Exact Simulation and Bayesian Inference for Jump-Diffusion

Exact Simulation and Bayesian Inference for Jump-Diﬀusion Processes

Fl´avioB. Gon¸calves, Gareth O. Roberts Department of Statistics, University of Warwick

Abstract The last 10 years have seen a large increase in statistical methodology for diffusions, and computationally intensive Bayesian methods using data augmentation have been particulary prominent. This activity has been fuelled by existing and emerging applications in economics, biology, genetics, chemistry, physics and engineering. However diffusions have continuous sample paths so may natural continuous time phenomena require more general classes of models. Jump-diffusions have considerable appeal as flexible families of stochastic models. Bayesian inference for jump-diffusion models motivates new methodological challenges, in particular requires the construction of novel simulation schemes for use within data augmentation algorithms and within discretely observed data. In this paper we propose a new methodology for exact simulation of jump-diffusion processes. Such method is based on the recently introduced Exact Algorithm for exact simulation of diffusions. We also propose a simulation-based method to make likelihood-based inference for discretely observed jump-diffusions in a Bayesian framework. Simulated examples are presented to illustrate the proposed methodology. Key Words: Jump-diffusion; exact simulation; Retrospective Rejection Sam- pling; Markov Chain Monte Carlo methods.

1 Introduction

Diffusion processes are continuous time stochastic processes extensively used for modelling phenomena that evolves continuously in time. They are used in many scientific areas, like economics (see Black and Scholes, 1973; Chan et al., 1992; Cox et al., 1985; Merton, 1971), biology (see McAdams and Arkin, 1997), genetics (see Kimura and Ohta, 1971; Shiga, 1985), chemistry (see Gillespie, 1976, 1977), physics (see Obuhov, 1959) and engineering (see Pardoux and Pignol, 1984). A diffusion process is formally defined as the solution of a stochastic differential equation (SDE) of the type:

dVt = µ(Vt, t)dt + σ(Vt, t)dBt,V0 = v0, (1)

1Address: Fl´avioB. Gon¸calves, Department of Statistics - University of Warwick, Coventry, UK, CV4 7AL. E-mail: [email protected]

1 where µ(Vt, t) and σ(Vt, t) are called the drift and diffusion coefficient, and define the in- stantaneous mean and variance of the process, respectively. They are presumed to satisfy the regularity conditions (locally Lipschitz, with a linear growth bound) that guarantee a weakly unique global solution. Bt is a Brownian Motion, which is a well-known Gaussian process. Due to the properties of Brownian motion, diffusion processes are continuous processes. Although they are very efficient to model a large number of phenomena, in some cases it is necessary to model processes that have jumps, that is, processes that have discontinuity points. To account for this feature, a natural solution is to use the so-called jump-diffusion processes. Jump-diffusions are continuous time stochastic processes that may jump and in between the jumps behave as a diffusion process. They take into account the fact that from time to time larger jumps in the process may occur and such jumps cannot be adequately modeled by pure diffusion-type processes. For example, in finance, a jump-diffusion may be useful to model stock prices where unexpected events may cause a jump in the price. The most common area where jump- diffusions are applied is financial economics (see Ball and Roma, 1993; Duffie et al., 2000a; Runggaldier, 2003; Eraker et al., 2003; Eraker, 2004; Johannes, 2004; Barndorff- Nielsen and Shephard, 2004; Feng and Linetsky, 2008; Kennedy et al., 2009). But other applications can be found, for example, in physics (see Chudley and Elliott, 1961; Ellis and Toennies, 1993), biomedicine (see Grenander and Miller, 1994) and object recognition (see Srivastava et al., 2002). Formally, a jump-diffusion is the solution V := {Vt : 0 ≤ t ≤ T } of the stochastic differential equation:

dVt = µ(Vt−)dt + σ(Vt−)dBt + dJt,V0 = v0 (2) where µ, σ : R → R are presumed to satisfy the regularity conditions (locally Lipschitz, with a linear growth bound) that guarantee a weakly unique global solution. Bt is again a Brownian motion and Jt is a jump process defined by two components: a Poisson process of rate λ(Vt−, t) that describes the jump times, and a function ∆(Vt−, z) that defines the + jump sizes, where z ∼ fz(·; t) and fz is a standard density function for every t ∈ R . For any r ∈ R, λ(r, t) is a positive real valued function on [0,T ] and is assumed to be absolutely continuous with respect to the Lebesgue measure. Moreover, Vt− is the state of the process at time t before the jump, if there is a jump at time t. Between any two jumps, the process behaves as a diffusion process with drift µ and diffusion coefficient σ. In this paper we consider only the case where the continuous part of the process is time homogeneous, as defined in (2). It is common to find applications of diffusions where the drift and/or the diffusion coefficient depend on unknown parameters. Making inference for these parameters based on discrete observations of the diffusion is then a very important and challenging problem which has been pursued in three main directions: considering alternative estimators to the MLE; using numerical approximations to the unknown likelihood function; and estimating an approximation to the likelihood by using Monte Carlo (MC) methods. Inference for discretely observed jump-diffusion processes is also a very important and extremely challenging problem. Moreover, it has an additional complication on the fact

2 that the jumps are not observed and add a set of latent variables to be dealt with in the models. There is much less work done in inference for jump-diffusions than there is for diffusions. Existing methods in the literature are based on path discretisation and data augmentation methods. Time-discretisation schemes like the Euler approximation are used to simulate from the conditional process and/or approximate the transition density of the process. Most of these methods rely on Monte Carlo techniques, in particular particle filters and MCMC. For example, Johannes et al. (2002) propose an algorithm which combines data augmentation and particle filtering. The authors use importance sampling to prop- agate the particles and Euler approximation to augment the data and approximate the likelihood. In Eraker et al. (2003) the authors propose an MCMC algorithm to estimate parameters in jump-diffusion models with stochastic volatility and allow for jumps in both returns and volatility. The models considered in the paper are introduced in Duffie et al. (2000b) and have state-independent jump rate and jump size distribution. The Euler scheme is directly applied to the observations to approximate the transition density and a Gibbs sampling algorithm is construct to sample from a Markov chain with invariant distribution given by the joint posterior of the parameters and latent variables. Johannes et al. (2009) propose an optimal filtering algorithm to estimate the latent variables in a stochastic volatility model with jumps keeping the values of the parameters constant. Two methods are proposed in the paper, based on the sampling-importance resampling (SIR) algorithm and on the auxiliary particle filtering (APF) algorithm, respectively. Euler scheme is used to augment the data and approximate the likelihood in these methods. Golightly (2009) also proposes particle filtering algorithms for sequential estimation and uses MCMC inside the particle filter. An algorithm is proposed to simulated data between the observations conditional on the jump times and sizes. The augmented data is combined with the Euler scheme to approximate the transition density of the process. The algorithms sample from the posterior distribution of the latent data and the model parameters online. Other methods can be found on Bekers (1981), Honoré(1998), Broadie and Kaya (2006), Jiang and Oomen (2007), Ramezani and Zeng (2007) and Yu (2007). Methods to test for jumps can also be found in the literature (see Johannes, 2004; Barndorff-Nielsen and Shephard, 2006; Rifo and Torres, 2009). The use of approximations based on path discretisation techniques have considerable implications on the inference process. For example, the discretised jump-diffusions assume that the jump times follow a Bernoulli distribution with parameter λ∆ in a time interval of length ∆, where λ is the jump rate in the original process. It implies that the maximum number of jumps in each interval is 1. Such assumption is reasonable as ∆ gets very small, but may not be a very good approximation depending on the process, data and ∆ considered. Furthermore, the Bayesian methods based on Monte Carlo techniques aim to sample from the posterior distribution of the unknown quantities. If discrete approximations are

