Filtering of a Brownian motion with respect to its local time ^{∗}

Giovanna Nappo ^{†}  Barbara Torti ^{‡}

Abstract We consider a filtering problem when the state process is a Brownian motion W_{t} and the observation process is its local time Λ_{s}, for s ≤ t. For this model we derive an approximation scheme based on a suitable interpolation of the observation process Λ_{t}. The convergence of the approximating filter to the original one combined with an explicit construction of the approximating filter allows us to derive the explicit form of the original filter. Some connections with the Azéma martingale are discussed.

MSC: 60J55; 60J65; 60F99; 60G35; 93E11; 60G55

Keywords: Brownian motion; Local time; Nonlinear filtering; Approximation; Counting processes; Reflection principle

1 Introduction

Let (X_{t}, Y_{t}) be a stochastic system. We assume that the state process X_{t} of the system cannot be directly observed, while the other component Y_{t} is completely observable and is referred to as the observation process. The aim of stochastic filtering is the computation of the conditional law of the state process at time t, given the observation process up to time t. Equivalently the aim is the computation of

E[g(X_{t})/F_{t}^{Y} ],

for all functions g belonging to a determining class, i.e. the best estimate of g(X_{t}) given the σ−algebra of the observations up to time t, F_{t}^{Y} = σ{Y_{s}, s ≤ t}.

The classical situation arises when the observation process Y_{t} is a noisy function of the state X_{t}, and when the noise can be represented via martingales. The typical example is

Y_{t} = ∫_{0}^{t} h(X_{s})ds + B_{t},

with B_{t} a Brownian motion, and is a generalization of the Kalman filter. In this case the problem can be solved either by the “innovation” method or by the “reference probability” method to nonlinear filtering (see, for example [5], [11]). These methods are based on martingale representation properties and allow to characterize the conditional distribution of X_{t} given F_{t}^{Y} as the solution of a system of equations, usually called the filter equation. Another situation of interest is the so-called singular filtering. It arises when the observation is a deterministic functional of the state. A first example arises when

Y_{t} = ∫_{0}^{t} h(X_{s})ds,

∗Partially supported by MURST project ”Processi Stocastici, Modelli Stocastici ed Applicazioni”

†Dipartimento di Matematica - Università di Roma ”La Sapienza”. Italy

‡Dipartimento di Matematica - Università di Roma ”Tor Vergata”, Italy

or more generally when Y_{t} is a functional of the state X up to time t and is absolutely continuous in time. A second example arises when the observation Y_{t} is a function of the state X at a fixed time t, i.e. when

Y_{t} = h(X_{t}).

In these cases the classical methods cannot be applied because the necessary martingale representation properties are not satisfied. There is not yet a general theory to solve these kind of problems. On the other hand much work has been done in order to give good approximations both for the first kind of example ([13] and references therein) and the second one ([9], [10]). In this paper we deal with the problem of filtering a reflected Brownian motion W_{t} with respect to its local time Λ_{t}, in the sense of the Skorohod problem (we recall the exact definition in §2). This kind of problem naturally comes into the frame of singular filtering since it is a deterministic functional of whole history of the state W up to time t as in the first kind of example, but with the non trivial difference that in our model Λ_{t} is singular w.r.t. the Lebesgue measure. Therefore, to our knowledge, the techniques in the frame of singular filtering do not apply.

This model is interesting since it can be viewed as the diffusive approximation, in heavy traffic conditions, of a queueing system, see, for example, [7], [8], and, more recently [3]. To be more precise, suppose we can observe, up to time t, whether a queue is busy or idle, but we cannot observe the size of the queue. Then the problem is to evaluate the size of a queue at time t, given this information. In the setup of heavy traffic limits this means to compute the conditional law of a reflected Brownian motion W_{t} + Λ_{t} when the observation process is its local time Λ_{t}. Formally our aim is the computation of E[g(W_{t} + Λ_{t})/F_{t}^{Λ}]. One can equivalently compute

π_{t}(g) = E[g(W_{t})/F_{t}^{Λ}],  (1)

since Λ_{t} is F_{t}^{Λ}-measurable.

In this paper we provide an approximation of π_{t}(g), and, by a limiting procedure, we also derive an explicit expression for π_{t}(g).

Our approach consists in constructing an L^{1}-approximation for π_{t}(g) where g ∈ C_{b}(ℝ), with compact support by conditioning to F_{t}^{Λ^{n}}, where Λ_{t}^{n} is a suitable approximation for the observation Λ_{t}. The approximating sequence Λ_{t}^{n} has the following properties.

- The process Λ_{t}^{n} is proportional to a counting process and therefore we can use the techniques of nonlinear filtering with counting observations (see, for instance [4]).

- The sequence of processes Λ_{t}^{n} converges to Λ_{t} for all ω in the space D_{ℝ}[0, ∞) w.r.t. the topology of the uniform convergence.

- The information F_{t}^{Λ^{n}} carried by Λ^{n} up to time t is increasing in n and strictly contained in the information F_{t}^{Λ} carried by Λ up to time t. Moreover it can be uniquely determined just looking at the observation process Λ_{t} on a suitable sequence of stopping times.

- It is possible to construct an explicit version for the approximating filter

π_{t}^{n}(g) = E[g(W_{t})/F_{t}^{Λ^{n}}].  (2)

(see Theorem 3.3) converging P -a.s. and in L^{1} to the original one (see Theorem 4.4).

The paper is organized as follows. In Section 2 we construct the approximating model and we show the convergence result for the filters. In Section 3 we study the approximating filter π_{t}^{n}(g), and in particular in Theorem 3.3 we provide its explicit expression

π_{t}^{n}(g) = \frac{1}{2Φ (\frac{1}{2^{n}\sqrt{s}}) − 1} ∫_{− \frac{1}{2^{n}}}^{+∞} g(−\frac{j}{2^{n}} + x) [\frac{e^{− \frac{x^{2}}{2s}}}{\sqrt{2πs}} − \frac{e^{− \frac{( \frac{2}{2^{n}} +x)^{2}}{2s}}}{\sqrt{2πs}} ] dx|_{s=t−σ_{j}^{n}},

when σ_{j}^{n} < t < σ_{j+1}^{n}, where {σ_{j}^{n}, j ∈ ℕ} are the jump times of Λ^{n}, and where Φ is the distribution function of a standard gaussian random variable.

In Section 4 we identify the original filter π_{t}(g) as the limit of the previous approximation. In Theorem 4.4 we get its explicit expression

E[g(W_{t})/F_{t}^{Λ}] = ∫_{0}^{∞} g (−l + y\sqrt{s}) ye^{\frac{−y^{2}}{2}} dy|_{l=Λ_{t},s=ζ_{t}},

where ζ_{t} is the elapsed time from the last visit to 0 for the process W_{t} + Λ_{t}, that is

ζ_{t} = t − sup{u < t s.t. W_{u} + Λ_{u} = 0},  (3)

which is F_{t}^{Λ}−measurable, since

sup{u < t s.t. W_{u} + Λ_{u} = 0} = sup{u < t s.t. Λ_{u} < Λ_{t}}.

As a consequence, we get immediately that

E[g(W_{t} + Λ_{t})/F_{t}^{Λ}] = ∫_{0}^{∞} g (y\sqrt{s}) ye^{\frac{−y^{2}}{2}} dy|_{s=ζ_{t}}.

The previous steps can be extended to a general Brownian motion with drift coefficient c and diffusion coefficient a^{2}. In this case the approximating filters and the exact filter are modifications of the corresponding filters for the standard Brownian motion. We obtain them starting from the standard Brownian motion case, via a deterministic time change, Girsanov Theorem and Kallianpur Striebel formula (see Section 6). Moreover, in Section 5, we point out some connections with the Azéma’s martingale. Starting from these connections we provide another approach for the computation of E[g(W_{t} + Λ_{t})/F_{t}^{Λ}].

