Simulated Maximum Likelihood in Nonlinear Continuous-Discrete State Space Models: Importance Sampling by Approximate Smoothing

Summary

The likelihood function of a continuous-discrete state space model is computed recursively by Monte Carlo integration, using importance sampling techniques. A functional integral representation of the transition density is utilized and importance densities are obtained by smoothing. Examples are the likelihood surfaces of an AR(2) process, a Ginzburg-Landau model and stock price models with stochastic volatilities.

Appendix

In this appendix it is shown, that the EKF update of the a priori density (21) leads to the correct variance reduced Monte Carlo estimator of the likelihood

$${L_{i + 1}} = p({z_{i + 1}}{\rm{\vert}}{Z^i})\;\; = \;\;\int {p({z_{i + 1}}{\rm{\vert}}{y_{i + 1}})p({y_{i + 1}}{\rm{\vert}}{Z^i})d{y_{i + 1}}} $$

((62))

Using the Bayes formula this can be rewritten as

$$p({z_{i + 1}}{\rm{\vert}}{Z^i})\;\; = \;\;\int {p({z_{i + 1}}{\rm{\vert}}{\eta _{{J_i} - 1}})p({\eta _{{J_i} - 1}}, \ldots ,{\eta _0}{\rm{\vert}}{Z^i})d\eta } $$

where dη = dηJ_i−1 … dη₀ and for this the optimal variance reducing density is

$$\begin{array}{lll}{{p_{2,opt}}}&=& {p({z_{i + 1}}{\rm{\vert}}{\eta _{{J_{i - 1}}}})p(\eta {\rm{\vert}}{Z^i}){\rm{/}}p({z_{i + 1}}{\rm{\vert}}{Z^i})}\\ {}&= & {p(\eta {\rm{\vert}}{Z^{i + 1}})}\end{array}$$

Therefore the optimal estimator is

$$\begin{array}{lll} {\hat p({z_{i + 1}}{\rm{\vert}}{Z^i})} & = & {{N^{ - 1}}\sum\limits_n {p({z_{i + 1}}{\rm{\vert}}{\eta _{n,{J_i} - 1}}){{p(\eta {\rm{\vert}}{Z^i})} \over {p(\eta {\rm{\vert}}{Z^{i + 1}})}}} }\\{}&= & {{N^{ - 1}}\sum\limits_n {p({z_{i + 1}}{\rm{\vert}}{\eta _{n,{J_i} - 1}}){{p(\eta {\rm{\vert}}{Z^i})} \over {p(\eta {\rm{\vert}}{Z^{i + 1}})}}} }\\{}&= & {\sum\limits_n {p({z_{i + 1}}{\rm{\vert}}{\eta _{n,{J_i} - 1}}){\alpha _{n,i + 1{\rm{\vert}}i}}} }\end{array}$$

which is the same as inserting (21) into (62). It. remains to note that

$$\begin{array}{lll} {p({z_{i + 1}}{\rm{\vert}}{\eta _{J - 1}})}& \approx & {\phi ({z_{i + 1}};h({y_{i + 1{\rm{\vert}}i}},\;{t_{i + 1}}),{H_{i + 1}}{P_{i + 1{\rm{\vert}}i}}H_{i + 1}^\prime + {R_{i + 1}})}\\{\;\;\;\;\;\;\;\;\;\;\;{y_{i + 1{\rm{\vert}}i}}}& = & {{\eta _{{J_i} - 1}} + f({\eta _{{J_i} - 1}},{\tau _{{J_i} - 1}})\delta t}\\{\;\;\;\;\;\;\;\;\;\;{P_{i + 1{\rm{\vert}}i}}}& = & {\Omega ({\eta _{{J_i} - 1}},{\tau _{{J_i} - 1}})\delta t,}\end{array}$$

since

$$\begin{array}{lll} z_{i+1}&=& h(y_{i+1}, t_{i+1})+\epsilon_{i+1} \\ &\approx & h(\eta_{J_{i}-1}+f(\eta_{J_{i}-1}, \tau_{J_{i}-1}) \delta t+.\\ &&.g(\eta_{J_{i}-1}, \tau_{J_{i}-1}) \delta W(\tau_{J_{i}-1}), t_{i+1})+\epsilon_{i+1} \\ &\approx & h(\eta_{J_{i}-1}+f(\eta_{J_{i}-1}, \tau_{J_{i}-1}) \delta t, t_{i+1})+ \\ && H_{i+1} g(\eta_{J_{i}-1}, \tau_{J_{i}-1}) \delta W(\tau_{J_{i}-1})+\epsilon_{i+1} \\ H_{i+1}&=& h_{y}(\eta_{J_{i}-1}+f(\eta_{J_{i}-1}, \tau_{J_{i}-1}) \delta t, t_{i+1}). \end{array}$$