z = x, θ, γ >. Z t is not a branching process : the quadratic term introduces interactions between the branches. However there is still an interpretation in terms of the evolution of a population of the solution of the Fellog process Z t, see A. Lambert (25). Consider dh s = H =, ( ) θ 2 γl s(h s ) ds + db s dl s(), s, (2) where B is a standard Brownian motion. In this SDE, the term dl s ()/2 takes care of the reflection of H at the origin. Etienne Pardoux (Marseille) MANEGE, 18/1/1 2 / 16

3 Feller s branching diffusion with logistic growth We consider the diffusion dz t = Z t (θ γz t )dt + 2 Z t dw t, t, (1) with Z = x, θ, γ >. Z t is not a branching process : the quadratic term introduces interactions between the branches. However there is still an interpretation in terms of the evolution of a population of the solution of the Fellog process Z t, see A. Lambert (25). Consider dh s = H =, ( ) θ 2 γl s(h s ) ds + db s dl s(), s, (2) where B is a standard Brownian motion. In this SDE, the term dl s ()/2 takes care of the reflection of H at the origin. Etienne Pardoux (Marseille) MANEGE, 18/1/1 2 / 16

4 Feller s branching diffusion with logistic growth We consider the diffusion dz t = Z t (θ γz t )dt + 2 Z t dw t, t, (1) with Z = x, θ, γ >. Z t is not a branching process : the quadratic term introduces interactions between the branches. However there is still an interpretation in terms of the evolution of a population of the solution of the Fellog process Z t, see A. Lambert (25). Consider dh s = H =, ( ) θ 2 γl s(h s ) ds + db s dl s(), s, (2) where B is a standard Brownian motion. In this SDE, the term dl s ()/2 takes care of the reflection of H at the origin. Etienne Pardoux (Marseille) MANEGE, 18/1/1 2 / 16

5 Results Using Girsanov s theorem, one can show Proposition The SDE (2) has a unique weak solution. H being the solution of (2), L s (t) denoting its local time, let S x := inf{s > : L s () > x}. Our main result is the Theorem L Sx (t), t, solves the Fellog SDE (1). Etienne Pardoux (Marseille) MANEGE, 18/1/1 3 / 16

6 Results Using Girsanov s theorem, one can show Proposition The SDE (2) has a unique weak solution. H being the solution of (2), L s (t) denoting its local time, let S x := inf{s > : L s () > x}. Our main result is the Theorem L Sx (t), t, solves the Fellog SDE (1). Etienne Pardoux (Marseille) MANEGE, 18/1/1 3 / 16

7 Results Using Girsanov s theorem, one can show Proposition The SDE (2) has a unique weak solution. H being the solution of (2), L s (t) denoting its local time, let S x := inf{s > : L s () > x}. Our main result is the Theorem L Sx (t), t, solves the Fellog SDE (1). Etienne Pardoux (Marseille) MANEGE, 18/1/1 3 / 16

8 Discrete approximation Our proof of the above extended Ray Knight theorem is based on an approximation by finite population. For N N, let Z N,x be the total mass of a population of individuals, each of which has mass 1/N. The initial mass is Z N,x = Nx /N, and Z N,x t follows a Markovian jump dynamics : from its current state k/n, { (k + 1)/N at rate 2kN + kθ Z N,x jumps to (k 1)/N at rate 2kN + k(k 1)γ/N. For γ =, this is a GW process in cont. time : each individual independently spawns a child at rate 2N + θ, and dies (childless) at rate 2N. For γ, the quadratic death rate destroys independence. Viewing the individuals alive at time t as being arranged from left to right, and by decreeing that each of the pairwise fights (which happens at rate 2γ) is won by the individual to the left, we arrive at the additional death rate 2γL i (t)/n for individual i, where L i (t) denotes the number of indiv. currently living to the left of i at time t. Etienne Pardoux (Marseille) MANEGE, 18/1/1 4 / 16