2 Approximation results

Let (Ω, F, P) be a probability space and let W be a standard Brownian motion defined on it. The local time Λ_{t} at level 0 of the process W_{t}, in the sense of solution of the Skorohod problem (see, e.g., [12]) is defined by

Λ_{t} = ℓ_{t}(W),

where ℓ is the deterministic functional

ℓ : D_{ℝ}[0, +∞) → D_{ℝ}[0, +∞);  ℓ_{t}(x) = − inf_{0≤s≤t} (x(s) ∧ 0) = − inf_{0≤s≤t} (x(s)) ∧ 0. (4)

Remark 2.1. It is easy to see that ℓ is a continuous function w.r.t. the topology of the uniform convergence on compact sets.

Observe that W_{0} = 0, so that

Λ_{t} = − inf_{0≤s≤t} (W_{s}).  (5)

For each n ∈ ℕ consider the sequence of stopping times σ_{j}^{n}, j ≥ 0 defined as the first time the process Λ_{t} crosses the threshold \frac{j}{2^{n}}, that is

σ_{0}^{n} = 0, σ_{j}^{n} = inf \{t s.t. Λ_{t} ≥ \frac{j}{2^{n}} \},  (6)

or, equivalently, as the first time the process W_{t} crosses the threshold −\frac{j}{2^{n}}, that is

σ_{0}^{n} = 0, σ_{j}^{n} = inf \{t s.t. W_{t} ≤ −\frac{j}{2^{n}} \}.  (7)

The approximating observation process Λ^{n} is the D_{ℝ}[0, +∞)-valued process

Λ_{t}^{n} = ∑_{j=0}^{∞} \frac{j}{2^{n}} 𝕀(σ_{j}^{n} ≤ t < σ_{j+1}^{n}) = \frac{1}{2^{n}} ∑_{j=0}^{∞} 𝕀(σ_{j}^{n} ≤ t),  (8)

where 𝕀(A) denotes the indicator function of A.

Lemma 2.2. Let Λ^{n} be defined as in (8). Then

Λ^{n} →_{n→∞} Λ a.s.

w.r.t. the topology of the uniform convergence.

Proof. The local time of a Brownian motion is continuous and nondecreasing almost surely. The continuity property implies Λ_{σ_{j+1}^{n}} = \frac{j+1}{2^{n}} = Λ_{σ_{j}^{n}} + \frac{1}{2^{n}} and therefore

Λ_{t}^{n} = Λ_{σ_{j}^{n}}^{n} = Λ_{σ_{j}^{n}}, for any t ∈ [σ_{j}^{n}, σ_{j+1}^{n}).

The nondecreasing property implies

Λ_{σ_{j}^{n}} ≤ Λ_{t} ≤ Λ_{σ_{j+1}^{n}} = Λ_{σ_{j}^{n}} + \frac{1}{2^{n}} , for any t ∈ [σ_{j}^{n}, σ_{j+1}^{n}).

Then, almost surely,

Λ_{t}^{n} ≤ Λ_{t} ≤ Λ_{t}^{n} + \frac{1}{2^{n}} , for any t ≥ 0,  (9)

that is the thesis.

For notational convenience from now on we set

G_{t}^{n} = F_{t}^{Λ^{n}} = σ(Λ_{s}^{n}, s ≤ t).

Moreover we introduce G_{t}^{∞} = ⋁_{n} G_{t}^{n}. The following theorem is the most important tool for showing the convergence result.

Lemma 2.3. Let t ∈ ℝ^{+}. Then {G_{t}^{n}, n ∈ ℕ} is an increasing sequence of σ-algebras such that G_{t}^{∞} = F_{t}^{Λ}.

Proof. The choice of the thresholds \frac{j}{2^{n}} yields Λ_{t}^{n} to be G_{t}^{n+1}-measurable for each n ∈ ℕ. In fact (6)( or (7) ) easily provides σ_{2j}^{n+1} = σ_{j}^{n}, for each j ∈ ℕ, and this, together with (8), implies Λ_{t}^{n} to be G_{t}^{n+1}-measurable for each n ∈ ℕ. Therefore G_{t}^{n} is an increasing sequence of σ-algebras. To complete the proof we show successively the inclusions G_{t}^{∞} ⊇ F_{t}^{Λ} and G_{t}^{∞} ⊆ F_{t}^{Λ}.

To prove the first inclusion we note that, for each m ∈ ℕ, for each choice of t_{1} ≤ t_{2} ≤ ... ≤ t_{m} ≤ t, and for each ω ∈ Ω it occurs (Λ_{t_{1}}^{n}, ..., Λ_{t_{m}}^{n}) → (Λ_{t_{1}}, ..., Λ_{t_{m}}) so that (Λ_{t_{1}}, ..., Λ_{t_{m}}) ∈ G_{t}^{∞}.

On the other hand Λ_{t}^{n} is F_{t}^{Λ}-measurable for each n ∈ ℕ. Indeed by (6) it is trivial to see that {σ_{j}^{n}, j ∈ ℕ} is a sequence of F_{t}^{Λ}-stopping times for each n ∈ ℕ. Then, by (8), G_{t}^{n} ⊆ F_{t}^{Λ} for each n ∈ ℕ, and the second inclusion is achieved.

Theorem 2.4. Let g be a bounded measurable function. Then

π_{t}^{n}(g) → π_{t}(g) a.s. and in L^{1}  (10)

Proof. Theorem 2.3 guarantees that π_{t}(g) = E[g(W_{t})/F_{t}^{Λ}] = E[g(W_{t})/G_{t}^{∞}]. Moreover, for each t ∈ ℝ^{+} and g ∈ C_{b}(ℝ), π_{t}^{n}(g) coincides with M^{n} = E[g(W_{t})/G_{t}^{n}]. The sequence {M_{n}, n ≥ 1} is a bounded uniformly integrable G_{t}^{n}-martingale in discrete time and then

M_{n} → M_{∞} = E[g(W_{t})/G_{t}^{∞}] a.s. and in L^{1},

that is (10).

3 The approximating filter

In order to derive the explicit expression of

π_{t}^{n}(g) = E[g(W_{t})/G_{t}^{n}]

we need the following preliminary result.

Proposition 3.1. The conditional law of W_{t} given G_{t}^{n} admits the following representation P -a.s.

E[g(W_{t})/G_{t}^{n}] = ∑_{j=0}^{∞} \frac{E[g (−\frac{j}{2^{n}} + W_{s+σ_{j}^{n}} − W_{σ_{j}^{n}}) 𝕀 (S_{j+1}^{n} > s) ]_{s=t−σ_{j}^{n}}}{E[𝕀(S_{j+1}^{n} > s)]_{s=t−σ_{j}^{n}}} 𝕀\{σ_{j}^{n} ≤ t < σ_{j+1}^{n}\}, (11)

where {σ_{j}^{n}, j ∈ ℕ} are defined by (7), and {S_{j}^{n} = σ_{j}^{n} − σ_{j−1}^{n}, j ∈ ℕ}. Moreover, let W_{s}^{∗} be a standard Brownian motion, let Λ_{s}^{∗} be its local time at level 0, then

E[g(W_{t})/G_{t}^{n}] = ∑_{j=0}^{∞} E[g(−\frac{j}{2^{n}} + W_{s}^{∗})/Λ_{s}^{∗} < \frac{1}{2^{n}} ]_{s=t−σ_{j}^{n}}𝕀\{σ_{j}^{n} ≤ t < σ_{j+1}^{n}\}.  (12)

Proof. Observe that

E[g(W_{t})/G_{t}^{n}] = ∑_{j=0}^{∞} E[g(W_{t})/G_{t}^{n}]𝕀{σ_{j}^{n} ≤ t < σ_{j+1}^{n}},

since {{σ_{j}^{n} ≤ t < σ_{j+1}^{n}}, j ∈ ℕ} is a partition.

