Växjö University. Some recent simulation techniques for diffusion bridges. School of Mathematics and System Engineering

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1 School of Mathematics and System Engineering Reports from MSI - Rapporter från MSI Växjö University Some recent simulation techniques for diffusion bridges Yadigar Sekerci May 2009 MSI Växjö University SE VÄXJÖ Report ISSN ISRN VXU/MSI/MA/C/ /--SE

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3 Yadigar Sekerci Some recent simulation techniques for diffusion bridges Bachelor s thesis Mathematics 2009 Växjö University

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5 Abstract We apply some recent numerical solutions to diffusion bridges written in Iacus (2008). One is an approximate scheme from Bladt and Sørensen (2007), another one, from Beskos et al (2006), is an algorithm which is exact: no numerical error at given grid points! Key-words: Brownian bridge, diffusion bridge, Brownian motion, stochastic differential equation, simulation, numerical methods, Euler-Maruyama s method, acceptencerejection method. Sammanfattning Vi tillämpar några nya numeriska lösningar till diffusionsbryggor beskrivna i Iacus (2008). En algoritm, från Bladt Sørensen (2007), är enkel och approximativ, den andra, från Beskos et al (2006), är exakt: inga numeriska fel vid givna gridpunkter! Nyckelord: Brownsk rörelse, diffusionsbro, Brownsk rörelse, stokastisk differentialekvation, simulering, numeriska metoder, diffusionsbro, Euler-Maruyama s method, acceptansförkastelsemetoden Acknowledgments I thank Roger Pettersson Contents 1 Introduction 1 2 Simulation of the Brownian motion 1 3 Simulation of the Brownian bridge 1 4 Simulation of stochastic differential equations 3 5 Simulation of diffusion bridges The approximate algorithm Ornstein-Uhlenbeck bridge One exact algorithm PDE-MCMC-algorithm References 11 A Appix: Definition of random variables etc 12 iii

6 B Appix: matlab codes 12 B.1 Euler-Maruyama s method for standard SDEs B.2 Brownian bridge B.3 Approximate diffusion bridge B.4 Code for simulation of Ornstein-Uhlenbeck bridge exact at grid points.. 13 iv

7 Figure 2.1: Twenty simulations of the Brownian motion where the number of gridpoints are Introduction Much of this work is inspired by nice examples of [9] where directions in the field of simulation of stochastic differential equations are presented. 2 Simulation of the Brownian motion A Brownian motion, also called Wiener process, is a continuous process {X t } t 0 starting at zero with indepent stationary increments that are normal distributed, more exactly, X t+h X t are normal distributed with zero mean and standard deviation h. One heuristic representation of the Brownian motion is that it satisfies the differential equation dx = dw, X 0 = 0 (2.1) where dw is dtn(0,1), a normal random variable with standard deviation dt! In fact the representation (2.1) is a particular case of stochastic differential equations (SDEs) often written in the form dx = µ(x)dt + σ(x)dw, X 0 = a for which there are stringent definitions in its integral form. It is simple to simulate the Brownian motion during a time interval [0,T ] at gridpoints 0 = t 0 <... < t n = T. The algorithm, with the miscellanous notation X i = X ti = X(t i ) is simply X i+1 = X i + i N(0,1), where i = ti = t i+1 t i. It means that for each time step you simulate a normal random variable with mean zero and standard deviation i. 3 Simulation of the Brownian bridge The Brownian bridge is a process {X t } that heuristically is just a Brownian motion with fixed points X 0 = a, X T = b. It satisfies the SDE dx = dw, X 0 = a, X T = b. (3.1) Note that X t is not adapted, it means X t not only deps on the information (the σ- algebra) of the Brownian motion W up till time t. X t deps on the future values. That means that even though (3.1) looks almost trivial it is actually not a standard SDE. However, it is well known that a solution to (3.1) is given by the closed formula, X t = a(1 t T ) + b t T + (W t t T W T ), 0 t T, (3.2) see any standard book, for example [2]. That makes is particularly simple to calculate the Brownain bridge: First you simulate the Brownian at grid points 0 = t 0 < t 1 <... < t n = T. Then you simulate the Brownian bridge at those grid points: X i = a(1 t i T ) + b t T + (W i t T W n), 0 t i T, (3.3) 1