3 used, the posterior distribution from which the algorithms sample from is not the exact posterior distribution. Therefore, if, for example, a MCMC algorithm is used to sample from the posterior distribution, apart from the convergence issues related to such method, the invariant distribution of Markov chain is just an approximation of the real posterior distribution of interest. Recently, Beskos et al. (2006) introduced a new direction based on a collection of algorithms for simulation of diffusions (the so-called Exact Algorithm, EA). The algorithms are exact in the sense that they involve no discretisation error and rely on techniques called Retrospective Rejection Sampling. Whilst a generalisation for jump-diffusions is al- ready available (see Casella and Roberts, 2008) to simulate jump-diffusions unconditional on the ending point, simulation-based inference methods are more efficient if the process can be simulated conditional on the observations and, due to the Markov property of these processes, it means to conditional on start and ending points. The aim of this work is, firstly, to generalise the Exact Algorithm for the simulation of conditional jump-diffusions. Then, to use such algorithm to develop efficient methods, free of discretisation error, to make inference for such processes by estimating the latent variables of the jump process (jump times and sizes) and parameters in the drift, diffusion coefficient, jump rate and jump size distribution. Section 2 presents the Exact Algorithm proposed in Beskos et al. (2006) and the Jump Exact Algorithm (JEA) proposed in Casella and Roberts (2008). The Jump Ex- act Algorithm for the simulation of jump-diffusions conditional on the ending point is proposed in Section 3. Section 4 presents an MCMC approach to perform inference in discretely observed jump-diffusions. Section 5 shows a simulated example with the proposed methodologies.

2 Exact Algorithm and Jump Exact Algorithm

The Exact Algorithm (EA) was proposed by Beskos et al. (2006) to perform exact simulation of a class of Itô’sdiffusions. The algorithm is exact in the sense that there is no discretisation error involved. The EA performs Rejection Sampling (RS) by proposing paths from processes that we can simulate and accepting them according to appropriate probability density ratios. The novelty lies in the fact that the paths proposed are unveiled only at finite (but random) time instances and the decision whether to accept or not the path can be easily taken. We will present here the EA for simulation of univariate and time homogeneous diffusions. Its extensions for both cases can be found in Beskos et al. (2008). Consider V := {Vt : 0 ≤ t ≤ T } a one-dimensional diffusion process solving the SDE:

dVt = b(Vt)dt + σ(Vt)dBt,V0 = V0 ∈ R, t ∈ [0,T ] To apply the Exact Algorithm it is necessary to work with diffusions with unit diffusion coefficient. This can be easily achieved by applying the 1-1 transformation Vt → η(Vt) = Xt given by Z v 1 η(v) = du z σ(u)

4 with z being some element of the state space of V . Assuming that σ(Vt) is continuously diﬀerentiable and applying Itˆo’sformula, we get the following transformed process:

dXt = α(Xt)dt + dBt,X0 = x = η(V0) ∈ R, t ∈ [0,T ] where b{η−1(u)} σ0{η−1(u)} α(u) = − σ{η−1(u)} 2 The function η−1 denotes the inverse transformation and σ0 is the derivative with respect to the state space variable. To recover the information on the original diﬀusion, one just have to apply the inverse transformation η−1 to the transformed process. Assuming that Q is the probability law of X and W is the probability law of a Brownian motion starting at x ∈ R, the Girsanov’s formula implies that

½Z T Z T ¾ dQ 1 2 = exp α(Xt)dXt − α (Xt)dt (3) dW 0 2 0 The following standard assumptions are now made:

(a) α(·) is continuously diﬀerentiable. (b) (α2 + α0)(·) is bounded below. (c) The Girsanov’s formula for X is given by expression (3) and is a martingale with respect to Wiener measure.

The next step is to get rid of the stochastic integral that appears in the Girsanov’s Z u formula. It is done by applying Itˆo’sformula to the transformation A(u) := α(y)dy, 0 u ∈ R, which gives an equality to the stochastic integral. Substituting the result in expression (3) results in

½ Z T ¾ dQ 1 ¡ 2 0 ¢ = exp A(XT ) − A(x) − α (Xt) + α (Xt) dt (4) dW 2 0 The idea is then to perform Rejection Sampling by proposing paths from a Brownian Motion. But the first problem we have is that, to perform Rejection Sampling, we need the ratio of the target density and the proposed density to be bounded for every point of the state space. And the term A(XT ) is typically unbounded. To circumvent this problem or just to simplify the Girsanov’s formula, we will propose paths from a process defined as Biased Brownian Motion which differs from a Brownian motion only in the distribution of the ending point XT . While in the Brownian Motion XT has a normal distribution with mean x and variance T , in the biased Brownian Motion the density h of XT is proportional to ½ ¾ 1 h ∝ exp A(X ) − (X − x)2 T 2T T

5 and we need this function to be integrable. Deﬁning Z as the probability measure of the biased Brownian motion, we use the following decomposition: dQ dQ dW = . dZ dW dZ dW It can be shown that the ratio depends only on the distribution of X and is dZ T given by dW f (X ; x, T ) = N T ∝ exp{−A(X )} dZ h T This implies in

½ Z T µ ¶ ¾ dQ 1 2 1 0 ∝ exp − α (Xt) + α (Xt) dt (5) dZ 0 2 2 It is possible to obtain a non-negative function φ such that ½ Z ¾ dQ T ∝ exp − φ(Xt)dt ≤ 1 (6) dZ 0 Analytically, φ is deﬁned as: α2(u) + α0(u) φ(u) = − k, u ∈ R, for k ≤ inf (α2 + α0)(u)/2. 2 u∈R If it was possible to draw complete continuous paths of Z on [0,T ] and calculate the integral involved in the acceptance probability analytically, the algorithm would be straightforward. However, the following theorem can be used to deﬁne an event with probability given by (6) which can be easily evaluated.

