### 1.1 Motivation

The fundamental dynamical laws of physics, both classical and quantum mechanical, are described in terms of variables continuous in time. The continuous nature of the dynamical variables has been verified at all length scales probed so far, even though the relevant dynamical variables, and the fundamental laws of physics, are very different in the microscopic and macroscopic realms. In practical situations, one often deals with macroscopic objects whose state is phenomenologically well-described by classical deterministic laws modified by external disturbances that can be modelled as random noise, or Langevin equations. Even when there is no underlying fundamental dynamical law, the Langevin equation provides an effective description of the state variables in many applications. It is therefore natural to consider the problem of the evolution of a state of a signal of interest described by a Langevin equation called the state process.

When the state model noise is Gaussian (or more generally multiplicatively Gaussian) the state process is a Markov process. Since the process is stochastic, the state process is completely characterized by a probability density function. The Fokker-Planck-Kolmogorov foward equation (FPKfe) describes the evolution of this probability density function (or equivalently, the transition probability density function) and is the complete solution of the state evolution problem.

However, in many applications the signal, or state variables, cannot be directly observed. Instead, what is measured is a nonlinearly related stochastic process called the measurement process. The measurement process can often be modelled by yet another continuous stochastic dynamical system called the measurement model. In other words, the observations, or measurements, are discrete-time samples drawn from a different Langevin equation called the measurement process.

The *conditional probability density function* of the state variables, given the observations, is the complete solution of the filtering problem. This is because it contains all the probabilistic information about the state process that is in the measurements and in the initial condition [1]. This is the Bayesian approach, i.e., the *a priori* initial data about the signal process contained in the initial probability distribution of the state is incorporated into the solution. Given the conditional probability density, optimality may be defined under various criterion. Usually, the conditional mean, which is the least mean-squares estimate, is studied due to its richness in results and mathematical elegance. The solution of the optimal nonlinear filtering problem is termed universal, if the initial distribution can be arbitrary.

### 1.2 Fundamental Sochastic Filtering Results

When the state and measurement processes are linear, the linear filtering problem was solved by Kalman and Bucy [2, 3]. The celebrated Kalman filter has been successfully applied to a large number of problems in many different areas.

Nevertheless, the Kalman filter suffers from some major limitations. The Kalman filter is not optimal even for the linear filtering case if the initial distribution is not Gaussian. It may still be optimal for a linear system under certain criteria, such as minimum mean square error, but not a general criterion. In other words, the Kalman filter is not a universal optimal filter, even when the filtering problem is linear. Secondly, the Kalman filter cannot be an optimal solution for the general nonlinear filtering problem since it assumes that the signal and measurement models are linear. The extended Kalman filter (EKF), obtained by applying the Kalman filter to a linearized model, cannot be a reliable solution, in general. Thirdly, even when the EKF estimates the state well in some cases, it gives no reliable indication of the accuracy of the state estimate, i.e., the conditional variance is unreliable. Finally, the Kalman filter assumes that the conditional probability distribution is Gaussian, which is a very restrictive assumption; for instance, it rules out the possibility of a multi-modal conditional probability distribution.

The continuous-continous nonlinear filtering problem (i.e., continuous-time state and measurement stochastic processes) was studied in [4–6] and [7]. This led to a stochastic differential equation, called the Kushner equation, for the conditional probability density in the continuous-continuous filtering problem. It was noted in [8, 9], and [10] that the stochastic differential equation satisfied by the unnormalized conditional probability density, called the Duncan-Mortensen-Zakai (DMZ) equation, is linear and hence considerably simpler than the Kushner equation. The robust DMZ equation, a partial differential equation (PDE) that follows from the DMZ equation via a gauge transformation, was derived in [11] and [12].

A disadvantage of the robust DMZ equation is that the coefficients depend on the measurements. Thus, one does not know the PDE to solve prior to the measurements. As a result, real-time solution is impossible. A fundamental advance was made in tackling the general nonlinear filtering problem by S-T. Yau and Stephen Yau. In [13], it was proved that the robust DMZ equation is equivalent to a partial differential equation that is independent of the measurements, which is referred to as the Yau Equation (YYe) in this paper. Specifically, the measurements only enter as initial condition at each measurement step. Thus, no on-line solution of a PDE is needed; all PDE computations can be done off-line.

However, numerical solution of partial differential equations presents several challenges. A naïve discretization may not be convergent, i.e., the approximation error may not vanish as the grid size is reduced. Alternatively, when the discretization spacing is decreased, it may tend to a different equation, i.e., be inconsistent. Furthermore, the numerical method may be unstable. Finally, since the solution of the YYe is a probability density, it must be positive which may not be guaranteed by the discretization.

A different approach to solving the PDE was taken in [14] and [15]. An explicit expression for the fundamental solution of the YYe as an ordinary integral was derived. It was shown that the formal solution to the YYe may be written down as an ordinary, but somewhat complicated, multi-dimensional integral, with an infinite series as the integrand. In addition, an estimate of the time needed for the solution to converge to the true solution was presented.

### 1.3 Outline of the Paper

In this paper, the (Euclidean) Feynman path integral (FPI) formulation is employed to tackle the continuous-continuous nonlinear filtering problem. Specifically, phrasing the stochastic filtering problem in a language common in physics, the solution of the stochastic filtering problem is presented. In particular, no other result in filtering theory (such as the DMZ equation, the robust DMZ equation, etc.) is used. The path integral formulation leads to a path integral formula for the transition probability density for the general additive noise case. A corollary of the FPI formalism is the path integral formula for the fundamental solution of the YYe and the Yau algorithm – a fundamental result of nonlinear filtering theory. It is noted that this paper provides a detailed derivation of results that were used in [16].

The following point needs to be emphasized to readers familiar with the discussion of standard filtering theory – the FPI is different from the Feynman-Kǎc path integral. In filtering theory literature, it is the Feynman-Kǎc formalism that is often used. The Feynman-Kǎc formulation is a rigorous formulation and has led to several rigorous results in filtering theory. However, in spite of considerable effort it has not been proven to be directly useful in the development of reliable practical algorithms with desirable numerical properties. It also obscures the physics of the problem.

In contrast, it is shown that the FPI leads to formulas that are eminently suitable for numerical implementation. It also provides a simple and clear physical picture. Many path integral manipulations have no counterpart in the Feynman-Kǎc approach. Finally, the theoretical insights provided by the FPI are highly valuable, as evidenced by numerous examples in modern theoretical physics (see, for instance, [17]), and shall be employed in subsequent papers.

The outline of this paper is as follows. In the following section, the filtering problem is reformulated in a language common in physics. In Section 3, the path integral formula for the transition probability density is derived for the general additive noise case. The Yau algorithm is then derived in the following section. In Sections 5 and 6 some conceptual remarks and numerical examples are presented. The conclusions are presented in Section 7. In Appendix 1, aspects of continuous-continuous filtering are reviewed.

For more details on the path integral methods, see any modern text on quantum field theory, such as [17], and especially [18] which discusses application of FPI to the study of stochastic processes.