8 Figure 3.1: The blue path (solid) is an exact simulation of the Brownian bridge at the grid points by the algorithm (3.3) starting and ing at zero. The green path (shadowed) is the Euler approximation (3.5) of (3.4). The number of gridpoints is The error seems to increase. For this simulation the error seems to be around the middle. When I simulated I found that the error is typically largest around the middle. It would be intersting to know a more exact description of the error. where we again used the miscellanous notation X i = X ti and W i = W ti. Now what is interesting, is that the Brownian bridge satisfies the SDE dx = b X t T t dt + dw t, 0 t T, X 0 = a. (3.4) Note that there is no -point condition X T = b. The SDE (3.4) looks like an SDE of standard type with solution X adapted to the Brownian motion, i.e. X t does not dep on future values of W. However, the drift coefficient (b X t )/(T t) is dangerously large as t approaches T. Now, an interesting question is, how bad is an Euler simulation of (3.4). In this bachelor thesis we don t reply that answer. Note that is not a trivial matter due to the non-boundedness of the drift coefficient for t near T. The Euler simulation is (again with miscellanous notation) X i+1 = X i + b X i T t i i + i N(0,1). (3.5) 4 Simulation of stochastic differential equations A generalization of the Brownian motion is a solution to the stochastic differential equation (SDE) dx t = µ(x t )dt + σ(x t )dw t, X 0 = a. (4.1) Nowadays there is a huge literature on such equations. One interpretation for tourists in this field is simply that it is the limit of the following naïve Euler-Maruyama algorithm: 2

9 X 0 = a and X i+1 = X i + µ(x i ) i + σ(x i ) i N(0,1),i = 1,2,... As before X i is X ti where t 0 = 0 < t 1 <..., i = t i+1 t i. In the next chapter on simulation of diffusion bridges, Euler-Maruyama simulation of SDEs will be used in an approximative algorithm for diffusion bridges. There are other more sophisticated methods, see e.g [9] and [5]. Throughout we assume there exists a unique solution of (4.1). Classical assumptions that give existence and uniqueness of a solution to (4.1) is linear growth and a Lipschitz continuity of b and σ 1. 5 Simulation of diffusion bridges A generalization of (2.1) is dx t = µ(x t )dt + σ(x t )dw t, 0 < t < T, X(0) = a, X(T ) = b (5.1) Given µ and σ it is often convenient to call a solution to (5.1) a (0,a,T,b)-bridge. It is not a trivial matter how to simulate (5.1). In the literature it seems that one important reason why diffusion bridge simulation is needed is that it is used for likelihood inference of discretely observed diffusion processes. That means that simulation of diffusion bridges is used for estimating parameters of SDEs observed at some grid points. In [3] it is suggested a particular technique based on Markov Chain Monte Carlo (MCMC) methods. In [7] there is a technique emaneting from the observation that limit solutions of certain stochastic partial differential equations are solutions to diffusion bridges! In [4] it is suggested a technique which is exact at any given time points the time interval [0,T ]! In this thesis we apply the approximate technique from [8] and the technique from [4]. 5.1 The approximate algorithm In [8] there is a simple algorithm to a simulate an approximate solution to (5.1), it means an approximate diffusion bridge. For the algorithm to work, time-reversibility of the diffusion process {X t [0,T ] } is assumed, that means {X t X 0 : t [0,T ]} and the process {X T t X T : t [0,T ]}, have the same statistical properties: the process behaves similarly if the time is running backwards. We impose the following condition, which ensures that the diffusion X is time-reversible, see [8], [5]. For that condition we let (l,r), l < r be the state space of {X t [0,T ] }, that means l is the smallest possible value for {X t [0,T ] } which may be and r is the largest possible value for {X t [0,T ] } which may be. Condition 5.1. Let {X t [0,T ] } be the solution to the SDE (4.1). Let x µ(y) s(x) = exp( 2 x# σ 2 (y) dy) where x# denotes an arbitrary point in the state space (l,r). Let m(x) = 1 σ 2 (x)s(x). 1 f Lipschitz continuous means f (x) f (y) L x y for all x,y. 3