Theorem 1 Let X be any continuous mapping from [0,T ] to R, and M(X) an upper bound for the mapping t 7→ φ(Xt), t ∈ [0,T ]. If Φ is a homogeneous Poisson process of unit intensity on [0,T ] × [0,M(X)] and N is the number of points of Φ found below the graph {(t, φ(Xt)); t ∈ [0,T ]}, then

½ Z T ¾ P (N = 0|X) = exp − φ(Xt)dt . 0 Theorem 1 gives us an event with probability equal to the acceptance probability of the Rejection Sampling algorithm proposed. Furthermore, in order to evaluate if such event happened or not it is enough to reveal the proposed path only in the time instances where there is a realisation of the Poisson process. The three versions of the Exact algorithm diﬀer in the way the upper bound M(X) for the mapping t 7→ φ(Xt), t ∈ [0,T ] is obtained. In EA1, φ(u) is bounded for every u ∈ R by a constant M and the algorithm is the following.

Exact Algorithm 1

6 1. Produce a realisation {x1, x2, . . . , xτ }, of a Poisson process on [0,T ] × [0,M], where xi = (xi,1, xi,2), 1 ≤ i ≤ τ;

2. Simulate a skeleton of X ∼ Z at the time instances {x1,1, x2,1, . . . , xτ,1}; 3. Evaluate N;

4. If N = 0, go to step 5, else go to step 1;

5. Output the currently constructed skeleton S(X) of X;

Note that the EA1 returns a “Skeleton” of the process X deﬁned over the time interval [0,T ] with starting point x. We represent it as:

S0(x; 0,T ) := {(t0, xt0 ), (t1, xt1 ),..., (tM , xtM )} with t0 = 0 and tM = T . We can simulate further information on the process X using the following representa- tion, immediately derived from elementary Brownian motion constructions:

MO−1

X|S0(x; 0,T ) ∼ BB(ti, xti ; ti+1, xti+1 ) i=0 where BB(s1, a; s2, b) is the measure of a Brownian bridge starting in a at time s1 and ending in b at time s2. If one of the following limits is satisﬁed, we have EA2.

lim sup φ(u) < ∞ u→∞ or lim sup φ(u) < ∞ u→−∞

In this case, it is possible to identify an upper bound M(m) for φ(Xt), with t ∈ [0,T ], after decomposing the proposed path Xt at its minimum or maximum. If lim sup φ(u) < ∞ and m is the minimum of Xt in [0,T ] u→∞ M(m) = sup{φ(u); u ≥ m}

If lim sup φ(u) < ∞ and m is the maximum of Xt in [0,T ] u→−∞

M(m) = sup{φ(u); u ≤ m}

The EA2 is then given by:

Exact Algorithm 2

1. Initiate a path Xt ∼ Z on [0,T ] by drawing XT ∼ h;

7 2. Simulate its minimum or maximum m and the moment tm when it is achieved;

3. Find an upper bound M(m) for φ(Xt);

4. Produce a realization {x1, x2, . . . , xτ } of a Poisson process on [0,T ] × [0,M(m)];

5. Simulate a skeleton of X|(m, tm) at the time instances {x1,1, x2,1, . . . , xτ,1}; 6. Evaluate N;

7. If N = 0, go to step 8, else go to step 1;

8. Output the currently constructed skeleton S(X) of X.

The algorithms to perform steps 2 and 5 are described in Beskos et al. (2006). In the case of EA 3 no hypothesis other than (a), (b), (c) has to be made. It will then be necessary to have lower and upper bounds for Xt between 0 and T . These bounds are obtained using the idea of a layered Brownian bridge. Deﬁne two increasing sequences of positive real numbers {ai}i≥1 and {bi}i≥1 with a0 = b0 = 0. Given X0 = x and XT = y, setx ¯ = x ∧ y andy ¯ = x ∨ y and deﬁne the following events: ½ ¾ \ ½ ¾ Ui = sup Xt ∈ [¯y + bi−1, y¯ + bi] inf Xt > x¯ − ai 0≤t≤T 0≤t≤T ½ ¾ \ ½ ¾ Li = inf Xt ∈ [¯x − ai, x¯ − ai−1] sup Xt < y¯ + bi 0≤t≤T 0≤t≤T [ Di = Ui Li, i ≥ 1

Deﬁne the random variable I = I(X) such that {I = i} = Di. {I = i} implies that {x¯ − ai < Xt < y¯ + bi, ∀ t ∈ [0,T ]}. An illustration of the layered Brownian bridge is given in Figure 1 with ai = bi and I(X) = 4.

Figure 1: Example of the construction of the layered Brownian bridge, ai = bi and I(X) = 4.

8 The random variable I provides the needed bounds for the process X in [0,T ] and the algorithm goes like this:

Exact Algorithm 3

1. Initiate a path Xt ∼ Z on [0,T ] by drawing XT ∼ h; 2. Simulate I(X);

3. Find an upper bound M(I) for φ(Xt);

4. Produce a realization {x1, x2, . . . , xτ }, of a Poisson process on [0,T ] × [0,M(I)];

5. Simulate a skeleton of X|I at the time instances {x1,1, x2,1, . . . , xτ,1}; 6. Evaluate N;

7. If N = 0, go to step 8, else go to step 1;

8. Output the currently constructed skeleton S(X) of X.

The algorithms to perform steps 2 and 5 are described in Beskos et al. (2008). Once again, for EA2 and EA3, we can simulate further information on the process X using the construction of the proposed path. The simulation is performed as it is done in step 5 of the EA2 and the EA3, accordingly. An interesting case of the Exact algorithm is to simulate diffusion processes also conditional on the ending point XT = y. In this case, paths can be proposed directly from a Brownian bridge starting in x at time 0 and finishing in y at time T . The Radon–Nikodym derivative of the conditional diffusion Q˜ and the Brownian bridge W˜ is given by ½ ¾ ˜ Z T dQ 1 ¡ 2 0 ¢ N(y; x, T ) = exp A(y) − A(x) − α (Xt) + α (Xt) dt × , (7) dW˜ 2 0 qT (x, y) where N(y; x, T ) is the density of a Normal distribution with mean x and variance T evaluated at y and qT (x, y) is the transition probability of going from x to y in a time interval of length T under the probability measure of Q. This way, the target Radon–Nikodym derivative will be: ½ ¾ ˜ Z T dQ 1 ¡ 2 0 ¢ ∝ exp − α (Xt) + α (Xt) dt (8) dW˜ 2 0 The conditional Exact Algorithm is then performed like the unconditional one but proposing from a Brownian bridge. This version of the algorithm is particulary interesting when the Exact Algorithm is used to make inference in diffusion processes. Beskos et al. (2006) propose several methods for inference in diffusions using the EA.