Moreover (see [4] chapter III T5)

G_{t}^{n} ∩ {σ_{j}^{n} ≤ t < σ_{j+1}^{n}} = σ{σ_{0}^{n}, ..., σ_{j}^{n}} ∩ {σ_{j}^{n} ≤ t < σ_{j+1}^{n}} = G_{σ_{j}^{n}}^{n} ∩ {σ_{j}^{n} ≤ t < σ_{j+1}^{n}}

so that, for any G_{t}^{n}-measurable random variable Y, there exists a G_{σ_{j}^{n}}^{n}-measurable random variable Y′ s.t.

Y 𝕀{σ_{j}^{n} ≤ t < σ_{j+1}^{n}} = Y′𝕀{σ_{j}^{n} ≤ t < σ_{j+1}^{n}}.

In particular

E[g(W_{t})/G_{t}^{n}]𝕀{σ_{j}^{n} ≤ t < σ_{j+1}^{n}} = Y′𝕀{σ_{j}^{n} ≤ t < σ_{j+1}^{n}}

and, by conditioning both sides on G_{σ_{j}^{n}}^{n},

E[E[g(W_{t})/G_{t}^{n}]𝕀{σ_{j}^{n} ≤ t < σ_{j+1}^{n}}/G_{σ_{j}^{n}}^{n}] = Y′E[𝕀{σ_{j}^{n} ≤ t < σ_{j+1}^{n}}/G_{σ_{j}^{n}}^{n}],

that is

E[E[g(W_{t})𝕀{σ_{j+1}^{n} > t}/G_{t}^{n}]/G_{σ_{j}^{n}}^{n}]𝕀{σ_{j}^{n} ≤ t} = Y′𝕀{σ_{j}^{n} ≤ t}P (σ_{j+1}^{n} > t/G_{σ_{j}^{n}}^{n}).

Then, recalling that σ_{j+1}^{n} = σ_{j}^{n} + S_{j+1}^{n},

Y′ = \frac{E[g(W_{t})𝕀 (S_{j+1}^{n} > t − σ_{j}^{n}) /G_{σ_{j}^{n}}^{n}]}{P (S_{j+1}^{n} > t − σ_{j}^{n}/G_{σ_{j}^{n}}^{n})}𝕀\{σ_{j}^{n} ≤ t\}.

Moreover on \{σ_{j}^{n} ≤ t < σ_{j+1}^{n}\}, W_{σ_{j}^{n}} = −\frac{j}{2^{n}} and W_{t} = −\frac{j}{2^{n}} + W_{t} − W_{σ_{j}^{n}}. So

\frac{E[g(W_{t})𝕀 (S_{j+1}^{n} > t − σ_{j}^{n}) /G_{σ_{j}^{n}}^{n}]}{P (S_{j+1}^{n} > t − σ_{j}^{n}/G_{σ_{j}^{n}}^{n})}𝕀\{σ_{j}^{n} ≤ t < σ_{j+1}^{n}\} =

= \frac{E[g (−\frac{j}{2^{n}} + W_{s+σ_{j}^{n}} − W_{σ_{j}^{n}}) 𝕀 (S_{j+1}^{n} > s) |_{s=t−σ_{j}^{n}}/G_{σ_{j}^{n}}^{n}]}{E[𝕀(S_{j+1}^{n} > s)|_{s=t−σ_{j}^{n}}/G_{σ_{j}^{n}}^{n}]}𝕀\{σ_{j}^{n} ≤ t < σ_{j+1}^{n}\}.

The process W_{s+σ_{j}^{n}} − W_{σ_{j}^{n}} is a standard Brownian motion independent of G_{σ_{j}^{n}}^{n}, S_{j+1}^{n} is a random variable independent of G_{σ_{j}^{n}}^{n} and t − σ_{j}^{n} is G_{σ_{j}^{n}}^{n}-measurable. Then (11) follows since

\frac{E[g (−\frac{j}{2^{n}} + W_{s+σ_{j}^{n}} − W_{σ_{j}^{n}}) 𝕀 (S_{j+1}^{n} > s) |_{s=t−σ_{j}^{n}}/G_{σ_{j}^{n}}^{n}]}{E[𝕀(S_{j+1}^{n} > s)|_{s=t−σ_{j}^{n}}/G_{σ_{j}^{n}}^{n}]}𝕀\{σ_{j}^{n} ≤ t < σ_{j+1}^{n}\} =

= \frac{E[g (−\frac{j}{2^{n}} + W_{s+σ_{j}^{n}} − W_{σ_{j}^{n}}) 𝕀 (S_{j+1}^{n} > s) ]_{s=t−σ_{j}^{n}}}{E[𝕀(S_{j+1}^{n} > s)]_{s=t−σ_{j}^{n}}} 𝕀\{σ_{j}^{n} ≤ t < σ_{j+1}^{n}\}.

Finally (12) follows by noting that

E[g(−\frac{j}{2^{n}} + W_{s+σ_{j}^{n}} − W_{σ_{j}^{n}})𝕀 (S_{j+1}^{n} > s) ] =E[g(−\frac{j}{2^{n}} + W_{s}^{∗})𝕀 (σ_{1}^{∗n} > s) ] =  (13)
=E[g(−\frac{j}{2^{n}} + W_{s}^{∗})𝕀 (Λ_{s}^{∗} < \frac{1}{2^{n}} ) ],

where σ_{1}^{∗n} is first exit time of W_{t}^{∗} from the region (−\frac{1}{2^{n}} , ∞).

Our next step is to give a more explicit expression of (12). First of all note that (12) can be rewritten as

E[g(W_{t})/G_{t}^{n}] = E[g(−l + W_{s}^{∗})/Λ_{s}^{∗} < \frac{1}{2^{n}} ]|_{l=Λ_{t}^{n}, s=ζ_{t}^{n}}  (14)

= E[g(−l + W_{s}^{∗})/Λ_{s}^{∗n} = 0]|_{l=Λ_{t}^{n}, s=ζ_{t}^{n}},

where Λ_{t}^{n} is defined in (8),

ζ_{t}^{n} = t − sup{u ≤ t s.t. Λ_{u}^{n} < Λ_{t}^{n}},  (15)

and Λ^{∗n} is obtained from W^{∗} as Λ^{n} from W. Then, denoting

Π^{n}(s, l; g) = E[g(−l + W_{s}^{∗})/Λ_{s}^{∗} < \frac{1}{2^{n}} ] = E[g(−l + W_{s}^{∗})/Λ_{s}^{∗n} = 0],  (16)

the conditional expectation (12) can be shortly rewritten as

E[g(W_{t})/G_{t}^{n}] = Π^{n}(ζ_{t}^{n}, Λ_{t}^{n}; g).  (17)

In order to give the explicit expression of the above conditional expectation, and taking into account the definition of Π^{n}(s, l), we need to compute the conditional law of W_{s}^{∗} given {Λ_{s}^{∗} < k}, for k > 0, where (W_{s}^{∗}, Λ_{s}^{∗}) are the processes of Proposition 3.1.

Lemma 3.2. When s > 0, the conditional law of W_{s}^{∗} given {Λ_{s}^{∗} < k}, has density

f_{W_{s}^{∗}}(x/Λ_{s}^{∗} < k) = \frac{1}{2Φ (\frac{k}{\sqrt{s}}) − 1} [\frac{e^{− \frac{x^{2}}{2s}}}{\sqrt{2πs}} − \frac{e^{− \frac{(2k+x)^{2}}{2s}}}{\sqrt{2πs}} ] 𝕀(−x < k),  (18)

where Φ is the distribution function of a normal standard random variable. When s = 0 the conditional law of W_{0}^{∗} given {Λ_{0}^{∗} < k} is the Dirac measure δ_{{0}}(dx) in 0.