9 Discrete approximation Our proof of the above extended Ray Knight theorem is based on an approximation by finite population. For N N, let Z N,x be the total mass of a population of individuals, each of which has mass 1/N. The initial mass is Z N,x = Nx /N, and Z N,x t follows a Markovian jump dynamics : from its current state k/n, { (k + 1)/N at rate 2kN + kθ Z N,x jumps to (k 1)/N at rate 2kN + k(k 1)γ/N. For γ =, this is a GW process in cont. time : each individual independently spawns a child at rate 2N + θ, and dies (childless) at rate 2N. For γ, the quadratic death rate destroys independence. Viewing the individuals alive at time t as being arranged from left to right, and by decreeing that each of the pairwise fights (which happens at rate 2γ) is won by the individual to the left, we arrive at the additional death rate 2γL i (t)/n for individual i, where L i (t) denotes the number of indiv. currently living to the left of i at time t. Etienne Pardoux (Marseille) MANEGE, 18/1/1 4 / 16

10 The just described reproduction dynamics gives rise to a forest F N of trees of descent. At any branch point, we imagine the new branch being placed to the right of the mother branch. Because of the asymmetric killing, the trees further to the right have a tendency to stay smaller : they are under attack by the trees to their left. For a given realization of F N, we read off a continuous and piecewise linear R + -valued path H N (called the exploration path associated with F N ) in the following way : Starting (, ), H N goes upwards at speed 2N until the top of the first mother branch (which is the leftmost leaf of the tree) is hit. There H N turns and goes downwards, again at speed 2N, until arriving at the closest branch point (which is the last birth time of a child of the first ancestor before his death). From there one goes upwards into the (yet unexplored) next branch, and proceeds in a similar fashion until being back at height, which means that the exploration of the leftmost tree is completed. Then explore the next tree, etc., until the exploration of the forest F N is completed. Etienne Pardoux (Marseille) MANEGE, 18/1/1 5 / 16

11 The just described reproduction dynamics gives rise to a forest F N of trees of descent. At any branch point, we imagine the new branch being placed to the right of the mother branch. Because of the asymmetric killing, the trees further to the right have a tendency to stay smaller : they are under attack by the trees to their left. For a given realization of F N, we read off a continuous and piecewise linear R + -valued path H N (called the exploration path associated with F N ) in the following way : Starting (, ), H N goes upwards at speed 2N until the top of the first mother branch (which is the leftmost leaf of the tree) is hit. There H N turns and goes downwards, again at speed 2N, until arriving at the closest branch point (which is the last birth time of a child of the first ancestor before his death). From there one goes upwards into the (yet unexplored) next branch, and proceeds in a similar fashion until being back at height, which means that the exploration of the leftmost tree is completed. Then explore the next tree, etc., until the exploration of the forest F N is completed. Etienne Pardoux (Marseille) MANEGE, 18/1/1 5 / 16

12 The just described reproduction dynamics gives rise to a forest F N of trees of descent. At any branch point, we imagine the new branch being placed to the right of the mother branch. Because of the asymmetric killing, the trees further to the right have a tendency to stay smaller : they are under attack by the trees to their left. For a given realization of F N, we read off a continuous and piecewise linear R + -valued path H N (called the exploration path associated with F N ) in the following way : Starting (, ), H N goes upwards at speed 2N until the top of the first mother branch (which is the leftmost leaf of the tree) is hit. There H N turns and goes downwards, again at speed 2N, until arriving at the closest branch point (which is the last birth time of a child of the first ancestor before his death). From there one goes upwards into the (yet unexplored) next branch, and proceeds in a similar fashion until being back at height, which means that the exploration of the leftmost tree is completed. Then explore the next tree, etc., until the exploration of the forest F N is completed. Etienne Pardoux (Marseille) MANEGE, 18/1/1 5 / 16

13 M 3 M 2 m 2 M 1 * M 4 m 1 * * m 3 Etienne Pardoux (Marseille) MANEGE, 18/1/1 6 / 16

14 The exploration process H N in case θ = γ = We have the following SDE ((H N, V N ) takes values in IR + { 1, 1}) dh N s ds = 2NVs N, H N =, dvs N = 21 {V N s = 1} dpn s 21 {V N r =1} dpn r + 2NdL N s (), V N = 1, where {P N s, s } is a Poisson processes with intensity 4N 2, i. e. M N s = P N s 4N 2 s is a martingale, and L N s (t) := the local time accumulated by H N at level t up to time s 1 := lim ε ε s 1 {t H N u <t+ε}du. Etienne Pardoux (Marseille) MANEGE, 18/1/1 7 / 16