9 2.1 Jump Exact Algorithm The Jump Exact Algorithm was proposed in Casella and Roberts (2008) to perform exact simulation of a class of jump-diffusions. Their algorithm simulates jump-diffusions unconditional on the ending point and requires the jump rate λ(Vt−, t) to be bounded for every Vt− in the state space of V and for every t in [0,T ]. Consider V := {Vt : 0 ≤ t ≤ T } a one-dimensional jump-diffusion process solving the SDE: dVt = b(Vt− )dt + σ(Vt− )dBt + dJt,V0 = v0 (9)

Jt is a jump process where the jump times follow a Poisson process of rate λ1(Vt− , t) and − the jump sizes are given by a function g1(zt,Vt ), where zt has a distribution fZt . Once more, we apply the transformation Vt → η(Vt) = Xt to work with

dXt = α(Xt− )dt + dBt + dJt,X0 = x, (10) where b{η−1(u)} σ0{η−1(u)} α(u) = − , g(z, u) = η(η−1(u) + g (z, η−1(u))) − u σ{η−1(u)} 2 1 −1 and λ(u, t) = λ1(η (u), t). As mentioned before, we assume that λ(Xt− , t) is bounded. To break down the state- dependency of the jump times we use the following result.

Poisson Thinning To simulate a Poisson process of rate λ(Xt− , t) ≤ λ on [0,T ] do: 1. Simulate a Poisson process of rate λ on [0,T ].

λ(X − , ti) ti 2. Keep each point with probability R(Xt− , ti) = , for i = 1, . . . , n, where n i λ is the number of points simulated in step 1.

The Jump Exact Algorithm is performed as follows.

Jump Exact Algorithm

1. Simulate a Poisson process of rate λ on [0,T ];

2. Apply the Exact Algorithm to [0,T ], also outputting Xt on the time instants of the PP simulated on step 1; 3. Make j=1;

4. Perform Poisson thinning until one point tj is kept or no point is kept;

5. If a point is kept, simulate the jump at tj, apply EA to [tj, T ], make j = j + 1 and go to 4; else go to 6; 6. Output the skeleton S(X) of X.

10 3 Jump Exact Algorithm for conditional jump-diﬀusions

In this section we extend the JEA to simulate jump-diﬀusions conditional on the initial and ending points. It will allow us to simulate a jump-diﬀusion conditional on discrete observations, since the Markov property implies that the distribution of Xt, for a given t ∈ [0,T ], depends only on the two observations which t is in between.

3.1 Preliminaries Consider the jump-diffusion defined in (10) and let P be the probability measure associated to this jump-diffusion. Note firstly that it is not possible to use the strategy used in the JEA for unconditional jump-diffusions. If we try to simulate from the conditional jump- diffusion by, for example, simulating a Brownian bridge B(0, x; T, y) and then performing the Poisson thinning, and keep doing it until there is no more jump, it would not be a realization of the conditional jump-diffusion with law P given XT = y. In this case we are only conditioning the continuous part of the jump-diffusion. The solution is to propose a jump process that we can simulate from and perform Rejection Sampling to decide whether or not to accept the proposed path. The following theorem is crucial for the rejection sampling algorithm.

Theorem 2 Let P be the probability law of a jump-diffusion as defined in (10). Let also Q be the probability law of a jump-diffusion which is the solution of the SDE dXt = dBt +Jt, where J is a jump process of constant rate λ and jump size distribution f that depends neither on the state of the process nor on the time. Then ( Ã ! ) NJ Z X Z T dP X tj 1 = exp A(X ) − A(x) − α(u)du − (α2 + α0)(X )dt dQ T 2 t− j=1 Xtj − 0 ½Z ¾ T YNJ λ(X , t )f (X − X ) × exp −(λ(X , t) − λ)dt tj − j g tj tj − (11) t− λf(X − X ) 0 j=1 tj tj − where NJ is the number of jumps in [0,T ] and the tj’s are the jumping times.

Proof Omitted

Now, let Q be the probability measure associated to the jump-diffusion which is the solution to the SDE dXt = dBt + dJt,X0 = x (12) where J is a jump process with constant jump rate λ and jump size distribution f which depends neither on the state of the process nor on the time instance. ˜ Now define P to be the law of the jump-diffusion with law P, given XT = y and, ˜ analogously, Q to be the law of the jump-diffusion with law Q, given XT = y. The first idea would be to propose paths from Q˜ to performed Rejection Sampling. But sampling

11 from Q˜ turns out to be very hard. We would have to know the marginal distributions of both the jump process and the continuous part given the ending point. Define H to be the sum of the jumps of the jump-diffusion with law Q and B = y −H. If we could find the distribution of H|B + H = y and J|H, where J is the jump process, it would be feasible to simulate from Q˜ by using the factorization ˜ Q(X) = Q(X|XT ∈ dy) = Q(Xt∈(0,T ),H,B|XT ∈ dy)

= Q(Xt∈(0,T )|H,B,XT ∈ y)Q(B|H,XT ∈ dy)Q(H|XT ∈ dy)

= Q(Bt∈(0,T ) + Jt∈(0,T )|H,B,XT ∈ dy)Q(B|H,H + B ∈ dy)Q(H|H + B ∈ dy) The simulation of Q˜ would then start by simulating H from Q(H|H + B ∈ dy), then simulating B from Q(B|H,H + B ∈ dy) and finally simulating Xt, for any desired t, from Q(Bt∈(0,T ) + Jt∈(0,T )|H,B,XT ∈ dy). XNJ The first difficult with this approach is to simulate H. Note that, under Q, H = Zj, j=1 where NJ ∼ P oisson(λ) and the variables Zj have a probability density function f, and B ∼ N(0,T ). This implies Q(H + B ∈ dy|H ∈ dh)Q(H ∈ dh) Q(H ∈ dh|H + B ∈ dy) = R Q(H + B ∈ dy|H ∈ dh)Q(H ∈ dh) Q(B ∈ dy − dh)Q(H ∈ dh) = R Q(B ∈ dy − dh)Q(H ∈ dh) and also X Q(H ∈ dh) = Q(H ∈ dh|NJ = n)Q(N = n) n

