Filtered Azéma martingales

We study the optional projection of a standard Brownian motion on the natural filtration of certain kinds of observation processes. The observation process, Y , is defined as a solution of a stochastic differential equation such that it reveals some (possibly noisy) information about the signs of the Brownian motion when Y hits 0. As such, the associated optional projections are related to Azéma’s martingales which are obtained by projecting the Brownian motion onto the filtration generated by observing its signs.


Introduction
Let (Ω, F, (F t ) t≥0 , P) be a filtered probability space satisfying the usual conditions and W be a standard Brownian motion with W 0 = 0 and adapted to (F t ) t≥0 .Define G 0 t := σ(sgn(W s ); s ≤ t), where sgn(x) = 1, if x > 0; −1, if x ≤ 0, and let (G t ) t≥0 be the augmentation of G 0 t with the P-null sets.Azéma's martingale is obtained by projecting W onto G.We will denote the (G, P)-optional projection of W with µ.This martingale first appeared in [1] and was further studied in a series of papers such as [2], [6] and [12].Our presentation follows [13].
By construction Azéma's martingale is closely related to the excursions of Brownian motion away from 0. In fact, if we set then (see, e.g.[13]) (1.2) Thus, Azéma's martingale is the best estimate, in a mean-square sense, for the value of a Brownian motion when one only observes its zeroes and the signs of its excursions.
The above interpretation of µ was used by [4] to model the default probabilities of a firm under incomplete information.Assuming cash balances follow a Brownian motion, [4] defines the default time for the firm as the first time that its cash balances have remained negative for a certain amount of time and doubled in absolute value.On the other hand, the market's only information regarding the cash balances is whether the firm is in financial distress, i.e. the cash balance is negative, or not.This information set thus corresponds to G in above notation.Using certain properties of Azéma's martingale and some results from excursion theory the authors explicitly compute the G-predictable compensator of the default indicator process.The use of Azéma's martingale in Mathematical Finance Theory is not limited to default risk.It is also the key process in models for Parisian barrier options (see [5]).Motivation of this paper comes from the following question: What happens to the optional projection of Brownian motion when we observe its signs, possibly with some noise, at the zeroes of another process which we can observe continuously?Clearly, the answer to this question depends on how one defines the observation process.The most common approach in applications is to model the observation process as a solution of a stochastic differential equation.In this paper we will look at two different types of stochastic differential equations for the observation process.
The first formulation that we will consider corresponds to the case when one imperfectly observes the signs of Brownian motion at the zeroes of an observation process.
Here imperfection corresponds to the case when the true signal is contaminated with some noise.In view of the standard nonlinear filtering theory one can model the observation process as a (weak) solution to the following stochastic differential equation (SDE): where α ∈ R, B is a standard Brownian motion independent of W , and g t (Y ) := sup{s ≤ t : Y s = 0}.(1.4)In Section 2 we study the existence and uniqueness of (weak) solutions of (1.3) and the projection of W onto the natural filtration of the solution.The methods employed are standard techniques from nonlinear filtering theory.On the other hand, the existence of a strong solution to (1.3) remains as an interesting open problem.
Another possibility for modeling the observation process is to introduce the knowledge on the sign of W through the local times of Y whose support is contained in the zero set of Y .In this case the corresponding SDE is the following: where L is the symmetric local time (see Exercise VI.1.25 in [14] for a definition) of Y at 0. We will see in Section 3 that the solution to the above equation is closely related to the skew Brownian motion which we recall next.
Theorem 1.1.(Harrison and Shepp [8]) There is a unique strong solution, called skew Brownian motion, to where L(X) is the symmetric local time of X at the level 0 if and only if |α| ≤ 1.
First appearances of skew Brownian motion in the literature goes back to as early as [9] and [15].Formally it is obtained by changing the sign of a Brownian motion in Filtered Azéma martingales every excursion depending on the value of an independent Bernoulli random variable.A related SDE introduced by Sophie Weinryb is whose pathwise uniqueness is established in [16] when α is a deterministic function taking values in [−1, 1] (see [7] for a recent work on the existence of solutions and related issues) .
The reader is referred to the recent survey in [10] where one can find a discussion of different constructions of skew Brownian motion and its properties.In Section 3 we will prove that there exists a unique strong solution to (1.5) and see how it is connected to the solutions of (1.6).This connection will be helpful in the characterisaton of the natural filtration of the solution of (1.5) and the associated projection of W , which is our main concern.We will see that this projection changes only by jumps which may only occur at the end of an excursion interval of a skew Brownian motion.

