Scalar Filtering of a Constant Hidden Signal

A Constant Signal Observed in Brownian Noise

Suppose an unknown signal is constant in time, but its value is not known exactly. We observe that signal continuously, with independent Brownian noise added to the observation. Each observation increment contains information about the constant, so uncertainty decreases over time. In this special case, the Kalman–Bucy equations can be solved explicitly: the estimation variance and gain are deterministic, and the conditional mean has a closed form involving the observation at the current time.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a random variable \(X\) and a standard Brownian motion \(B_t\) independent of \(X\). Assume \(X\) is measurable at time zero and has distribution \(N(m_0,P_0)\), where \(m_0\in\mathbb{R}\) and \(P_0\ge0\). The signal is \(X_t=X\) for every \(t\). For a fixed observation-noise variance \(R>0\), the observation process is

The integrand \(1\) is adapted, progressively measurable and belongs to \(\mathcal{L}^2\) on every finite time interval. The observation filtration \(\mathcal{F}_t^Y\) is the completed filtration generated by \((Y_s)_{0\le s\le t}\). Filtering means finding the conditional mean \(\mathbb{E}[X\mid\mathcal{F}_t^Y]\) and conditional variance of \(X\) given \(\mathcal{F}_t^Y\). Here \(P_t\) will denote that estimation variance; it is not the probability measure \(\mathbb{P}\).

Why These Formulas Give the Conditional Distribution

For \(P_0>0\), the pair consisting of \(X\) and any finite collection of observations \(Y_{s_1},\ldots,Y_{s_n}\) is jointly Gaussian. Indeed, each observation is a linear combination of \(X\) and Brownian values. To identify the conditional mean and variance from the full observation history, set \(a_t=P_0/(R+P_0t)\) and define the candidate residual \(E_t=X-m_0-a_t(Y_t-m_0t)\). We verify that it is independent of all observations up to time \(t\).

Proof. For \(0\le s\le t\), independence of \(X-m_0\) and \(B\), together with \(\mathbb{E}[(X-m_0)^2]=P_0\) and \(\mathbb{E}[B_tB_s]=s\), gives \(\operatorname{Cov}(X,Y_s)=P_0s\) and \(\operatorname{Cov}(Y_t,Y_s)=P_0ts+Rs=s(P_0t+R)\). Therefore \(\operatorname{Cov}(E_t,Y_s)=P_0s-a_t s(P_0t+R)=0\). For any finite set of observation times, \(E_t\) and the observations at those times are jointly Gaussian. Zero cross-covariances imply their independence. Brownian paths and \(Y\) are continuous, so observations at a countable dense set of times generate the same completed observation sigma-field as all observations up to \(t\). Thus \(E_t\) is independent of \(\mathcal{F}_t^Y\), and \(\mathbb{E}[E_t\mid\mathcal{F}_t^Y]=0\). This proves that \(m_0+a_t(Y_t-m_0t)\) is the conditional mean.

For the variance, the same Gaussian conditioning calculation gives \(\operatorname{Var}(X\mid Y_t)=P_0-\operatorname{Cov}(X,Y_t)^2/\operatorname{Var}(Y_t)\). Here \(\operatorname{Cov}(X,Y_t)=P_0t\) and \(\operatorname{Var}(Y_t)=P_0t^2+Rt=t(P_0t+R)\) for \(t>0\). Hence the conditional variance is \(P_0-P_0^2t/(P_0t+R)=P_0R/(P_0t+R)\). The formula at \(t=0\) is \(P_0\), as required. Finally, \(m_0+a_t(Y_t-m_0t)=(Rm_0+P_0Y_t)/(R+P_0t)\). All the claimed conditional formulas follow. \(\square\)

This argument also shows why the entire observation history can be summarized by \(Y_t\) in this particular model: after subtracting the proposed mean, the residual is independent of every observation up to \(t\). This sufficiency property depends on the signal being constant and the observation noise having constant variance rate \(R\).

Solving the Riccati Equation and Mean Equation

The Riccati equation has no signal process-noise term because the signal does not move. Its only uncertainty reduction comes from observing the signal through noise. With \(P_t\) denoting conditional variance and \(R\) the observation-noise variance rate, the scalar equation is \(\dot P_t=-P_t^2/R\). The following computation verifies its explicit solution and uniqueness among nonnegative differentiable solutions.