Even in the most simple cases, for example, when Zj ∼ N(0, 1), the two distributions above are untractable. The distribution Q(B|H ∈ dh, H + B ∈ dy) is just a point mass at y − h, which means that B = y − h with probability one. To simulate Xt from Q(Bt + Jt|H,B,XT ∈ dy), it is enough to simulate Bt and Jt separately from Q(Bt|B) and Q(Jt|H), respectively, since they are independent given H and B. Q(Bt|B) is a Brownian bridge and, therefore, can be easily simulated. On the other hand, it is quite complex to simulate the jump process given the sum of the jumps. The candidate paths will then be proposed from a stochastic process that has a probability law D˜ and is composed by the sum of two dependent process J and B∗, where J is the jump process from the jump-diﬀusion with law Q and B∗ is the Brownian bridge B(0, x; T, y − JT ). Our aim is to simulate exact skeletons of the jump-diﬀusion with probability law P˜ using Rejection Sampling and proposing from the stochastic process with law D˜. In order dP˜ to do that, we need to evaluate the Radon-Nikodym derivative . We do that by using dD˜ the following results: dP˜ dP˜ dQ˜ = × dD˜ dQ˜ dD˜

12 Lemma 1 Under conditions (a) − (c) from Section 2, P˜ is absolutely continuous with respect to Q˜ with density dP˜ Q(T, y) = G × dQ˜ P(T, y) where Q(T, y) is the transition density of the process with probability law Q at (0, x; T, y) and analogue for P(T, y), and ( Ã ! ) NJ Z X Z T X tj 1 G = exp A(y) − A(x) − α(u)du − (α2 + α0)(X ) + 2(λ(X , t) − λ)dt 2 t t j=1 Xtj − 0

YNJ λ(X , t )f (X − X ; ·) × tj − j g tj tj − λf(X − X , ·) j=1 tj tj −

Proof Omitted

dQ˜ The formula for is obtained based on the following three propositions: dD˜ ˜ ˜ Proposition 1 Given the ﬁnal state JT of the jump process, Q and D have the same distribution, i.e. ˜ d ˜ (Q|JT ) = (D|JT )

Proof Omitted ˜ dQ ˜ Proposition 2 The ratio is given by the ratio of the density of JT under Q and the dD˜ same density under D˜, i.e. dQ˜ Q˜ (J ) f˜ = T = Q (J ) ˜ ˜ T dD D(JT ) fD˜

Proof Omitted

Proposition 3 Q˜ (J ) f (y − J ) T = BT T ˜ D(JT ) Q(T, y) where fBT (u) is the density of a normal distribution with mean x and variance T at u.

Proof Omitted

13 We finally get the result: dP˜ Q(T, y) f (y − J ) f (y − J ) = G BT T = G BT T (13) dD˜ P(T, y) Q(T, y) P(T, y) ½ Z µ ¶ ¾ T α2 + α0 ∝ exp − (X ) + λ(X , t)dt 2 t t 0 ³ ´ R X tj ½ ¾ NJ λ(X , t )f (X − X ; ·) exp − α(u)du Y tj − j g tj tj − Xt − 1 × j exp − (y − J − x)2 λf(X − X , ·) 2T T j=1 tj tj − which can be rewritten as ( ) ½ Z T µ 2 0 ¶ ¾ α + α 1 2 exp − (Xt) + λ(Xt, t)dt exp − (y − JT − x) (14) 0 2 | 2T {z }    NJ µ ¶ Z X  X λ(X , t )f (X − X ; ·) tj × exp log tj − j g tj tj − − α(u)du    λf(Xtj − Xtj −, ·) Xt −  |j=1 {z j } and defining the first underbraced part as F1 and the second one as F2, we have ½ Z µ ¶ ¾ dP˜ T α2 + α0 1 ∝ exp − (Xt) + λ(Xt, t) − (F1 + F2)dt (15) dD˜ 0 2 T Assuming that the function inside the integral in (15) is bounded below, we define µ ¶ α2 + α0 1 l = inf (Xt) + λ(Xt, t) − (F1 + F2) (16) X 2 T This implies that ½ Z µ ¶ ¾ dP˜ T α2 + α0 1 ∝ exp − (Xt) + λ(Xt, t) − (F1 + F2) − l dt ≤ 1 (17) dD˜ 0 2 T ˜ Once the proposal is sampled from D, we have the values of F1 and F2 and Equation (17) can then be written as ½ Z ¾ dP˜ T ∝ exp − φ(Xt)dt ≤ 1 (18) dD˜ 0 To evaluate this probability we use Theorem 1 with ·µ ¶ ¸ α2 + α0 1 M(X) = sup (Xt) + λ(Xt, t) − (F1 + F2) − l. (19) 0≤t≤T 2 T The Rejection Sampling method requires (15) to be bounded. But it will only happen when, additionally to the assumptions of the EA, the part inside the product in (13) is equal or smaller then 1 for every j. This will be true when λ(Xt, t) and the ratio of densities fg/f are bounded and one of the following is true:

14 ¯Z ¯ ¯ ∞ ¯ ¯ ¯ 1. ¯ α(u)du¯ < ∞; −∞

2. |α(u)| ≤ a, ∀u ∈ R and |Xt − Xt−| ≤ b for some a, b ∈ R.

Each of these conditions guarantees that the exponential term inside the product in (13) is bounded. Besides, λ is chosen to be big enough do make each term of the product smaller than 1. In order to relax the hypothesis that at least one of the conditions 1 or 2 above is satisfied, we propose another stochastic process to simulate the proposed paths from in the Rejection Sampling algorithm. Define F˜ to be the law of the process that differs from ˜ s c the one with law D in the distribution of (NJ, J ,XJ ), where NJ is the number of jumps s c ∗ in J, J are the jump sizes and XJ is the process B in the time instances where there is a jump. In the process with law D˜ we have YNJ YNJ s c s c s s c c f(NJ, J ,XJ ) = f(NJ)f(J |NJ)f(XJ |J ,NJ) = f(NJ) f(Jtj ) f(Xtj |Xtj−1 ,JT ) j=1 j=1 where NJ ∼ P oisson(λ) s f(Jtj ) = f c c c (Xtj |Xtj−1 ,JT ) ∼ BB(tj−1,Xtj−1 ; T, y − JT ) c with t0 = 0 and Xt0 = x. ˜ s For the process with law F, assuming Jt0 = 0, we have ( P ) NJ Z Xc + j Js X tj k=0 tk f (NJ, J s,Xc ) ∝ f (NJ, J s,Xc ) exp − α(u)du (20) F˜ J D˜ J P Xc + j−1 Js j=1 tj k=0 tk which has to be integrable. dP˜ To obtain the Radon-Nikodym derivative , we use the fact that dF˜ dP˜ dP˜ dQ˜ dD˜ = × × dF˜ dQ˜ dD˜ dF˜ and the following proposition. dD˜ Proposition 4 The ratio is given by the ratio of the density of (NJ, J s,Xc ) under dF˜ J D˜ and the same density under F˜, i.e. dD˜ D˜(NJ, J s,Xc ) f = J = D˜ (NJ, J s,Xc ), ˜ ˜ s c J dF F(NJ, J ,XJ ) fF˜

15 which is ( ) ( Ã !) c Pj s NJ Z X + J NJ Z Xt dD˜ X tj k=0 tk X j = κ exp α(u)du = κ exp α(u)du ˜ c Pj−1 s dF X + J Xt − j=1 tj k=0 tk j=1 j where κ is a constant.