10 Figure 5.1: The figures describes the approximate algorithm for the diffusion bridge dx = 0.5Xdt + dw, X 0 = 0, X T = 1. In the upper figure the blue one (solid line) starts at 0, the green one (dotted) starts at 1 (!) and the red one (dashed) is the flipped version of the green one, it s at 1. The bridge starts with the blue (solid) one and if it crosses the red (dashed) one the bridge continues with the red one which finishes at 1. Assume and x x# r l s(x)dx = m(x)dx < x# l s(x)dx =. (5.2) The approximative algorithm from [8] for simulation of diffusion bridges is as follows. Let W 1 and W 2 be two indepent Wiener processes, and let X 1 and X 2 as the solutions to dxt 1 = µ(xt 1 )dt + σ(xt 1 )dwt 1, X0 1 = a and dxt 2 = µ(xt 2 )dt + σ(xt 2 )dwt 2, X0 2 = b. The idea is to use the time-reversibility of X 2 and realize a (0,a,T,b) -bridge by simulating the process X 1 from a forward in time and X 2 from b backward in time! If the samples paths of the two process intersect, they can be combined into a realization of the bridge. Fact 5.1 ([8]). Let τ = inf(0 t 1Xt 1 = X1 t 2 ), where the infimum over empty set is taken to be infinite. Define { X 1 Z t = t, if 0 t τ X1 t 2 if τ < t 1 Assume Condition 5.1. Then, the distribution of {Z t : 0 t T } conditional on the event {τ T } is equal to the conditional distribution of {X t : 0 t T } given X 0 = a and X T = b; i.e., Z is a (0,a,T,b)-diffusion bridge. The fact 5.1 points towards a natural algorithm for simulation of a diffusion bridge.the algorithm consists in simulating two indepent diffusion process X 1 and X 2 using one numerical method, for example the Euler-Maruyama on the time interval [0, T ] with discretization step = T /n applying a rejection sampling procedure. Let Yi 1 and Y i 2, i = 0,1,2,...,n be indepent simulations of X 1 and X 2. If either Y 1 i Y 2 Y 1 (i+1) Y 2 (n (i+1)) or Y 1 i Y 2 (n i) and Y 1 (i+1) Y 2 (n (i+1)), a crossing has been realized. Hence, let ν = min{i (1,2,...,n) Yi 1 Y (n i) 2 } if Y 0 1 Y T 2 (1,2,...,n) Yi 1 Y (n i) 2 } if Y 0 1 Y T 2, and define { Y 1 B i = i, if i = 0,1,...,ν 1 if i = ν,...,n Y 2 (n i) (n i) and and v = min{i Then B is simulation of a (0,a,T,b)-diffusion bridge. If no crossing happened,start again by simulating Yi 1 and Y i 2 and iterate until a crossing of the two trajectories is realized. 4

11 Figure 5.2: 20 simulated OU-bridges with the algorithm exact at the grid points. Figure 5.3: 20 simulated OU-bridges with the approximate algorithm 5.2 Ornstein-Uhlenbeck bridge For the Ornstein-Uhlenbeck bridge (OU-bridge) dx t = θx t dt + σdw t, X 0 = a, X T = b (5.3) there is a simple algorithm that is exact at the grid points, [8]. According to [8], from the transition densities of the Ornstein-Uhlenbeck process we can calculate the transition densities of the OU-bridge. Thus we could in principle simulate the OU-bridge by sampling transitions from these densities. The following alternative method is, however, numerically more stable. For this particular equation there exists an algorithm which at the grid points t 1,t 2 is exactly an OU-bridge. That means that at the grid points the simulated process has the same distribution as the OU-bridge. The algorithm from the lemma below is from [8]: Lemma 5.1. For a partition 0 = t 0 < t 1 <... < t n = T, define X i = X ti and Y i = Y ti by X 0 = a, X i+1 = e θ(t i+1 t i ) X ti +Y i, i = 1,...n where Y i+1 = ( σ 2 2θ (1 e 2θ(t i+1 t i ))) 1/2 N(0,1), i = 1,...n where N(0,1) represents indep N(0,1) random variables. Let Z i be defined by Z i = X i + (b X n ) eθt i e θt i e θt n e θt n,, i = 1,...n. Then Z ti = Z i has the same distribution as a Ornstein-Uhlenbeck bridge at the grid points 0 = t 0 < t 1 <... < t n = T, with Z 0 = a, Z T = b. Lemma 5.1 makes it possible to compare the approximate algorithm with the OUalgorithm exact at the grid points. See figures Figure 5.4: Kernel density estimate of the Ornstein-Uhlenbeck bridge at midpoint time t = 1/2 with the algorithm exact at the grid points and the kernel density estimate of the OU-bridge with the approximate algorithm. For the approximate algorithm the number of time steps is 500 and the number of simulations is = For the exact algorithm only the mid-point time is simulated (because the random draw is exact) and the number of simulations is also , due to slowness of the exact algorithm. 5