15 The exploration process H N in case θ = γ = We have the following SDE ((H N, V N ) takes values in IR + { 1, 1}) dh N s ds = 2NVs N, H N =, dvs N = 21 {V N s = 1} dpn s 21 {V N r =1} dpn r + 2NdL N s (), V N = 1, where {P N s, s } is a Poisson processes with intensity 4N 2, i. e. M N s = P N s 4N 2 s is a martingale, and L N s (t) := the local time accumulated by H N at level t up to time s 1 := lim ε ε s 1 {t H N u <t+ε}du. Etienne Pardoux (Marseille) MANEGE, 18/1/1 7 / 16

16 If we let S N x It is easily seen that := inf{s > : L N s () [Nx]/N}, Zt N,x = L N S (t), t. x N Etienne Pardoux (Marseille) MANEGE, 18/1/1 8 / 16

17 If we let S N x It is easily seen that := inf{s > : L N s () [Nx]/N}, Zt N,x = L N S (t), t. x N Etienne Pardoux (Marseille) MANEGE, 18/1/1 8 / 16

18 Taking the limit As N, V N oscillates faster and faster. In order to study the limit of H N, we consider for f C 2 (IR), the perturbed test function, see e. g. Ethier, Kurtz (1986) f N (h, v) = f (h) + v 4N f (h). Implementing this with f (h) = h, we get where M 1,N s = 1 2N s H N s + V s N 4N = M1,N s Ms 2,N LN s (), 1 {V N r = 1} dmn r and Ms 2,N = 1 s 1 2N {V N r =1} dmn r are two mutually orthogonal martingales. Etienne Pardoux (Marseille) MANEGE, 18/1/1 9 / 16

19 Taking the limit As N, V N oscillates faster and faster. In order to study the limit of H N, we consider for f C 2 (IR), the perturbed test function, see e. g. Ethier, Kurtz (1986) f N (h, v) = f (h) + v 4N f (h). Implementing this with f (h) = h, we get where M 1,N s = 1 2N s H N s + V s N 4N = M1,N s Ms 2,N LN s (), 1 {V N r = 1} dmn r and Ms 2,N = 1 s 1 2N {V N r =1} dmn r are two mutually orthogonal martingales. Etienne Pardoux (Marseille) MANEGE, 18/1/1 9 / 16

20 Taking the limit as N We obtain the following joint convergence Theorem For any x >, as n, ( {H N s, Ms 1,N, Ms 2,N, s }, {L N s (t), s, t }, Sx N ) ( {Hs, Bs 1, Bs 2 ), s }, {L s (t), s, t }, S x, where B 1 and B 2 are two mutually independent B. M.s, if B = ( 2) 1 (B 1 B 2 ), H is B reflected above, L its local time, and S x = inf{s > ; L s () > x}. Etienne Pardoux (Marseille) MANEGE, 18/1/1 1 / 16

21 A Girsanov transformation 1 Let s Xs N,1 := X N,2 s := s X N := X N,1 + X N,2, θ 2N 1 {V N r = 1}dMN r, γl N r (Hr N ) 1 N {V N r =1} dmn r, and Y N := E(X N ) denote the Doléans exponential of X N, i. e. Y N solves the SDE s Ys N = 1 + Yr dx N r N, s. We prove Proposition Y N is a martingale ( EY N s = 1, s > ). Etienne Pardoux (Marseille) MANEGE, 18/1/1 11 / 16

22 A Girsanov transformation 1 Let s Xs N,1 := X N,2 s := s X N := X N,1 + X N,2, θ 2N 1 {V N r = 1}dMN r, γl N r (Hr N ) 1 N {V N r =1} dmn r, and Y N := E(X N ) denote the Doléans exponential of X N, i. e. Y N solves the SDE s Ys N = 1 + Yr dx N r N, s. We prove Proposition Y N is a martingale ( EY N s = 1, s > ). Etienne Pardoux (Marseille) MANEGE, 18/1/1 11 / 16