Proof Omitted

This gives the result: ( Ã !) NJ Z X dP˜ Q(T, y) f (y − J ) X tj = G BT T κ exp α(u)du (21) dF˜ P(T, y) Q(T, y) j=1 Xtj − ½ Z µ ¶ ¾ T α2 + α0 ∝ exp − (Xt) + λ(Xt, t)dt 0 2 ½ ¾ YNJ λ(X , t )f (X − X ; ·) 1 × tj − j g tj tj − exp − (y − J − x)2 λf(X − X , ·) 2T T j=1 tj tj − which can be rewritten as ( ) ½ Z T µ 2 0 ¶ ¾ α + α 1 2 exp − (Xt) + λ(Xt, t)dt exp − (y − JT − x) (22) 0 2 | 2T {z }    µ ¶ XNJ λ(X , t )f (X − X ; ·)  × exp log tj − j g tj tj −  λf(X − X , ·)   j=1 tj tj −  | {z } and defining the first underbraced part as F1 and the second one as F2, we have ½ Z µ ¶ ¾ dP˜ T α2 + α0 1 ∝ exp − (Xt) + λ(Xt, t) − (F1 + F2)dt (23) dF˜ 0 2 T Assuming that the function inside the integral in (23) is bounded below, we define µ ¶ α2 + α0 1 l = inf (Xt) + λ(Xt, t) − (F1 + F2) (24) X 2 T which implies in ½ Z µ ¶ ¾ dP˜ T α2 + α0 1 ∝ exp − (Xt) + λ(Xt, t) − (F1 + F2) − l dt ≤ 1 (25) dF˜ 0 2 T

16 3.2 The Conditional JEA algorithm Considering all the results from Section 5.1, the Conditional Jump Exact Algorithm (CJEA) follows like this: start by choosing a proposal, that is, a jump rate and a jump size distribution that guarantees that the Radon-Nikodym derivative between the target jump-diﬀusion and the proposed process is bounded. Then, decide if the proposed paths will be from D˜ or F˜. Make the required calculations and proceed as following:

1. If proposing from D˜, simulate the jump process with jump rate λ and jump size ˜ c distribution f. If proposing from F, simulate the jump process and XJ according s c to fF˜(NJ, J ,XJ ). µ ¶ α2 + α0 2. If (X ) + λ(X , t) is not bounded above, simulate r(X) to ﬁnd an upper 2 t t bound M(X) for φ(X). r(X) can be the minimum of X, the maximum of X or I, depending on the situation;

3. Simulate a Poisson process of unit rate on [0,T ] × [0,M(X)];

4. Simulate the process Xt on the same time instances of the Poisson processes simulated in step 3 and evaluate φ(X) in these points;

5. Evaluate N. If N = 0 go to next step, else, go back to 1;

6. Output the skeleton of the accepted path.

We deﬁne CJEA1, CJEA2 and CJEA3 following the deﬁnitions of EA1, EA2 and EA3. For example, CJEA2 is the case where r(X) is either the minimum or the maximum of the path in (0,T ).

4 A Markov Chain Monte Carlo approach

In this section, a Bayesian method for inference in discretely observed jump-diffusion processes is proposed. The method is based on an MCMC algorithm where a Markov chain that has the exact posterior distribution of the parameters of the process as its equilibrium distribution is constructed. The Conditional Jump Exact Algorithm plays a crucial role in this method as it is shown next. Once again, consider a parameter set θ of a jump-diffusion process V like the one defined in (9) for which inference has to be made based on discrete observations of the process at time instances 0 = t0, . . . , tn = T . Suppose it is possible to simulate exact samples from this process using the CJEA. The aim here is to construct a Markov Chain with stationary distribution given by the exact posterior distribution of θ. Differently from competitive existing MCMC algorithms, which are based on path augmentation, the method proposed here uses the variables involved in the CJEA algorithm and obtains the posterior distribution of θ as a marginal

17 of the joint posterior of these variables and θ. Such idea was introduced in Beskos et al. (2006) to perform Bayesian inference for diffusion processes using the Exact Algorithm. We present here the MCMC algorithm only for the CJEA1 case. The extension for CJEA2 has also been derived, but will be omitted. Let’s start by defining (J ∗, ω∗, Φ∗) as the accepted random elements of CJEA1. Since this is a Bayesian approach, we have to assign prior distributions to the parameters of interest. Let π(θ) be the prior density of θ and suppose that the jump-diffusion V is ∗ ∗ ∗ observed at times t0, . . . , tn. Now define (Ji , ωi , Φi ) as the accepted random elements of ∗ ∗ CJEA1 in the time interval [ti−1, ti], and S(Ji , ωi ) as the corresponding output skeleton, for i = 1, . . . , n and xi = xi(θ) := η(Vti ; θ), for i = 0, . . . , n. We now use the components we have to construct the following Bayesian hierarchical model: θ ∼ π(θ), ¡ ¢ Vti |Vti−1 , θ ∼ p∆ti Vti−1 ,Vti ; θ , ∗ ∗ ∗ ∗ ∗ ∗ (J , ω , Φ )|θ, xi−1(θ), xi(θ) ∼ π (J , ω , Φ |θ, xi−1(θ), xi(θ)) Beskos et al. (2006) define this model as hierarchical simulation model to emphasize that it represents the order in which the variables must be simulated to ensure that the ∗ ∗ (∆ti,xi−1,xi) output of the CJEA, {J , ω }, is indeed from Qθ .