12 5.3 One exact algorithm Let us first describe an algorithm for SDEs (4.1). We later describe a modified algorithm for diffusion bridges (5.1). In [4] is suggested a general technique for simulation of SDE (4.1) for which the simulated process has the same distribution as the process given by (4.1). It is not, as for example, the Euler-Maruyama method only an approximative algorithm. The conditions we apply here is the following condition. Condition 5.2. (i) The derivative µ x of µ exists (ii) There exists k 1 and k 2 such that k µ2 (x) µ x(x) k 2 for any x R Condition 5.2 can be relaxed but then the algorithm is more technical. In the algorithm below we use the function φ(x) = 2 1µ2 (x)+ 2 1 µ x(x) k 1. Note that 0 φ(x) M = k 2 k 1 The exact algorithm for simulating (4.1) is briefly as follows. For a partition {0 = t 0 < t 1 < < t n = T }, given that X ti is already simulated, X ti+1 is first simulated in a certain way, typically by a acceptance-rejection rule (not a variant of the Euler-Maruyama method). To simulate the process in [t i,t i+1 ], a Poisson random number (an integer) is simulated. If that Poisson random integer, N say, is positive, N couples of uniformly random variables {(τ 1,v 1 ),...,(τ N,v N )}, where τ 1,...,τ N are uniform random numbers in [t i,t i+1 ] and (v 1,...,v N ) are uniform random in the set [0,M]. Then simulate a standard Brownian bridge in [t i,t i+1 ] beginning at X ti and ing at X ti+1. Consider the values y 1,...y N of the Brownian bridge at the time points τ 1,...,τ N. The Brownian bridge is accepted as a simulation of {X t : t [t i,t i+1 ]} if φ(y 1 ) v 1,...,φ(y N ) v N. If we only want values of {X t } at the grid points 0 = t 1 < t 2 <... < t n = T, the connecting Brownian bridges are not needed. An alternative would instead be to only simulate the point X T exactly by an acceptance-rejection rule, and then simulate the Brownian bridge in the whole interval [0,T ] and, if the Brownian bridge is accepted, the values of the Brownian bridge at the grid points is obtained as simulated values of the process {X t : t [0,T ]}. However, the rejection probability of the X T -value and the Brownian bridge is in that situation probably high which should imply an inefficient algorithm. For diffusion bridges of type (5.1) the exact algorithm is simpler. Given X 0 = a, X T does not need to be simulated by a time consuming acceptance-rejection rule: X T is just given the value b. Instead the skeleton time points τ 1,...,τ k can be uniformly picked in the whole time interval [0,T ]. It is applied to diffusion bridges of the type dx t = µ(x t )dt + dw t, X 0 = a, X T = b, (5.4) i.e. σ 1. Note that (5.4) is more general than one may think at the first sight. The equation (5.1) with possibly non-constant σ can, under non-degeneracity conditions of σ (for example σ(x) > ε > 0 for any x) be transformed into one with diffusion coefficient one by applying the Lamperti transform, Xt 1 Y t = F(X t ) = z σ(u) du. Here z is any arbitrary value in the state space of {X t }. Indeed, the process {Y t } solves the SDE dy t = µ Y (Y t )dt + dw t, where µ Y (Y t ) = µ(f 1 (y)) σ(f 1 (y) 1 2 σ x(f 1 (y)) 6

13 Figure 5.5: 20 simulated bridges of dx = sin(x)dt +dw, X 0 = X 1 = 0 with the algorithm exact at the grid points. Figure 5.6: 20 simulated bridges of dx = sin(x)dt + dw, X 0 = X 1 = 0 with the approximate algorithm which can also be written as dy t = ( µ(x t) σ(x t ) 1 2 σ x(x t ))dt + dw t. The exact algorithm to simulate the diffusion bridge (5.4), from [4], see also [9]: 1. Let X T = b. 2. Simulate a non-negative integer τ = N from the Poisson distribution with intensity λ = T M. 3. Draw (if N 1) k pairs of uniformly distributed random numbers {(τ i,v i ) : i = 1,...,N} on [0,T ] [0,M],i=1,2,...,N. 4. Generate a Brownian bridge X starting at x at time 0 and ing in b at time T at time instants t i ; i.e., generate X τi = y i, i = 1,...,N. 5. Compute the indicator function I = N i=1 1 {φ(yi ) ν i }. 6. If I = 1, the trajectory {Y 0 = a,y t1 = y 1,...,Y tk = y k,y T = b} is accepted as an exact draw of a solution X to (5.4) at the points 0 < t 1 <... < t k < T. Otherwise, restart from step 1. Exact draws from X given by (5.4) at pre-determined grid-points 0 = t 1 < t 2 <... < t n = T is obtained by noting the values at these points of the connecting Brownian bridges between the points τ 1 <... < τ k. Figure 5.7: Kernel density estimate of the bridge dx = sin(x)dt + dw, X 0 = X 1 = 1 at midpoint time t = 1/2 with the algorithm exact at the grid points and the kernel density estimate of the bridge with the approximate algorithm. For the exact algorithm the number of time steps is only 3 (we only need the midpoint!) and the number of simulations is 500. For the approximate algorithm the number of time steps is 501 and the number of simulations only 500. In this situation I don t know a closed formula for the density of X 1/2, the midpoint value. 7

