Exact Simulation and Bayesian Inference for Jump-Di®usion 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 dif- fusions, 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 engi- neering. However di®usions have continuous sample paths so may natural continu- ous time phenomena require more general classes of models. Jump-di®usions have considerable appeal as flexible families of stochastic models. Bayesian inference for jump-di®usion models motivates new methodological challenges, in particular requires the construction of novel simulation schemes for use within data augmen- tation algorithms and within discretely observed data. In this paper we propose a new methodology for exact simulation of jump-di®usion processes. Such method is based on the recently introduced Exact Algorithm for exact simulation of di®usions. We also propose a simulation-based method to make likelihood-based inference for discretely observed jump-di®usions in a Bayesian framework. Simulated examples are presented to illustrate the proposed methodology. Key Words: Jump-di®usion; exact simulation; Retrospective Rejection Sam- pling; Markov Chain Monte Carlo methods. 1 Introduction Di®usion processes are continuous time stochastic processes extensively used for modelling phenomena that evolves continuously in time. They are used in many scienti¯c 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 di®usion process is formally de¯ned as the solution of a stochastic di®erential equation (SDE) of the type: dVt = ¹(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 ¹(Vt; t) and σ(Vt; t) are called the drift and di®usion coe±cient, and de¯ne 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, di®usion processes are continuous processes. Although they are very e±cient 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-di®usion processes. Jump-di®usions are continuous time stochastic processes that may jump and in between the jumps behave as a di®usion 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 di®usion-type processes. For example, in ¯nance, a jump-di®usion may be useful to model stock prices where unexpected events may cause a jump in the price. The most common area where jump- di®usions are applied is ¯nancial economics (see Ball and Roma, 1993; Du±e et al., 2000a; Runggaldier, 2003; Eraker et al., 2003; Eraker, 2004; Johannes, 2004; Barndor®- 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-di®usion is the solution V := fVt : 0 · t · T g of the stochastic di®erential equation: dVt = ¹(Vt¡)dt + σ(Vt¡)dBt + dJt;V0 = v0 (2) where ¹; 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 de¯ned by two components: a Poisson process of rate ¸(Vt¡; t) that describes the jump times, and a function ¢(Vt¡; z) that de¯nes the + jump sizes, where z » fz(¢; t) and fz is a standard density function for every t 2 R . For any r 2 R, ¸(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 di®usion process with drift ¹ and di®usion coe±cient σ. In this paper we consider only the case where the continuous part of the process is time homogeneous, as de¯ned in (2). It is common to ¯nd applications of di®usions where the drift and/or the di®usion coe±cient depend on unknown parameters. Making inference for these parameters based on discrete observations of the di®usion 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-di®usion 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-di®usions than there is for di®usions. Existing methods in the literature are based on path discretisation and data augmen- tation 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 ¯lters and MCMC. For example, Johannes et al. (2002) propose an algorithm which combines data augmentation and particle ¯ltering. 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 pa- rameters in jump-di®usion models with stochastic volatility and allow for jumps in both returns and volatility. The models considered in the paper are introduced in Du±e 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 ¯ltering 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 ¯ltering (APF) algorithm, re- spectively. Euler scheme is used to augment the data and approximate the likelihood in these methods. Golightly (2009) also proposes particle ¯ltering algorithms for sequential estimation and uses MCMC inside the particle ¯lter. 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), Honor¶e(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; Barndor®-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-di®usions assume that the jump times follow a Bernoulli distribution with parameter ¸¢ in a time interval of length ¢, where ¸ 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 algo- rithms for simulation of di®usions (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-di®usions is al- ready available (see Casella and Roberts, 2008) to simulate jump-di®usions unconditional on the ending point, simulation-based inference methods are more e±cient 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.