## Bayesian Analysis

### Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach

#### Abstract

In this article we examine two relatively new MCMC methods which allow for Bayesian inference in diffusion models. First, the Monte Carlo within Metropolis (MCWM) algorithm (O'neil, et al. 2000) uses an importance sampling approximation for the likelihood and yields a Markov chain. Our simulation study shows that there exists a limiting stationary distribution that can be made arbitrarily close'' to the posterior distribution (MCWM is not a standard Metropolis-Hastings algorithm, however). The second method, described in Beaumont (2003) and generalized in Andrieu and Roberts (2009), introduces auxiliary variables and utilizes a standard Metropolis-Hastings algorithm on the enlarged space; this method preserves the original posterior distribution. When applied to diffusion models, this pseudo-marginal (PM) approach can be viewed as a generalization of the popular data augmentation schemes that sample jointly from the missing paths and the parameters of the diffusion volatility. The efficacy of the PM approach is demonstrated in a simulation study of the Cox-Ingersoll-Ross (CIR) and Heston models, and is applied to two well known datasets. Comparisons are made with the MCWM algorithm and the Golightly and Wilkinson (2006) approach.

#### Article information

Source
Bayesian Anal., Volume 6, Number 2 (2011), 231-258.

Dates
First available in Project Euclid: 13 June 2012

https://projecteuclid.org/euclid.ba/1339612045

Digital Object Identifier
doi:10.1214/11-BA608

Mathematical Reviews number (MathSciNet)
MR2806243

Zentralblatt MATH identifier
1330.60092

#### Citation

Stramer, Osnat; Bognar, Matthew. Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach. Bayesian Anal. 6 (2011), no. 2, 231--258. doi:10.1214/11-BA608. https://projecteuclid.org/euclid.ba/1339612045