14 Figure 5.8: Simulation of a numerical solution to (5.6),[6]. The solution to (5.6) converges to a diffusion bridge. 5.4 PDE-MCMC-algorithm From [7], for X given by the diffusion bridge SDE dx(x) = µ(x(x))dx + σdw(x), X(0) = X(1) = 0 (5.5) (here the time variable is x; the reason will be clear from the arguments below) can be sampled by MCMC-sampling of u given by stochastic partial differential equation u t = 1 ( 2 u µ(u)µ (u) 1 σ 2 x 2 2 µ (u)) W t x u(0,t) = u(1,t) = 0, t [0,T ] u(x,0) = u 0 (x), x [0,1]. (5.6) where 2 W t x is a time-space Gaussian white noise. MCMC (Markov Chain Monte Carlo) is a particular class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its equilibrium distribution. The state of the chain after a large number of steps is then used as a sample from the desired distribution. The quality of the sample improves as a function of the number of steps. What is used here is that (for a fixed x [0,1], as t, u(x,t) converges to a process X(x) which solves (5.5). What is needed is to simulate u(t,x) for large t in a clever way. Normally when simulations are needed the time interval is bounded, but here the focus is on for large times. For that the, for me complicated MCMC-algorithm, is needed. References [1] S: Iacus, Simulation and inference for stochastic differential equations, with R examples, Springer, (2008). [2], I Karatzas, I Shreve, Brownian motion and stochastic calculus, Springer (1991). [3] G. Roberts, O. Stramer, On inference for partial observations from partial observed nonlinear diffusion model using Metropolis-Hastings algorithms, Biometrika 88:3 (2001), [4] A.Beskos, O.Papaspiliopoulos and G.O. Roberts, Retrospective exact simulation of diffusion sample paths with applications, Bernoulli 12:6 (2006), [5] J. Kent, Time-reversible diffusions, Adv Appl Probab 10 (1978), [6] R. Pettersson, Projection scheme for a reflected stochastic heat equation with additove noise, Foundations of Probability and Physics-3, AIP Conference Proceedings, ed: A. Yu Khrennikov, (2005), [7] A.M. Stuart, J. Voss, and P. Wiberg Conditional path sampling of SDEs and the Langevin MCMC method. Comm. Math. Sci. 2 (2004),

15 [8] M.Bladt and M.Sorensen, Simple simulation of diffusion bridges with application to likelihood inference for diffusions,centre for Analytical Finance, Working Paper, University of Copenhagen. February (2007) [9] S. M. Iacus, Simulation and interference for Stochastic Differential Equations, with R Examples, Springer Series in Statistics, (2008) A Appix: Definition of random variables etc A random variable X is a function Ω ω X(ω) measurable in a certain sense so that probabilities can be defined for sets of the type {X(ω) x}. The ω represents a random choice. A real-valued random variable simply takes its values in R. A continuous stochastic proces X = {X t : t 0} is a random variable with values in the set of continuous functions C(R). The map then is Ω = C(R) X t (ω) = ω(t), i.e. the random selection is then a continuous function. A Brownian motion is a continuous stochastic process {W t : t 0}, W(0) = 0, with indepent increments W t+h W t that are hn(0,1) for h > 0. An Ornstein-Uhlenbeck process is a process {X t 0 } satisfying dx t = θx t dt + σdw t, X 0 = a i.e. as the OU-bridge (5.3) but without restrictions on X T. B Appix: matlab codes B.1 Euler-Maruyama s method for standard SDEs Matlab code for Euler simulation of SDE of the type dx = µ(x)dt + σ(x)dw, X 0 = x. In the code below mu(x) = sin(x), σ(x) = 1. x0=1; % initial value n=1000; % number of partition points t=linspace (0,1,n); %gridpoints dt=1/(n-1); %partition step N=500; % N number of simulations x=x0*ones (1,N); mu=@(x) sin(x); sigma=@(x)1; dw=sqrt (dt)*randn (n,n); for i=1:n-1; x (i+1,:)=x (i,:)+mu(x (i,:))*dt+sigma (x (i,:)).*dw (i,:); B.2 Brownian bridge Matlab code for Brownian bridge X t = a(1 t T ) + b t T + (W t t T W T ), 0 t T, 9