Proof. The case s = 0 is trivial. When s > 0, set M_{s}^{∗} = max_{u≤s} W_{u}^{∗}. By the reflection principle one can compute the joint law of (W_{s}^{∗}, M_{s}^{∗}) (see [6], for example)

P (W_{s}^{∗} ≤ x, M_{s}^{∗} ≤ k) = \begin{cases}Φ (\frac{x}{\sqrt{s}}) − Φ (\frac{x − 2k}{\sqrt{s}} )  if x ≤ k \\ P (M_{s}^{∗} ≤ k) = Φ (\frac{k}{\sqrt{s}}) − Φ (\frac{−k}{\sqrt{s}})  if x > k. \end{cases}

Then, by the symmetry of W, the joint law of (W_{s}^{∗}, Λ_{s}^{∗}) is

P (W_{s}^{∗} ≤ x, Λ_{s}^{∗} ≤ k) = \begin{cases}0  if −x ≥ k and k < 0 \\ 2Φ (\frac{k}{\sqrt{s}}) − Φ (\frac{−x}{\sqrt{s}}) − Φ (\frac{x + 2k}{\sqrt{s}} )  if −x < k and k ≥ 0. \end{cases}

Then the thesis trivially follows by observing that

P (W_{s}^{∗} ≤ x/Λ_{s}^{∗} < k) = \frac{2Φ (\frac{k}{\sqrt{s}}) − Φ (\frac{−x}{\sqrt{s}}) − Φ (\frac{x+2k}{\sqrt{s}})}{2Φ (\frac{k}{\sqrt{s}}) − 1}𝕀(−x < k).

Now we have all the tools to give a more explicit expression for E[g(W_{t})/G_{t}^{n}].

Theorem 3.3. Let

f_{n}(x, s) = \frac{1}{2Φ (\frac{1}{2^{n}\sqrt{s}}) − 1} [\frac{e^{− \frac{x^{2}}{2s}}}{\sqrt{2πs}} − \frac{e^{− \frac{( \frac{2}{2^{n}} +x)^{2}}{2s}}}{\sqrt{2πs}} ] 𝕀(x > −\frac{1}{2^{n}}), for s > 0.  (19)

Then

E[g(W_{t})/G_{t}^{n}] = ∑_{j=0}^{∞} \{∫_{− \frac{1}{2^{n}}}^{+∞} g(−\frac{j}{2^{n}} + x)f_{n}(x, s)dx\}|_{s=t−σ_{j}^{n}} 𝕀\{σ_{j}^{n} ≤ t < σ_{j+1}^{n}\},  (20)

with the convention that f_{n}(x, 0) = δ_{0}(x), the Dirac function at the point 0. Equivalently, the conditional law of W_{t} given G_{t}^{n} admits the representation (17) P -a.s., and

Π^{n}(s, l; g) = \begin{cases}∫_{−∞}^{+∞} g(−l + x)f_{n}(x, s)dx if s > 0 \\ g(−l) if s = 0. \end{cases}  (21)

Proof. From Lemma 3.2 we get that, for s > 0,

E[g(−\frac{j}{2^{n}} + W_{s}^{∗})/Λ_{s}^{∗} < \frac{1}{2^{n}} ] = ∫_{−∞}^{+∞} g(−\frac{j}{2^{n}} + x)f_{W_{s}^{∗}}(x/Λ_{s}^{∗} < \frac{1}{2^{n}})dx.

Comparing (19) and (18) we see that

f_{n}(x, s) = f_{W_{s}^{∗}}(x/Λ_{s}^{∗} < \frac{1}{2^{n}}),

then (20) follows by substituting the previous expression into (12), and (21) is an immediate consequence of (16).

4 The original filter as an L^{1} limit

In this section we compute explicitly the L^{1}-limit of the approximating filter π_{t}^{n}(g) = E[g(W_{t})/G_{t}^{n}]. Thanks to the convergence result stated in Theorem 2.4, this limit is a version of the original filter

π_{t}(g) = E[g(W_{t})/F_{t}^{Λ}].

By (17) and by Theorem 3.3 we know that

π_{t}^{n}(g) = Π^{n}(ζ_{t}^{n}, Λ_{t}^{n}; g),

where Π^{n}(s, l; g) is defined in (16), its expression is given in (21), ζ_{t}^{n} is defined in (15), and Λ_{t}^{n} in (8). We have already shown in Lemma 2.2 the uniform convergence of Λ_{t}^{n} to Λ_{t}. The idea is now to show that ζ_{t}^{n} is converging to ζ_{t}, the elapsed time from last visit to 0 for the process W_{t} +Λ_{t} (see (3)), and that Π^{n}(s_{n}, l; g) is converging to a limit Π(s, l; g), to be computed explicitly, whenever s_{n} → s. Taking into account (16), the natural candidate for Π(s, l; g) is

E [g(−l + W_{s}^{∗})/Λ_{s}^{∗} = 0] := ∫_{ℝ} g(−l + x)f_{W_{s}^{∗}|Λ_{s}^{∗}}(x|0)dx

This is the case, as explained in Remark 4.2 below by using the following Lemma.

Lemma 4.1. Denote by

Π(s, l; g) = ∫_{0}^{∞} g(−l + y\sqrt{s})ye^{− \frac{y^{2}}{2}} dy s ≥ 0.  (22)

Let g be a continuous function with compact support, then

Π^{n}(s, l; g) →_{n→∞} Π(s, l; g) for s ≥ 0.  (23)

Moreover if s_{n} →_{n→∞} s and s > 0, then

Π^{n}(s_{n}, l; g) →_{n→∞} Π(s, l; g).  (24)

Proof. The proof of (23) when s = 0 is trivial. Let s be strictly positive, and let s_{n} converge to s. Note that Π^{n}(s_{n}, l; g) is the integral of the product of g(−l + x) and f_{n}(x, s_{n}), where f_{n} is defined in (19), so that

f_{n}(x, s_{n}) = \frac{1}{\sqrt{2πs_{n}}} \frac{1}{2 (Φ (\frac{1}{2^{n}\sqrt{s_{n}}} ) − Φ(0))} [e^{− \frac{x^{2}}{2sn}} − e^{− \frac{( \frac{2}{2^{n}} +x)^{2}}{2sn}} ] 𝕀 (x > −\frac{1}{2^{n}} ) .

Moreover note that, since s > 0 we may rewrite

Π(s, l; g) = ∫_{0}^{∞} g(−l + x)\frac{x}{s}e^{− \frac{x^{2}}{2s}} dx.  (25)

If we prove that f_{n}(x, s_{n}) is a bounded sequence and is pointwise converging to \frac{x}{s}e^{− \frac{x^{2}}{2s}}, then the dominated convergence theorem yields (24), and therefore (23), since g(−l + x) is integrable. The bp-convergence of f_{n}(x, s_{n}) is easy to prove. Indeed, first of all

Φ (\frac{1}{2^{n}\sqrt{s_{n}}} ) − Φ (0) = \frac{1}{\sqrt{2π}} e^{− \frac{θ^{2}_{n}}{2}} \frac{1}{2^{n}\sqrt{s_{n}}} ,

where

θ_{n}^{2} ∈ (0, \frac{1}{2^{n}\sqrt{s_{n}}} ) ,

and therefore

θ_{n}^{2} →_{n→∞} 0.