23 A Girsanov transformation 1 Let s Xs N,1 := X N,2 s := s X N := X N,1 + X N,2, θ 2N 1 {V N r = 1}dMN r, γl N r (Hr N ) 1 N {V N r =1} dmn r, and Y N := E(X N ) denote the Doléans exponential of X N, i. e. Y N solves the SDE s Ys N = 1 + Yr dx N r N, s. We prove Proposition Y N is a martingale ( EY N s = 1, s > ). Etienne Pardoux (Marseille) MANEGE, 18/1/1 11 / 16

24 A Girsanov transformation 2 Define new probability measures P N s. t. for all s >, d P N Fs dp Fs = Y N s, s Consider the 2-variate point process ( r (Qr 1,N, Qr 2,N ) = 1 {V N u = 1} dpn u, Under P N, r 1 {V N u =1} dpn u Q 1,N r has intensity (4N 2 + 2θN)1 {V N r = 1} dr Q 2,N r has intensity 4[N 2 + γnl N r (H N r )]1 {V N r =1} dr. ), r, This means that under P N, H N has the requested approximate Log Feller dynamics. Etienne Pardoux (Marseille) MANEGE, 18/1/1 12 / 16

25 A Girsanov transformation 2 Define new probability measures P N s. t. for all s >, d P N Fs dp Fs = Y N s, s Consider the 2-variate point process ( r (Qr 1,N, Qr 2,N ) = 1 {V N u = 1} dpn u, Under P N, r 1 {V N u =1} dpn u Q 1,N r has intensity (4N 2 + 2θN)1 {V N r = 1} dr Q 2,N r has intensity 4[N 2 + γnl N r (H N r )]1 {V N r =1} dr. ), r, This means that under P N, H N has the requested approximate Log Feller dynamics. Etienne Pardoux (Marseille) MANEGE, 18/1/1 12 / 16

26 A Girsanov transformation 2 Define new probability measures P N s. t. for all s >, d P N Fs dp Fs = Y N s, s Consider the 2-variate point process ( r (Qr 1,N, Qr 2,N ) = 1 {V N u = 1} dpn u, Under P N, r 1 {V N u =1} dpn u Q 1,N r has intensity (4N 2 + 2θN)1 {V N r = 1} dr Q 2,N r has intensity 4[N 2 + γnl N r (H N r )]1 {V N r =1} dr. ), r, This means that under P N, H N has the requested approximate Log Feller dynamics. Etienne Pardoux (Marseille) MANEGE, 18/1/1 12 / 16

27 A Girsanov transformation 2 We have that as N, X N X = θ B 1 + 2γ L r (H r )dbr 2 2 ( s [ ] ) θ Y N 2 Y = exp X s 4 + γ2 L 2 r (H r ) dr. Define the probability P s. t. for all s >, Under P, d P Fs dp Fs = Y s, s. H s = θ s 2 s γ L r (H r )dr + B r L s(), s. Etienne Pardoux (Marseille) MANEGE, 18/1/1 13 / 16

28 A Girsanov transformation 2 We have that as N, X N X = θ B 1 + 2γ L r (H r )dbr 2 2 ( s [ ] ) θ Y N 2 Y = exp X s 4 + γ2 L 2 r (H r ) dr. Define the probability P s. t. for all s >, Under P, d P Fs dp Fs = Y s, s. H s = θ s 2 s γ L r (H r )dr + B r L s(), s. Etienne Pardoux (Marseille) MANEGE, 18/1/1 13 / 16

29 A Girsanov transformation 2 We have that as N, X N X = θ B 1 + 2γ L r (H r )dbr 2 2 ( s [ ] ) θ Y N 2 Y = exp X s 4 + γ2 L 2 r (H r ) dr. Define the probability P s. t. for all s >, Under P, d P Fs dp Fs = Y s, s. H s = θ s 2 s γ L r (H r )dr + B r L s(), s. Etienne Pardoux (Marseille) MANEGE, 18/1/1 13 / 16

30 Conclusion Our main theorem follows taking the limit as N in the identity Z N,x t = L N S N x (t), Lemma under the measures P N and P, thanks to the following elementary Let (ξ N, η N ), (ξ, η) be random pairs defined on a probability space (Ω, F, P), with η N, η nonnegative scalar random variables, and ξ N, ξ taking values in some complete separable metric space X. Assume that E[η N ] = E[η] = 1. Write ( ξ N, η N ) for the random pair (ξ N, η N ) defined under the probability measure P N which has density η N with respect to P, and ( η, ξ) for the random pair (η, ξ) defined under the probability measure P which has density η with respect to P. Then ( ξ N, η N ) converges in distribution to ( η, ξ), provided that (ξ N, η N ) converges in distribution to (ξ, η). Etienne Pardoux (Marseille) MANEGE, 18/1/1 14 / 16