16 n=10000; % number of partition points T=1; t=linspace(0,t,n) ; dt=1/(n-1); N=5; dw=sqrt (dt)*randn (n,n); W=cumsum(dW); a=0; b=0; X=a*(1-t/T)+b*t/T+(W-t/T.*W(,:) ); B.3 Approximate diffusion bridge The matlab code for the approximate diffusion bridge that appeared in [8]. n=501; %number of partition points N=500; %number of simulations t=linspace(0,t,n) ; ok=0; dt=t(2)-t(1); bridges=zeros(n,n); for j=1:n ok=0; while ok==0; x=a;y=b; z=a; dwx=sqrt (dt)*randn (n,1); dwy=sqrt (dt)*randn (n,1); for i=1:n-1; x (i+1)=x (i)+mu(x (i))*dt+sigma (x (i)).*dwx (i); y (i+1)=y (i)+mu(y (i))*dt+sigma (y (i)).*dwy (i); z=fliplr(y); dummy=(x(1:-1)>=z(1:-1)).*(x(2:)<=z(2:))... (x(1:-1)<=z(1:-1)).*(x(2:)>=z(2:)); if max(dummy)==1 ok=1; %they meet index=min(find(dummy==1)); bridges(:,j)=[x(1:max(index-1,1)) z(max(index,2):)]; B.4 Code for simulation of Ornstein-Uhlenbeck bridge exact at grid points Matlab code for simulation of exact at the grid points, [8]. dx = θ(x t )dt + σ(dw t ), X 0 = a, X T = b n=100; % number of partition points N=n^2; %number of simulations 10

17 T=1; t=linspace(0,t,n) ; %grid points dt=1/(n-1); a=0; b=1; theta=.5; sig=1; Y=(sig/sqrt(2*theta)*sqrt(1-exp(-2*theta*diff(t)))*ones(1,N)).*... randn(n-1,n); X=a*ones(1,N); for i=1:n-1 X(i+1,:)=exp(-theta*dt)*X(i,:)+Y(i,:); Z=X+(exp(theta*t)-exp(-theta*t))/(exp(theta*t(n))-... exp(-theta*t(n))*(b-x(n,:)); Algorithm for more general diffusion bridges exact at grid points Matlab code for dx t = µ(x t )dt + dw t, X 0 = a,x T = b, i.e. σ 1, where k µ2 (x) µ x(x) k 2 N=500;%number of simulations n=3;%number of partition points mu=@(x)sin(x); mup=@(x)cos(x); sigma=@(x)1; a=0; b=0; T=1;%final time prephi=@(x)1/2*mu(x).^2+1/2*mup(x); mprephi=@(x)-prephi(x); k1=prephi(fminsearch(prephi,-pi));%-1/2; k2=prephi(fminsearch(mprephi,pi/2));%5/8; phi=@(x)prephi(x)-k1; M=k2-k1; lambda=t*m; Ndifb=zeros(n,N); t=linspace(0,t,n) ; for j=1:n accept=0; while accept==0 k=poissrnd(lambda); if k>0 pretskel=sort(t*rand(k,1)); v=m*rand(k,1); tskel=[0; pretskel; T]; W=[0; cumsum(sqrt(diff(tskel)).*randn(length(tskel)-1,1))]; skeleton=a+w-tskel/t*(w()-b+a); I=sum((phi(skeleton(2:-1))<=v));%prod 11

18 if I==k accept=1; % [tskel skeleton] difb=a; for i=2:length(skeleton) if max((t>tskel(i-1)&t<tskel(i)))>0 s=t(t>tskel(i-1)&t<tskel(i)); difb=[difb; Bbfn(tskel(i-1),skeleton(i-1),tskel(i),... difb=[difb;b]; Ndifb(:,j)=difb; %plot(t,difb) %plot(ti,x); 12

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