Moreover

e^{− \frac{x^{2}}{2sn}} − e^{− \frac{( \frac{2}{2^{n}} +x)^{2}}{2sn}} = e^{−y_{n}(x)} \frac{1}{2s_{n}} [(\frac{2}{2^{n}} + x)^{2} − x^{2}]
= e^{−y_{n}(x)} \frac{2}{2^{n}s_{n}} [x + \frac{1}{2^{n}} ] ,

where

y_{n}(x) ∈ (\frac{1}{2s_{n}} x^{2}, \frac{1}{2s_{n}} (\frac{1}{2^{n−1}} + x)^{2}) ,

and therefore

y_{n}(x) →_{n→∞} \frac{x^{2}}{2s}.

Then

f_{n}(x, s_{n}) = e^{\frac{θ^{2}_{n}}{2}} e^{−y_{n}(x)} \frac{1}{s_{n}} [x + \frac{1}{2^{n}} ] 𝕀 (x > −\frac{1}{2^{n}} ) ,

so that f_{n}(x, s_{n}) is bounded uniformly in x, and

lim_{n→∞} f_{n}(x, s_{n}) = \frac{x}{s}e^{− \frac{x^{2}}{2s}} 𝕀 (x ≥ 0) ,  (26)

and the thesis follows.

Remark 4.2. The joint density of (W_{s}^{∗}, Λ_{s}^{∗}) is

f_{W_{s}^{∗},Λ_{s}^{∗}}(x, k) = \frac{2}{s\sqrt{2πs}}(x + 2k) exp (−\frac{(x + 2k)^{2}}{2s} ) 𝕀(k ≥ (−x) ∨ 0), for s > 0,

and then

f_{W_{s}^{∗}|Λ_{s}^{∗}}(x|k) = \frac{1}{s}(x + 2k) exp (−\frac{(x + 2k)^{2}}{2s} + \frac{k^{2}}{2s}) 𝕀(k ≥ (−x) ∨ 0).

As noted in the proof of Theorem 3.3, f_{n}(x, s) = f_{W_{s}^{∗}}(x/Λ_{s}^{∗} < \frac{1}{2^{n}}), and therefore (26) is equivalent to

f_{W_{sn}^{∗}} (x/Λ_{s_{n}}^{∗} < \frac{1}{2^{n}}) →_{n→∞} f_{W_{s}^{∗}|Λ_{s}^{∗}}(x|0) = \frac{x}{s}e^{− \frac{x^{2}}{2s}} 𝕀 (x ≥ 0) .

Taking into account (25) we get the announced interpretation

Π(s, l; g) = ∫_{0}^{∞} g(−l + x)f_{W_{s}^{∗}|Λ_{s}^{∗}}(x|0)dx.

It is also interesting to note that when g is differentiable, then

∫_{0}^{∞} g(−l + x)f_{W_{s}^{∗}|Λ_{s}^{∗}}(x|0)dx = g(−l) + ∫_{0}^{∞} g′(−l + x)e^{− \frac{x^{2}}{2s}} dx  (27)

since −\frac{x}{s}e^{− \frac{x^{2}}{2s}} = \frac{d}{dx}e^{− \frac{x^{2}}{2s}}.

The last observation is a key point in the alternative probabilistic proof of the result stated in Lemma 4.1, and that we postpone to the Appendix to this section.

The next result concerns the limit of the processes ζ_{t}^{n}.

Lemma 4.3. Let ζ_{t}^{n} be defined by (15) and let ζ_{t} be the elapsed time from last visit to 0 for the process W_{t} + Λ_{t} (see (3)). Then, for all ω ∈ Ω

ζ_{t}^{n}(ω) ↘_{n→∞} ζ_{t}(ω).  (28)

Proof. Note that ζ_{t} = t − η_{t} and ζ_{t}^{n} = t − η_{t}^{n}, where

η_{t} = sup{u < t s.t. Λ_{u} < Λ_{t}}

and

η_{t}^{n} = sup\{u < t s.t. Λ_{u}^{n} < Λ_{t}^{n}\} = sup\{u < t s.t. Λ_{u}^{n} = Λ_{t}^{n} − \frac{1}{2^{n}} \},

with the convention that sup{∅} = 0. Observe that

η_{t}^{n} = σ_{j}^{n} if and only if − \frac{j + 1}{2^{n}} < inf_{s≤t} W_{s} ≤ −\frac{j}{2^{n}}

and

η_{t}^{n+1} = \begin{cases}σ_{2j}^{2n} if and only if − \frac{2j+1}{2^{n+1}} < inf_{s≤t} W_{s} ≤ −\frac{2j}{2^{n+1}} = −\frac{j}{2^{n}} \\ σ_{2j+1}^{2n} if and only if \frac{j+1}{2^{n}} = −\frac{2j+2}{2^{n+1}} < inf_{s≤t} W_{s} ≤ −\frac{2j+1}{2^{n+1}}. \end{cases}

This observation implies η_{t}^{n} to be increasing in n. Moreover η_{t}^{n} ≤ t and so η_{t}^{n} ↑ η_{t}^{∞}.

To complete the proof we show successively the inequalities η_{t}^{∞} ≤ η_{t} and η_{t}^{∞} ≥ η_{t}.

The first inequality is equivalent to η_{t}^{n} ≤ η_{t}, for any n ∈ ℕ. In case t < σ_{1}^{n} this inequality is trivial since η_{t}^{n} = 0 ≤ η_{t}. When t ≥ σ_{1}^{n} observe that η_{t}^{n} ∈ {σ_{j}^{n}, j ∈ ℕ} and Λ_{σ_{j}^{n}}^{n} = Λ_{σ_{j}^{n}} for all n, j ∈ ℕ and so

Λ_{η_{t}^{n}} = Λ_{η_{t}^{n}}^{n} = Λ_{t}^{n} − \frac{1}{2^{n}} .

Moreover, by (9), Λ_{t}^{n} ≤ Λ_{t}, so that Λ_{η_{t}^{n}} ≤ Λ_{t} − \frac{1}{2^{n}}. The last inequality implies η_{t}^{n} ≤ η_{t} for all n ∈ ℕ.

On the other hand, if η_{t}^{∞} < s ≤ t then η_{t}^{n} ≤ η_{t}^{∞} < s ≤ t for all n ∈ ℕ, and therefore Λ_{s}^{n} = Λ_{t}^{n} for all n. Passing to the limit in the previous equality we get Λ_{s} = Λ_{t} and so s ≥ η_{t}. Therefore η_{t}^{∞} ≥ η_{t}.

Finally we can state our main result

Theorem 4.4. Let ζ_{t}, Λ_{t} and Π(s, l) be defined in (3), (5) and (22), respectively. Then, for all continuous functions g, with compact support,

∫_{0}^{T} E|π_{t}^{n}(g)−Π(ζ_{t}, Λ_{t}; g)|dt ⟶_{n→∞} 0,

and the process Π(ζ_{t}, Λ_{t}; g) is a cadlag version of π_{t}(g) = E[g(W_{t})/F_{t}^{Λ}], for all bounded measurable functions g.

Proof. Taking into account (2) and (17), our first aim is to show that, for all continuous functions g with compact support,

∫_{0}^{T} E|Π^{n}(ζ_{t}^{n}, Λ_{t}^{n}; g)−Π(ζ_{t}, Λ_{t}; g)|dt ⟶_{n→∞} 0.

Note that

E|Π^{n}(ζ_{t}^{n}, Λ_{t}^{n}; g)−Π(ζ_{t}, Λ_{t}; g)| ≤
≤E|Π^{n}(ζ_{t}^{n}, Λ_{t}^{n}; g) − Π^{n}(ζ_{t}^{n}, Λ_{t}; g)| + E|Π^{n}(ζ_{t}^{n}, Λ_{t}; g) − Π(ζ_{t}, Λ_{t}; g)|.