Proof. The candidate \(P_t=P_0R/(R+P_0t)\) has \(P_{t=0}=P_0\) and derivative \(\dot P_t=-P_0^2R/(R+P_0t)^2=-P_t^2/R\). To check uniqueness, let \(P\) and \(Q\) be two nonnegative solutions with the same initial value. Their difference \(D=P-Q\) satisfies \(\dot D=-(P+Q)D/R\). Multiplying by \(\exp(\int_0^t(P_s+Q_s)/R\,ds)\) gives a function with derivative zero and initial value zero. Thus \(D=0\) throughout their common interval of existence. This includes the zero initial condition, for which the formula gives \(P_t=0\). \(\square\)

The gain is the variance divided by the observation-noise variance rate: \(K_t=P_t/R=P_0/(R+P_0t)\). To verify the mean equation, write the closed form as \(m_t=c(t)Y_t+d(t)\), where \(c(t)=P_0/(R+P_0t)\) and \(d(t)=Rm_0/(R+P_0t)\). Since these coefficients are deterministic and differentiable, the product rule with the semimartingale \(Y\) gives \(dm_t=c(t)dY_t+(c'(t)Y_t+d'(t))dt\); no second-order correction is needed for multiplication by a deterministic finite-variation function. As \(c'=-cK\) and \(d'=-Kd\), the drift term is \(-K_tm_tdt\). Therefore \(dm_t=K_t(dY_t-m_tdt)\), as asserted.

The integrals in this equation are well-defined on each finite interval. In particular, \(K_t\) and \(K_tm_t\) are adapted and progressively measurable; \(K_t\) is bounded by \(P_0/R\), while \(m_t\) has finite second moments on finite time intervals because it is an affine function of the Gaussian variable \(Y_t\). Thus the relevant Brownian integrands are in \(\mathcal{L}^2\). The equation may also be written using the innovations process \(I_t=R^{-1/2}(Y_t-\int_0^t m_s\,ds)\) as \(dm_t=\sqrt{R}K_t\,dI_t\). In the observation filtration, \(I_t\) is Brownian by the innovations result; this is the standard mean equation for the Kalman–Bucy filter.

How Observation Time Changes the Estimate

When \(P_0>0\), the variance \(P_t=P_0R/(R+P_0t)\) decreases as \(t\) increases and tends to zero as \(t\) tends to infinity. The gain also decreases: the data have already reduced uncertainty, so the filter puts less weight on new observations. This is not a steady-state gain; it continues to change with time and tends to zero. If \(P_0=0\), the signal is already known, so observations do not change the estimate.

The closed form can also be read through precision. At time \(t\), the prior contributes precision \(1/P_0\), and the observation history contributes \(t/R\). The corresponding posterior mean combines prior information and the total observation \(Y_t\): \[ \frac{m_0/P_0+Y_t/R}{1/P_0+t/R} =\frac{Rm_0+P_0Y_t}{R+P_0t}. \] This interpretation is valid for \(P_0>0\); the degenerate case \(P_0=0\) is handled separately rather than by dividing by zero.

The distinction between the signal and observation noise matters. The hidden signal has zero drift and zero diffusion because \(X_t=X\) is constant. The Brownian term \(\sqrt R\,dB_t\) belongs to the observation process, not to the signal dynamics. Accordingly, it is the observation-noise variance rate \(R\) that appears in the gain \(P_t/R\).

Common Mistake: Holding the Gain Fixed

A tempting but incorrect shortcut is to use the initial gain \(P_0/R\) at every time. For \(P_0=R=1\), that would use gain \(1\) forever. But the Riccati solution gives \(P_t=1/(1+t)\), so \(K_t=1/(1+t)\); at \(t=1\), the correct gain is \(1/2\), not \(1\). For example, with \(m_0=0\) and observation value \(Y_1=1\), the correct conditional mean is \(1/2\). Treating the gain as fixed and using the initial-value update \(m_0+(P_0/R)(Y_1-m_0)\) would give \(1\). The error is ignoring that each part of the observation history has already reduced the remaining variance.

The exact formulas here rely on a Gaussian prior, an independent Brownian observation noise, and a constant signal. Under those assumptions, the filter is completely explicit and its conditional distribution is determined by a mean and variance. When the signal itself evolves, its dynamics add terms to the variance equation; the present formulas should not be transferred to that setting without re-deriving the filter.