#### References

• Aït-Sahalia, Y. (1999). "Transition Densities for Interest Rate and Other Nonlinear Diffusions." Journal of Finance, 54(4): 1361–1395.
• –- (2002). "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach." Econometrica, 70(1): 223–262.
• –- (2008). "Closed-Form Likelihood Expansions for Multivariate Diffusions." The Annals of Statistics, 36(2): 906–937.
• Aït-Sahalia, Y. and Kimmel, R. (2007). "Maximum Likelihood Estimation of Stochastic Volatility Models." Journal of Financial Economics, 83: 413–452.
• Andrieu, C., Berthelsen, K., Doucet, A., and Roberts, G. (2010). "Posterior sampling in the presence of unknown normalising constants: An adaptive pseudo-marginal approach." Technical report, University of Bristol.
• Andrieu, C. and Roberts, G. (2009). "The pseudo-marginal approach for efficient Monte" Carlo computations. The Annals of Statistics, 37: 697–725.
• Bally, V. and Talay, D. (1996). "The Law of the Euler Scheme for Stochastic Differential Equations. II": Convergence Rate of the Density (STMA V38 2092). Monte Carlo Methods and Applications, 2: 93–128.
• Beaumont, M. (2003). "Estimation of population growth or decline in genetically monitored populations." Genetics, 164: 1139–1160.
• Beskos, A., Papaspiliopoulos, O., and Roberts, G. O. (2009). "Monte Carlo maximum likelihood estimation for discretely observed diffusion processes." Annals of Statistics, 37: 223–245.
• Beskos, A., Papaspiliopoulos, O., Roberts, G. O., and Fearnhead, P. (2006). "Exact and Computationally Efficient Likelihood-based Estimation for Discretely Observed Diffusion Processes (with Discussion)." Journal of the Royal Statistical Society, Series B: Statistical Methodology, 68(3): 333–382.
• Chernov, M., Gallant, A. R., Ghysels, E., and Tauchen, G. (2003). "Alternative models for stock price dynamics." Journal of Econometrics, 116: 225–257.
• Chib, S., Pitt, M. K., and Shephard, N. (2006). "Likelihood Based Inference for Diffusion Driven Models." Technical report, Olin School of Business, Washington University, St Louis, Missouri.
• Cox, J. C., Ingersoll, J. E., and Ross, S. A. (1985). "A Theory of the Term Structure of Interest Rates." Econometrica, 53(2): 385–407.
• Di Pietro, M. (2001). "Bayesian Inference for Discretely Sampled Diffusion Processes with Financial Applications." Ph.D. Thesis, Department of Statistics, Carnegie-Mellon University.
• Durham, G. B. and Gallant, A. R. (2002). "Numerical Techniques for Maximum Likelihood Estimation of Continuous-time Diffusion Processes." Journal of Business & Economic Statistics, 20(3): 297–338.
• Elerian, O., Chib, S., and Shephard, N. (2001). "Likelihood Inference for Discretely Observed Nonlinear Diffusions." Econometrica, 69(4): 959–993.
• Eraker, B. (2001). "MCMC" Analysis of Diffusion Models with Application to Finance." Journal of Business & Economic Statistics, 19(2): 177–191.
• –- (2004). "Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices." Journal of Finance, 59(3): 1367–1404.
• Golightly, A. and Wilkinson, D. J. (2006). "Bayesian Sequential Inference for Nonlinear Multivariate Diffusions." Statistics and Computing, 16(4): 323–338.
• –- (2008). "Bayesian inference for nonlinear multivariate diffusion models observed with error." Computational Statistics and Data Analysis, 52(3): 1674–1693.
• Heston, S. (1993). "A closed-form solution for options with stochastic volatility with applications to bonds and currency options." Review of Financial Studies, 6: 327–343.
• Johannes, M. S., Polson, N. G., and Stroud, J. R. (2009). "Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices." Review of Financial Studies, 22(7): 2759–2799.
• Jones, C. S. (1999). "Bayesian estimation of continuous-time finance models." Unpublished paper, Simon School of Business. University of Rochester.
• –- (2003). "The dynamics of stochastic volatility: Evidence from underlying and options markets." Journal of Econometrics, 116: 181–224.
• Kalogeropoulos, K. (2007). "Likelihood-based inference for a class of multivariate diffusions with unobserved paths." Journal of Statistical Planning and Inference, 137(10): 3092–3102.
• Kalogeropoulos, K., Roberts, G., and Dellaportas, P. (2010). "Inference for Stochastic Volatility Models Using Time Change Transformations." Annals of Statistics, 38(2): 784–807.
• Lamoureux, C. G. and Paseka, A. (2005). "Information in Option Prices and the Underlying Asset Dynamics." Working paper, Eller School of Business, University of Arizona.
• Milstein, G. N., Schoenmakers, J. G., and Spokoiny, V. (2004). "Transition density estimation for stochastic differential equations via forward-reverse representations." Bernoulli, 10(2): 281–312.
• O'Neil, P. D., Balding, D. J., Becker, N. G., Serola, M., and Mollison, D. (2000). "Analyses of Infectious Disease Data from Household Outbreaks by Markov chain Monte Carlo Methods." Applied Statistics, 49: 517–542.
• Pasarica, C. and Gelman, A. (2010). "Adaptively Scaling the Metropolis Algorithm Using Expected Squared Jumped Distance." Statistica Sinica, 20: 343–364.
• Roberts, G. O. and Stramer, O. (2001). "On Inference for Partially Observed Nonlinear Diffusion Models Using the Metropolis-Hastings Algorithm." Biometrika, 88(3): 603–621.
• Sørensen, H. (2004). "Parametric Inference for Diffusion Processes Observed at Discrete Points in Time: a Survey." International Statistical Review, 72(3): 337–354.
• Stramer, O., Bognar, M., and Schneider, P. (2010). "Bayesian Inference of Discretely Sampled Markov Processes with Closed-Form Likelihood Expansions." The Journal of Financial Econometrics, 8: 450–480.
• Stramer, O. and Yan, J. (2007). "Asymptotics of an Efficient Monte Carlo Estimation for the Transition Density of Diffusion Processes." Methodology and Computing in Applied Probability, 9(4): 483–496.