Consider separately the terms on the right hand side. Recall (9), so that |Λ_{t}^{n} − Λ_{t}| ≤ \frac{1}{2^{n}}. Then, by (21), a direct computation yields

|Π^{n}(ζ_{t}^{n}, Λ_{t}^{n}; g) − Π^{n}(ζ_{t}^{n}, Λ_{t}; g)| ≤ ω_{g} (\frac{1}{2^{n}} ) ,

where ω_{g}(δ) = sup_{|s−u|<δ} |g(s) − g(u)|.

As far as the second term is concerned, (24) and (28) imply that for each (t, ω) satisfying the condition ζ_{t}(ω) > 0

|Π^{n}(ζ_{t}^{n}, Λ_{t}; g) − Π(ζ_{t}, Λ_{t}; g)| ⟶_{n→∞} 0.

The observation that {(t, ω) ∈ [0, T] × Ω such that ζ_{t}(ω) = 0} is a zero measure set with respect to dt × dP, and an easy application of the dominated convergence theorem provide

E[g(W_{t})/G_{t}^{n}] ⟶_{L^{1}([0,T]×Ω)} Π(ζ_{t}, Λ_{t}; g).

This first result, combined with Theorem 2.4, yields that π_{t}(g) = Π(ζ_{t}, Λ_{t}; g) dt × dP -a.e., for continuous functions with compact support. Then the probability measures π_{t} and Π(ζ_{t}, Λ_{t}) coincide dt × dP -a.e., and the second statement is achieved.

We end this section by observing that the cadlag version of the filter π_{t}(g) can be also written as follows (see Remark 4.2)

Π(ζ_{t}, Λ_{t}; g) = g(−Λ_{t})𝕀(ζ_{t} = 0) + ∫_{0}^{∞} g(−Λ_{t} + x)f_{W_{s}^{∗}|Λ_{s}^{∗}}(x|0)dx|_{s=ζ_{t}}𝕀(ζ_{t} > 0),  (29)

and consequently we can write down the cadlag version for the filter of W_{t} + Λ_{t} w.r.t. F_{t}^{Λ}

E[g(W_{t} + Λ_{t})/F_{t}^{Λ}] = ∫_{0}^{∞} g(x)f_{W_{s}^{∗}|Λ_{s}^{∗}}(x|0)dx|_{s=ζ_{t}} = ∫_{0}^{∞} g(y\sqrt{s})ye^{− \frac{y^{2}}{2}} dy|_{s=ζ_{t}}.  (30)

Appendix to Section 4

We now give a different derivation of the limit Π(s, l; g), under the hypothesis that g has continuous derivative and is uniformly continuous. For the sake of simplicity we consider only the case s_{n} = s > 0, but the general case when s_{n} is converging to s > 0 can be managed in the same manner. By the definition (19) of f_{n}(x, s), we have

Π^{n}(s, l; g) = \frac{E[g(−l + W_{s}^{∗})𝕀(W_{s}^{∗} > −\frac{1}{2^{n}})] − E[g(−l + W_{s}^{∗} − 2\frac{1}{2^{n}})𝕀(W_{s}^{∗} − 2\frac{1}{2^{n}} > −\frac{1}{2^{n}})]}{2Φ (\frac{1}{2^{n}\sqrt{s}}) − 1}.

Note that the above expression could also be derived directly from (13) by means of the reflection principle. Assume that g is differentiable, and observe that

g(−l + x)𝕀(x > −\frac{1}{2^{n}}) − g(−l + x − 2\frac{1}{2^{n}})𝕀(x − 2\frac{1}{2^{n}} > −\frac{1}{2^{n}}) =

g(−l + x)𝕀(−\frac{1}{2^{n}} < x ≤ \frac{1}{2^{n}}) + (g(−l + x) − g(−l + x − 2\frac{1}{2^{n}}))𝕀(x > \frac{1}{2^{n}} ) =

g(−l + x)𝕀(−\frac{1}{2^{n}} < x ≤ \frac{1}{2^{n}}) + (g′(−l + x − ϑ^{n}(x)2\frac{1}{2^{n}}))2\frac{1}{2^{n}} 𝕀(x > \frac{1}{2^{n}} ) ,

where ϑ^{n}(x) ∈ (0, 1). Then

E[g(−l + W_{s}^{∗})𝕀(W_{s}^{∗} > −\frac{1}{2^{n}}) − g(−l + W_{s}^{∗} − 2\frac{1}{2^{n}})𝕀(W_{s}^{∗} − 2\frac{1}{2^{n}} > −\frac{1}{2^{n}})] =

= E[g(−l + W_{s}^{∗})𝕀(−\frac{1}{2^{n}} < W_{s}^{∗} ≤ \frac{1}{2^{n}}) + g′(−l + W_{s}^{∗} − ϑ^{n}(W_{s}^{∗})2\frac{1}{2^{n}}) 2\frac{1}{2^{n}} 𝕀(W_{s}^{∗} > \frac{1}{2^{n}})],

and therefore

Π^{n}(s, l; g) = \frac{E[g(−l + W_{s}^{∗})𝕀(−\frac{1}{2^{n}} < W_{s}^{∗} ≤ \frac{1}{2^{n}})]}{2Φ (\frac{1}{2^{n}\sqrt{s}}) − 1}

+\frac{E[g′(−l + W_{s}^{∗} − ϑ^{n}(W_{s}^{∗})2\frac{1}{2^{n}})𝕀(W_{s}^{∗} > \frac{1}{2^{n}})] 2\frac{1}{2^{n}}}{2Φ (\frac{1}{2^{n}\sqrt{s}}) − 1} .

The first addend is converging to g(−l) since 2Φ (\frac{1}{2^{n}\sqrt{s}}) − 1 = P (−\frac{1}{2^{n}} < W_{s}^{∗} ≤ \frac{1}{2^{n}})) and |g(−l + x) − g(−l)| ≤ ω_{g}(\frac{1}{2^{n}}) for x ∈ (−\frac{1}{2^{n}} , \frac{1}{2^{n}}]. For the second addend observe that the expectation in the numerator is converging to E[g′(−l + W_{s}^{∗})𝕀(W_{s}^{∗} > 0)], while

lim_{n→∞} \frac{2Φ (\frac{1}{2^{n}\sqrt{s}}) − 1}{2\frac{1}{2^{n}}} = \frac{1}{\sqrt{2πs}} ·

The expression (25) of Π(s, l; g) is achieved again since

E[g′(−l + W_{s}^{∗})𝕀(W_{s}^{∗} > 0)] = \frac{1}{\sqrt{2πs}} ∫_{0}^{∞} g′(−l + x)e^{− \frac{x^{2}}{2s}} dx

and since the following integration by part formula, which can be thought as the counterpart of (27), yields

E[g′(−l + W_{s}^{∗})𝕀(W_{s}^{∗} > 0)] = \frac{1}{\sqrt{2πs}} \{g(−l + x)e^{− \frac{x^{2}}{2s}} |_{0}^{∞} + ∫_{0}^{∞} g(−l + x)\frac{x}{s}e^{− \frac{x^{2}}{2s}} dx\} =

= \frac{1}{\sqrt{2πs}} \{−g(−l) + ∫_{0}^{∞} g(−l + x)\frac{x}{s}e^{− \frac{x^{2}}{2s}} dx\} .

5 Connections with the Azéma’s martingale

Let B_{t} be a standard Brownian motion. Consider the process L_{t}(B) defined by

L_{t}(B) = |B_{t}| − ∫_{0}^{t} sgn(B_{s})dB_{s}  (32)

where

sgn(x) = \begin{cases}1  if  x > 0 \\ −1  if  x ≤ 0 \end{cases}

The process L_{t}(B) is the usual definition of the local time of a standard Brownian motion B_{t} via the Tanaka formula.