We are interested in the distribution π(θ|v), for v = {Vt0 ,...,Vtn }. It is obtained as ∗ ∗ the marginal of the joint posterior π (θ, {S(Ji , ωi ), 1 ≤ i ≤ n}|v). We sample from this distribution via Gibbs Sampling by alternating between

(S(J ∗, ω∗)|θ, v) and (θ|S(J ∗, ω∗), v)

Samples of (S(J ∗, ω∗)|θ, v) are obtained via CJEA and to derive the density π (θ|S(J ∗, ω∗), v) we start with the following theorem:

Lemma 2 Consider any two ﬁxed points x and y. Let Φ = {Ψ, Υ} be the marked Poisson process on [0, t] × [0, 1] with rate r(θ) and number of points κ ∼ P o{r(θ)t}, which is (∆t,x,y) used in CJEA for simulating from Qθ . Let ω ∼ W(t,0,0),and I be the acceptance indicator which decides whether (ωs + (1 − s/t)x + (s/t)y, s ∈ [0, t]) is accepted as a path (∆t,x,y) from Qθ . Then, the conditional density of ω and Φ given {I = 1}, π(ω, Φ|θ, x, y, I = 1), is · ½ µ ¶ ¾ ¸ exp[t{1 − r(θ)}]r(θ)κ Yκ ψ ψ I φ ω + 1 − l x + l y; θ < υ , a(x, y, θ) ψl t t l l=1 with respect to the product measure Φ+ × W(t,0,0), where Φ+ is the measure of a unit rate Poisson process on [0, t] × [0, 1], and a(x, y, θ) is the acceptance probability of the CJEA1.

We derive now the calculations for the case where the CJEA1 proposes paths from D˜. For the case where the proposals come from F˜, the calculations are similar and we present only the ﬁnal result. It is very important, in this derivation, to keep track of all terms that depend on θ.

18 ˜ If the proposed measure in the CJEA1 is D, the output of the algorithm in [ti−1, ti] is µ µ ¶ µ ¶ ¶ s − ti−1 s − ti−1 Js + ωs + 1 − xi−1(θ) + (xi − Tti ) ∆ti ∆ti where Js is the sum of the jumps in [ti−1, ti] until time s and ωs ∼ BB(0, ti−1, 0, ti). The measure D˜ has two components: the jump process J and the Brownian bridge ˜ with measure W. J is a jump process with rate λi(θ) and jump size density fi(·; θ). NJ Therefore, we can write the measure of J as the product of measures JPP ⊗ Pf ; JPP is a Poisson process with rate λi(θ) and Pf is the probability measure of a random variable with density fi(·; θ). ∗ ∗ ∗ Using Lemma 2, we write the density of (J , ω , Φ |θ, xi−1(θ), xi(θ)) with respect to + NJ (t,0,0) + + the measure JPP ⊗L ⊗W ⊗Φ , where JPP is the measure of a unit Poisson process NJ on [0, t] and L is the NJ dimensional Lebesgue measure.

∗ ∗ ∗ π (J , ω , Φ |θ, xi−1(θ), xi(θ)) = (26) YNJ NJ κ exp {(∆ti)(1 − λi(θ))} [λi(θ)] fi(∆Jtj ) exp {(∆ti)(1 − ri(θ))} [ri(θ)] j=1 Yκ · µ µ ¶ µ ¶ ¶ ¸ ψl − ti−1 ψl − ti−1 1 φ Jψl + ωψl + 1 − xi−1(θ) + (xi(θ) − Jti < υl ∆ti ∆ti l=1 1

a (xi−1(θ), xi(θ), θ) ·µ ¶ ¸ α2 + α0 1 where r (θ) = sup (X ; θ) + λ(X , t; θ) − (F + F ) − l(θ), i 2 t t ∆t 1 2 ³ ti−1≤´t≤ti i α2+α0 1 l(θ) = infX (Xt; θ) + λ(Xt, t; θ) − (F1 + F2) and F1 and F2 are the ones deﬁned 2 ∆ti in (14). We now use the following decomposition: Yn ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ π (θ, J , ω , Φ |v) = π(θ|v)π (J , ω , Φ |θ, v) = π(θ|v) π (Ji , ωi , Φi |θ, v) (27) i=1 where the second equality is obtained by the Markov property. Furthermore, Yn 0 π(θ|v) ∝ π(θ) p˜ (x (θ), x (θ)) |η (v; θ)| −1 (28) ∆ti i−1 i v=η (xi(θ);θ) i=1

The next step is to ﬁnd an equality for the transition densityp ˜∆ti (xi−1(θ), xi(θ)). From (13), we have that dP˜ fN (xi − JT − xi−1) = G ∆ti ˜ dD p˜∆ti (xi−1(θ), xi(θ))

19 and taking the expectation with respect to measure D˜ on both sides, we get ³ ´ p˜ (x (θ), x (θ)) = E˜ Gf (x − J − x ) (29) ∆ti i−1 i D N∆ti i T i−1

Recalling that

½ NJ Z X Xtj G = exp A(xi(θ); θ) − A(xi−1(θ); θ) − α(u; θ)du j=1 Xtj − Z µ ¶ ¾ ti α2 + α0 = − (Xt; θ) + (λ(Xt; θ) − λi(θ))dt (30) ti−1 2 YNJ λ(Xt−; θ)fg(∆Xtj ; θ) = fN (xi − JT − xi−1) λ (θ)f (∆X ; θ) ∆ti j=1 i i tj and · ½ Z µ ¶ ti α2 + α0 a(x (θ), x (θ)) = E exp − (X ; θ) + λ(X ; θ) (31) i−1 i D˜ 2 t t ti−1 ¾¸ 1 − (F1 + F2) − l(θ)dt ∆ti we get the following identity

p˜∆ti (xi−1(θ), xi(θ)) = exp {A(xi(θ); θ) − A(xi−1(θ); θ) + ∆ti(λi(θ) − li(θ))} (32) 1 √ a(xi−1(θ), xi(θ)) 2π∆ti Replacing (32) in (28) and then (26) and (28) in (27), we obtain

π (θ, J ∗, ω∗, Φ∗|v) ∝ Yn · π(θ) exp {A(y(θ); θ) − A(x(θ); θ)} exp {∆ti(λi(θ) − li(θ))} i=1 YNJi κi NJi exp {∆ti(1 − ri(θ))} [ri(θ)] exp {∆ti(1 − λi(θ))} [λi(θ)] fi(∆Jitj ; θ) (33) j=1 Yκ · µ µ ¶ µ ¶ ¶ ¸ ψil − ti−1 ψil − ti−1 1 φ Jψil + ωψil + 1 − xi−1(θ) + (xi(θ) − Jti < υil ∆ti ∆ti l=1 ¸ 0 |η (v; θ)| −1 v=η (xi(θ);θ)

And integrating the υil’s out, we ﬁnally get

20 π (θ, S(J ∗, ω∗)|v) ∝ Yn · π(θ) exp {A(y(θ); θ) − A(x(θ); θ)} exp {∆ti(λi(θ) − li(θ))} i=1 YNJi κi NJi exp {∆ti(1 − ri(θ))} [ri(θ)] exp {∆ti(1 − λi(θ))} [λi(θ)] fi(∆Jitj ; θ) (34) j=1 Yκ · µ µ ¶ µ ¶ ¶¸ ψil − ti−1 ψil − ti−1 φ Jψil + ωψil + 1 − xi−1(θ) + (xi(θ) − Jti ∆ti ∆ti l=1 ¸ 0 |η (v; θ)| −1 v=η (xi(θ);θ)