31 Conclusion Our main theorem follows taking the limit as N in the identity Z N,x t = L N S N x (t), Lemma under the measures P N and P, thanks to the following elementary Let (ξ N, η N ), (ξ, η) be random pairs defined on a probability space (Ω, F, P), with η N, η nonnegative scalar random variables, and ξ N, ξ taking values in some complete separable metric space X. Assume that E[η N ] = E[η] = 1. Write ( ξ N, η N ) for the random pair (ξ N, η N ) defined under the probability measure P N which has density η N with respect to P, and ( η, ξ) for the random pair (η, ξ) defined under the probability measure P which has density η with respect to P. Then ( ξ N, η N ) converges in distribution to ( η, ξ), provided that (ξ N, η N ) converges in distribution to (ξ, η). Etienne Pardoux (Marseille) MANEGE, 18/1/1 14 / 16

32 Appendix : Girsanov s theorem for Poisson processes 1 Let {(Q (1) s,..., Q s (d) ), s } be a d-variate point process adapted to, s } be the predictable some filtration F, and let {λ (i) s (P, F) intensity of Q (i), 1 i d. In other words, M (i) r := Q (i) s s λ(i) r dr is a (P, F) martingale, 1 i d. Assume that none of the Q (i), Q (j), i j, jump simultaneously (so that the M (i) s are mutually orthogonal). Let {µ r (i), r }, 1 i d, be nonnegative F-predictable processes such that for all s and all 1 i d s µ (i) r λ (i) r dr < P -a.s. Etienne Pardoux (Marseille) MANEGE, 18/1/1 15 / 16

33 Appendix : Girsanov s theorem for Poisson processes 1 Let {(Q (1) s,..., Q s (d) ), s } be a d-variate point process adapted to, s } be the predictable some filtration F, and let {λ (i) s (P, F) intensity of Q (i), 1 i d. In other words, M (i) r := Q (i) s s λ(i) r dr is a (P, F) martingale, 1 i d. Assume that none of the Q (i), Q (j), i j, jump simultaneously (so that the M (i) s are mutually orthogonal). Let {µ r (i), r }, 1 i d, be nonnegative F-predictable processes such that for all s and all 1 i d s µ (i) r λ (i) r dr < P -a.s. Etienne Pardoux (Marseille) MANEGE, 18/1/1 15 / 16

34 Appendix : Girsanov s theorem for Poisson processes 1 Let {(Q (1) s,..., Q s (d) ), s } be a d-variate point process adapted to, s } be the predictable some filtration F, and let {λ (i) s (P, F) intensity of Q (i), 1 i d. In other words, M (i) r := Q (i) s s λ(i) r dr is a (P, F) martingale, 1 i d. Assume that none of the Q (i), Q (j), i j, jump simultaneously (so that the M (i) s are mutually orthogonal). Let {µ r (i), r }, 1 i d, be nonnegative F-predictable processes such that for all s and all 1 i d s µ (i) r λ (i) r dr < P -a.s. Etienne Pardoux (Marseille) MANEGE, 18/1/1 15 / 16

35 Appendix : Girsanov s theorem for Poisson processes 2 Theorem {T i k, k = 1, 2...} denoting the jump times of Q(i), let for s ( Y s (i) ) := µ (i) exp T i k 1:Tk i s k { s (1 µ (i) r )λ (i) r } dr and Y s = d j=1 Y (j) s. If E[Y s ] = 1, s, then, for each 1 i d, the process Q (i) has the (i) ( P, F)-intensity λ r = µ (i) r λ (i) r, r, where the probability measure P is defined by d P Fs = Y s, s. dp Fs Note that Y (i) = E(X (i) ), with X (i) s := s (µ (i) r 1)dM r (i), 1 i d, s. Etienne Pardoux (Marseille) MANEGE, 18/1/1 16 / 16