It is well-known that the processes (W_{t}+Λ_{t}, Λ_{t}) and (|B_{t}|, L_{t}(B)) have the same law on D_{ℝ^{2}}[0, ∞) endowed with the Skorohod topology (see e.g. [14]). This result implies that the filter of W_{t} + Λ_{t} given the history generated by the process Λ_{t} and the filter of |B_{t}| given the history generated by the process L_{t}(B) have the same law and that these filters are obtained just applying the same functional to the processes Λ_{.} and L_{.}(B) respectively.

In the sequel we see that the filter of |B_{t}| given the history generated by the process L_{t}(B) can be directly obtained just starting from some results related to the Azéma martingale.

Azéma and Yor ([2]) define the process

μ_{t} = sgn(B_{t})\sqrt{t − g_{t}(B)},  (33)

where g_{t}(B) = sup{s < t : B_{s} = 0}. They show that μ_{t} is a martingale w.r.t. a filtration containing F_{t}^{sgn(B)} = σ{sgn(B_{s}), s ≤ t}. Observe that μ_{t} is adapted to the filtration F_{t}^{sgn(B)} and therefore is an F_{t}^{sgn(B)}-martingale. The process μ_{t} coincides, up to a constant, with the martingale

E[B_{t}/F_{t}^{sgn(B)}],  (34)

which is known in the literature as the Azéma martingale (see, e.g. [14]).

From Lemma 1 and Proposition 4 in [2] one can also draw that

E[f(|B_{t}|)/F_{t}^{sgn(B)}] = ∫_{0}^{∞} f(y\sqrt{t − g_{t}(B)})ye^{− \frac{y^{2}}{2}} dy .  (35)

Observe now that g_{t}(B) is measurable w.r.t. the filtration F_{t}^{L(B)} generated by L(B) up to time t since

g_{t}(B) = sup{s < t s.t. |B_{s}| = 0} = sup{s < t s.t. L_{s}(B) < L_{t}(B)}  (36)

Finally, recalling that F_{t}^{L(B)} ⊆ F_{t}^{sgn(B)} (see e.g. [14]), from (35)

E[f(|B_{t}|)/F_{t}^{L(B)}] = ∫_{0}^{∞} f(y\sqrt{t − g_{t}(B)})ye^{− \frac{y^{2}}{2}} dy .  (37)

Then the connection with our filtering problem is evident by comparing the above equality with (30). Indeed one can obtain one from the other just by changing (W_{t} +Λ_{t}, Λ_{t}) with (|B_{t}|, L_{t}(B)). Indeed, if we define

γ : D_{ℝ}[0, ∞) → D_{ℝ}[0, ∞); x → γ(x), s.t. γ(x)(t) = t − sup{s < t, s.t. x_{s} < x_{t}}

the process t−g_{t}(B) in (36) and the elapsed time process ζ_{t} defined in Lemma 4.3 are respectively defined by

t − g_{t}(B) = γ (L(B)) (t),
ζ_{t} = γ (Λ) (t).

6 The case of a general Brownian motion

In this Section we derive the approximating and exact filters of a general Brownian motion W_{t} with diffusion coefficient a^{2} and drift c ∈ ℝ, with respect to its local time Λ_{t}.

Remark 6.1. The results and the proof of Proposition 3.1 are still valid for a general Brownian motion, since also in the general case the process W_{s}^{∗} = W_{s+σ_{j}^{n}} − W_{σ_{j}^{n}} is a Brownian motion independent of G_{σ_{j}^{n}}^{n}, with the same law as the process W_{s}. Therefore

E_{(a,c)}[g(W_{t})/G_{t}^{n}] = ∑_{j=0}^{∞} E_{(a,c)}[g(−\frac{j}{2^{n}} + W_{s}^{∗})/Λ_{s}^{∗} < \frac{1}{2^{n}} ]_{s=t−σ_{j}^{n}}𝕀\{σ_{j}^{n} ≤ t < σ_{j+1}^{n}\},  (38)

where we use the symbol E_{(a,c)} to recall the diffusion and the drift coefficient, and we use the same notations of Section 2.

The above remark and the representation (38) allow us to write the approximating filter as

E_{(a,c)}[g(−l + W_{s}^{∗})/Λ_{s}^{∗} < \frac{1}{2^{n}} ]|_{(s,l)=(ζ_{t}^{n},Λ_{t}^{n})} = E_{(a,c)}[g(−l + W_{s}^{∗})/Λ_{s}^{∗n} = 0]|_{(s,l)=(ζ_{t}^{n},Λ_{t}^{n})} , (39)

where Λ_{t}^{n} and ζ_{t}^{n} are obtained as in Section 2, and Λ^{∗n} is obtained from W^{∗} as Λ^{n} from W.

It is immediate to see that c = 0 corresponds to the case of a standard Brownian motion up to the deterministic time change s → a^{2}s. Indeed

E_{(a,0)}[g(−l + W_{s}^{∗})/Λ_{s}^{∗n} = 0] = E_{(1,0)}[g(−l + W_{a^{2}s}^{∗})/Λ_{a^{2}s}^{∗n} = 0] = Π^{n}(a^{2}s, l; g),  (40)

where W_{s}^{∗} is a standard Brownian motion under the expectation E_{(1,0)} = E, and Π^{n}(s, l; g) is defined in (16). Then by the results of Section 4

E_{(a,0)}[g(W_{t})/G_{t}^{n}] = Π^{n}(a^{2}ζ_{t}^{n}, Λ_{t}^{n}; g),  (41)

and

Π^{n}(a^{2}ζ_{t}^{n}, Λ_{t}^{n}; g) ⟶_{L^{1}([0,T]×Ω)} Π(a^{2}ζ_{t}, Λ_{t}; g),  (42)

where ζ_{t} is defined as in (3) and Π^{n}(s, l; g) is defined as in (22), and therefore

Π(a^{2}s, l; g) = E[g(−l + W_{a^{2}s}^{∗})/Λ_{a^{2}s}^{∗} = 0] = ∫_{0}^{∞} g (−l + ya\sqrt{s}) y exp (−\frac{y^{2}}{2} ) dy s ≥ 0. (43)

Theorem 6.2. Let W_{t} be a Brownian motion with diffusion coefficient a^{2} and drift coefficient c. Then the conditional law of W_{t} given G_{t}^{n} = F_{t}^{Λ^{n}} admits the following representation P -a.s.

E_{(a,c)}[g(W_{t})/G_{t}^{n}] = \frac{Π^{n}(a^{2}s, l; g(·) exp(\frac{c}{a^{2}} ·))}{Π^{n}(a^{2}s, l; exp(\frac{c}{a^{2}} ·))}|_{(s,l)=(ζ_{t}^{n},Λ_{t}^{n})}  (44)

Proof. The proof can be given in two ways. The first one is via Kallianpur-Striebel formula and the exponential martingale Z_{t} = exp\{\frac{c}{a^{2}} W_{t} − \frac{1}{2} \frac{c^{2}}{a^{2}} t\}

E_{(a,c)}[g(W_{t})/G_{t}^{n}] = \frac{E_{(a,0)}[g(W_{t})Z_{t}/G_{t}^{n}]}{E_{(a,0)}[Z_{t}/G_{t}^{n}]}

= \frac{E_{(a,0)}[g(W_{t}) exp\{\frac{c}{a^{2}} W_{t} − \frac{1}{2} \frac{c^{2}}{a^{2}} t\}/G_{t}^{n}]}{E_{(a,0)}[ exp\{\frac{c}{a^{2}} W_{t} − \frac{1}{2} \frac{c^{2}}{a^{2}} t\}/G_{t}^{n}]} = \frac{E_{(a,0)}[g(W_{t}) exp\{\frac{c}{a^{2}} W_{t}\}/G_{t}^{n}]}{E_{(a,0)}[ exp\{\frac{c}{a^{2}} W_{t}\}/G_{t}^{n}]}.