For the case where the CJEA1 proposes from F˜ we have

π (θ, S(J ∗, ω∗)|v) ∝ Yn · π(θ) exp {A(y(θ); θ) − A(x(θ); θ)} exp {∆ti(λi(θ) − li(θ))} i=1 κi exp {∆ti(1 − ri(θ))} [ri(θ)] (36) YNJi YNJi S C C fi1(NJi; θ) fi2(Jtj ; θ) fi3(Xtj |Xtj−1 , θ) (37) j=1 j=1 Yκ · µ µ ¶ µ ¶ ¶¸ ψil − ti−1 ψil − ti−1 φ Jψil + ωψil + 1 − xi−1(θ) + (xi(θ) − Jti ∆ti ∆ti l=1 ¸ 0 |η (v; θ)| −1 v=η (xi(θ);θ)

The MCMC algorithm will alternate between updating S(J ∗, ω∗) and θ. The skeletons ∗ ∗ S(Ji , ωi ) are conditionally independent given θ and are simulated via CJEA1. In the case of θ, it may or not be possible to simulate directly from (34) or (36). If it is not possible, we use a Metropolis-Hastings step. The MCMC algorithm gives “for free” the posterior distribution of the jump process since this is simulated on the step where (S(J ∗, ω∗)|θ, v) is sampled. This allows to compute, for example, the posterior probability of having a jump in a certain time interval, or the posterior distribution of a particular jump, given that it occurs.

21 5 Simulated examples

In this Section we apply the MCMC method to estimate the parameters of the following jump-diﬀusion:

dXt = sin (Xt − δ) dt + dBt + dJt; X0 = x (38)

λ(Xt, t) = λ 2 g(Z,Xt) = Z,Z ∼ N(µ, σ ) for 0 ≤ δ < 2π. The parameter θ set is composed by four parameters {δ, λ, µ, σ2}. The proposals in the CJEA algorithm are drawn from a process with law D˜ like the one dP˜ described in Section 3.1, with jump rate e2λ, that assures that the ratio is bounded, dD˜ and the jump size distribution N(µ, σ2). The acceptance probability of the CJEA in a given time interval [ti−1, ti] is ½ Z ¾ ti 2 1 + sin (Xt − δ) + cos(Xt − δ) 1 a(xti−1 , xti , θ) = exp − − (F1 + F2)dt (39) ti−1 2 ∆ti where:½ ¾ P ¡ ¢ 1 2 NJ F1 = − (xti − J∆ti − xti−1 ) , F2 = −2NJ + j=1 cos(Xtj − δ) − cos(Xtj − − δ) . 2∆ti Suppose we have discrete observations of X in [0,T ] and XT = y. The full log- likelihood function of the parameter set is given by

XNJ £ ¤ l(θ, X) ∝ cos(x − δ) − cos(y − δ) + cos(Xtj − δ) − cos(Xtj − − δ) j=1 Z T 2 sin (Xt − δ) + cos(Xt − δ) − dt − T λ + NJ log(λ) (40) 0 2 N 1 XNJ ¡ ¢ − J log(σ2) − ∆X − µ 2 2 2σ2 tj j=1

And the full conditional distribution π (θ|S(J ∗, ω∗), v) is

π (θ|S(J ∗, ω∗), v) ∝ P n N π(θ) exp {cos(x − δ) − cos(y − δ) − T λ} λ i=1 Ji ( ) P n NJi n N X X 2 − i=1 Ji 1 (i) 2 (σ ) 2 exp − (∆J − µ) (41) 2σ2 tj i=1 j=1 ( ) " #κ(i) (i) · ¸ Xn Yn (i) (i) Yn Yκ 2 (i) 9 (F1 + F2 ) 1 − sin (ψil − δ) − cos(ψil − δ) exp F2 − 8 ∆ti 2 i=1 i=1 i=1 l=1

22 We assume the following prior distributions:

π(θ) = π(δ)π(λ)π(µ)π(σ2) δ ∼ U[0, 2π) λ ∼ Gamma(0.001, 0.001) µ ∼ N(0, 1000) σ2 ∼ IGamma(0.001, 0.001)

The jump-diﬀusion is simulated in the time interval [0, 1000] with δ = π, λ = 0.04, µ = 5, and σ2 = 2. Figure 2 shows the simulated skeleton.

Figure 2: Path simulated from the jump-diﬀusion X via CJEA.

The data set is composed of observations in every integer time and therefore has 1001 equally spaced observations. The results are quite good. For all the four parameters, the posterior modes are very close to the real values. However, it can be noticed that the match is not perfect. This rises a very important issue concerning inference for jump- diffusions; an issue about information and identifiability. Given two consecutive observations, the inference procedure has to consider how much of the difference between these two observations are due to the continuous part of the process and how much is due to possible jumps. Then, there are cases where the observations are not fine enough to provide enough information to distinguish between these two components of the process. For example, consider a jump-diffusion where the jump size distribution is a standard Normal distribution. In this case, it is likely to have small jumps and, depending on the model assumed for the continuous part, the data may not provide enough information to distinguish if the difference between two consecutive observations is due only to the continuous part or also due to a small jump. Another important issue is whether a jump-diffusion model should be used instead of a pure diffusion to fit the data. This gives rise to questions about the existence or not of jumps in the process. For example, Barndorff-Nielsen and Shephard (2006) propose a

23 Figure 3: Posterior distribution of the parameters. test for jumps in ﬁnancial economics based on a decomposition of the quadratic variation (QV). They decompose the QV in two parts, one corresponds to the continuous part of the process and the other one to the jumps.

6 Conclusions

In this paper, we proposed a new methodology for exact simulation of jump-diffusions called Conditional Jump Exact Algorithm. The algorithm involves no discretisation and simulates from the exact probability measure of the jump-diffusion. Based on such algorithm, a Bayesian method to perform inference for discretely observed jump-diffusions is proposed. The method consists of an MCMC algorithm that converges to the exact posterior distribution of the parameters due to the properties of CJEA. We also presented a simulated example where the proposed method is used to estimate parameters in a jump-diffusion. The method performed quite well and gave rise to some important issues concerning information and identifiability. The main feature of the method proposed here is the fact that it involves no discretisation of the process like other existing methods. On the other hand, we acknowledge that they can be considerably restrictive in the sense that they are only applied to univariate and time-homogeneous jump-diffusions. Extensions for both cases are possible directions

24 for future work.

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