Then we apply (41) to g(x) exp(\frac{c}{a^{2}} x) and to exp(\frac{c}{a^{2}} x).

On the other hand, an elementary proof is based on the following observations. When W_{s} is a Brownian motion with diffusion coefficient a^{2} and drift c, the joint density f_{W_{s},Λ_{s}}^{(a,c)}(x, k) of

(W_{s}, Λ_{s}) is equal to f_{W_{s}}^{(a,c)}(x) · f_{Λ_{s}|W_{s}}^{(a,c)}(k|x), and the conditional density f_{Λ_{s}|W_{s}}^{(a,c)}(k|x) of Λ_{s} given W_{s} does not depend on c (see, for instance, [1]), i.e.

f_{Λ_{s}|W_{s}}^{(a,c)}(k|x) = f_{Λ_{s}|W_{s}}^{(a,0)}(k|x) = f_{Λ_{a^{2}s}|W_{a^{2}s}}^{(1,0)}(k|x).

Then

f_{W_{s},Λ_{s}}^{(a,c)}(x, k) =f_{W_{s}}^{(a,c)}(x) · f_{Λ_{s}|W_{s}}^{(a,0)}(k|x) = \frac{1}{a\sqrt{2πs}} exp (−\frac{(x − cs)^{2}}{2a^{2}s} ) · f_{Λ_{s}|W_{s}}^{(a,0)}(k|x) =
=f_{W_{s}}^{(a,0)}(x) · f_{Λ_{s}|W_{s}}^{(a,0)}(k|x) · exp (−\frac{2csx + c^{2}s^{2}}{2a^{2}s} ) =
=f_{W_{s},Λ_{s}}^{(a,0)}(x, k) exp (\frac{c}{a^{2}} x) exp (−\frac{c^{2}s}{2a^{2}} ) 𝕀(x + k > 0)𝕀(k > 0).

The general term in (39) can be written as

E_{(a,c)}[g(−l + W_{s})/Λ_{s} < \frac{1}{2^{n}} ] = \frac{∫0^{\frac{1}{2^{n}}} dk ∫_{−k}^{∞} g(−l + x)f_{W_{s},Λ_{s}}^{(a,0)}(x, k) exp (\frac{c}{a^{2}} x)dx}{∫0^{\frac{1}{2^{n}}} dk ∫_{−k}^{∞} f_{W_{s},Λ_{s}}^{(a,0)}(x, k) exp (\frac{c}{a^{2}} x)dx}

= \frac{∫0^{\frac{1}{2^{n}}} dk ∫_{−k}^{∞} g(−l + x)f_{W_{s},Λ_{s}}^{(a,0)}(x, k) exp (\frac{c}{a^{2}} (−l + x))dx}{∫0^{\frac{1}{2^{n}}} dk ∫_{−k}^{∞} f_{W_{s},Λ_{s}}^{(a,0)}(x, k) exp (\frac{c}{a^{2}} (−l + x))dx} ·

Since f_{W_{s},Λ_{s}}^{(a,0)}(x, k) = f_{W_{a^{2}s},Λ_{a^{2}s}(x,k)}^{(1,0)}, the thesis follows by (40).

Following the same lines as in Section 4, and observing that the convergence result for the filter as stated in Theorem 2.4 holds either for standard Brownian motion or for general Brownian motion we can finally state the main result of this Section.

Theorem 6.3. For all g continuous, with compact support

E_{(a,c)}[g(W_{t})/G_{t}^{n}] ⟶_{L^{1}([0,T]×Ω)} \frac{Π(a^{2}s, l; g(·) exp(\frac{c}{a^{2}} ·))}{Π(a^{2}s, l; exp(\frac{c}{a^{2}} ·))}|_{(s,l)=(ζ_{t},Λ_{t})} .

Moreover, for all g measurable and bounded, a cadlag version of E_{(a,c)}[g(W_{t})/F^{Λ_{t}}] is given by and then by (43),

E_{(a,c)}[g(W_{t})/F^{Λ_{t}}] = \frac{∫_{0}^{∞} g (−l + ya\sqrt{s}) exp(\frac{c}{a^{2}} (−l + ya\sqrt{s}))y exp (−\frac{y^{2}}{2} ) dy}{∫_{0}^{∞} exp(\frac{c}{a^{2}} (−l + ya\sqrt{s}))y exp (−\frac{y^{2}}{2} ) dy} |_{(s,l)=(ζ_{t},Λ_{t})} · (45)

Before giving the proof, we note that the above expression (45) can be obtained directly from Theorem 4.4, via the Kallianpur-Striebel formula and a deterministic time change as in the case of the conditional law given G_{n}.

Proof. Using the representation (44) instead of (17), the proof of Theorem 4.4 can be repeated verbatim for a general Brownian motion, since Lemma 4.3 still holds true, that is

1. ζ_{t}^{n}(ω) ↘ ζ_{t}(ω) P_{(a,c)} − a.s.,

2. the set {(t, ω) ∈ [0, T] × Ω such that ζ_{t}(ω) = 0} is a zero measure set with respect to the measure dt × dP_{(a,c)}. Indeed, by Girsanov theorem, there exists a probability measure P^{(a,c)}, equivalent to P_{(a,c)}, and such that W has the same law as a Brownian motion with diffusion coefficient a^{2} and zero drift under P^{(a,c)}. Then the previous properties 1. and 2. hold w.r.t to P^{(a,c)} instead of P_{(a,c)}, and the equivalence between the two probability measures guarantees that 1. and 2. hold.

Acknowledgements We wish to thank Anna Gerardi for her stimulating and helpful comments.

References

[1] Asmussen, S., Glynn, P., and Pitman, J. Discretization error in simulation of onedimensional reflecting Brownian motion. Ann. Appl. Probab. 5, 4 (1995), 875–896.

[2] Azéma, J., and Yor, M. Étude d’une martingale remarquable. In Séminaire de Probabilités, XXIII, Azéma, Meyer, Yor, Ed., Lecture Notes in Mathematics n. 1372. Springer, Berlin, 1989, pp. 88–130.

[3] Bramson, M., and Dai, J. G. Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11 (2001), 49–90.

[4] Brémaud, P. Point Process and Queues. Springer, New York, 1981.

[5] Elliott, R. J. Stochastic calculus and Applications. Springer-Verlag, 1982.

[6] Harrison, J. M. Brownian motion and stochastic flow systems. J. Wiley & Sons, 1985.

[7] Harrison, J. M., and Williams, R. J. On the quasireversibility of a multiclass brownian service station. Ann. Probab. 18 (1990), 1249–1268.

[8] Harrison, J. M., and Williams, R. J. Brownian models of feedforward queueing networks: Quasireversibility and product form solutions. Ann. Appl. Probab. 2 (1992), 263–293.

[9] Joannides, M., and Le Gland, F. Nonlinear filtering with perfect discrete time observations. In Proocedings of the 34-th conference on Decision and Control (December 1995), IEEE-CSS, pp. 4012–4017.

[10] Joannides, M., and Le Gland, F. Small noise asymptotics of the bayesian estimator in nonidentifiable models. Statistical Inference for Stochastic Processes 1 (2002), 95–130.

[11] Kallianpur, G. Stochastic Filtering Theory. Springer-Verlag, 1980.

[12] Karatzas, I., and Shreve, S. E. Brownian Motion and Stochastic Calculus. Springer, New York, 1988.

[13] Korezlioglu, H., and Runggaldier, W. J. Filtering for nonlinear systems driven by nonwhite noises: an approximation scheme. Stoch. and Stoch Rep. 44 (1993), 65–102.

[14] Protter, P. Stochastic integration and Differential Equations: A New Approach. Springer, Berlin, 1990.