Filtered Azéma martingale of the first kind
Observe that the drift coefficient of the SDE in (1.3) is path dependent and, thus, the classical results on the existence and uniqueness of strong solutions of SDEs do not apply.However, since sgn function is bounded, one can easily construct a weak solution to this equation on any interval [0, T ].Indeed, if β and W are two independent Brownian motions in some probability space, one can define a change of measure via the martingale and under the new measure β solves (1.3) while W stays a Brownian motion.The same Girsanov transform also implies that the law of any weak solution (W, Y ) of (1.3) is the same.Let F Y be the smallest filtration satisfying the usual conditions and containing the filtration generated by Y .In the remainder of this section we will fix a weak solution to (1.3) and compute the corresponding conditional probabilities for this pair.
However, the weak uniqueness of the solutions imply that the conditional laws of W on F Y computed in this section 1 do not depend on the choice of the weak solution.
In the computations performed in this and the subsequent section we will often make use of the balayage formula as given in the next lemma.
As a first application of the balayage formula, we will now see that sgn(W g(B (α) ) )B (α)   is a weak solution of (1.3) where B (α) is defined by

Moreover,
• 0 sgn(W gt(B (α) ) )dB t is a standard Brownian motion independent of W .The claim follows since by construction g(Y ) = g(B (α) ).Thus, by the uniqueness of weak 1 One should be careful in computing the conditional laws of random variables measurable with respect to F∞ since the martingale used for the change of measure is not uniformly integrable.
In other words, Y is obtained by changing the sign of a Brownian motion with drift via the sign of an independent Brownian motion sampled at the beginning of the current excursion (away from 0) of the drifting Brownian motion.As such, the resulting process in a sense is in the same spirit of a skew Brownian motion described in (1.6), which will be relevant to the filtered Azéma martingale of the second kind discussed in the next section.
An immediate consequence of the aforementioned equality in law is the following Proposition 2.2.Let (Y, W ) be the unique weak solution of (1.3).Then, ii) P(sup{t : Proof.i) follows from the fact that |B α) .Similarly, since B (α) transient, there is a last time that it hits 0. Since the zeroes of Y are the same as those of B (α) , the result follows.
The above result is another manifestation of that the law of Y is equivalent to the law of a Brownian motion only if they are stopped at a finite stopping time.Indeed, if the law of Y were equivalent to the Wiener measure, the zero set of Y would be unbounded with probability 1.This discrepancy also confirms that the martingale used to obtain the measure change is not uniformly integrable.(2.3)Then, there is a unique strong solution to this equation.Indeed, in view of the balayage formula, sgn(W gt(Z) )Z t = B t .Thus, the zeroes of Z are the zeroes of B and we have Z t = sgn(W gt(B) )B t .

On the other hand, similar arguments do not seem to work for (2.2). It is an open question whether this equation admits a strong solution.
We next obtain the semimartingale decomposition of Y with respect to its own filtration.
where B Y is an F Y -Brownian motion.
Proof.Note that ii) follows immediately from i) in view of the standard results on filtering, see, e.g.Theorem 8.1 in [11].To see why i) holds take a constant T > t and consider the measure Q ∼ P T under which (Y s ) s∈[0,T ] is a Brownian motion independent of (W s ) s∈[0,T ] where P T is the restriction of P to F T .Then, it follows from Girsanov's theorem that where the second equality follows from Lemma 2.1 and the last equality is due to the independence of W and Y (up to time T ) under Q along with the facts that g t (Y ) is F Y t -measurable and the probability that W s > 0 is 1/2 for any s.
Using the same technique as in the proof of the above proposition, we can obtain the conditional law of W . (2.4) i) The F Y t -conditional law of W t has a density, which is given by dx.
ii) The conditional moments of W are given by In particular, Proof.Let f : R → R be a bounded measurable function.Then, where Q is the measure defined in the proof of Proposition 2.4.Moreover, the numerator in the above fraction equals due to the independence of W and Y under Q.On the other hand, for any s ≤ t the distribution of W s conditional on W t = x is Gaussian with mean s t x and variance s(t−s) t .Thus, x .

Filtered Azéma martingales
Utilising once more the independence of Y and W , we see that (2.5) equals x e −αYt .
This completes the proof of the density.The conditional moments can be calculated by integrating this density, which is a lengthy task.However, since for any λ ∈ R exp(λW t − 1 2 λ 2 t) is a martingale independent of Y , and in particular of g t (Y ), one has Since we can differentiate with respect to λ under the integral sign, we have Moreover, one has Thus, due to the symmetry of p, we obtain In view of the above theorem we may define the filtered Azéma martingale of the first kind by μt = 2gt(Y ) π tanh(αY t ).Observe that, since tanh(0) = 0 and g t (Y ) changes value only when Y hits 0, μ is a continuous martingale in contrast to the discontinuous Azéma martingale, µ.
Although the Brownian motion W is clearly not independent of Y , observing Y does not tell us anything new regarding the process (γ t ).We will only prove γ 1 is independent of Y .The analogous statement can be proven for any γ t along the same lines.
for some bounded measurable real function f , where Q is as constructed in the proof of Proposition 2.4 for some T > 1. Observe that since conditional on γ 1 , (W t ) t∈[0,γ1] is a Brownian bridge (see Exercise XII.3.8 in [14]) and therefore Filtered Azéma martingales as well since g t (Y ) ≤ 1 and therefore sgn(W gt(Y ) ) is independent of γ 1 (see, e.g.Lemme 1 in [2]).Since [g t (Y ) = γ 1 ] is a Q-null set due to their independence and the continuity of the distribution of γ 1 , we deduce that ] for any f .To show the independence for t > 1, note that it suffices to consider ] the problem is reduced to the previous case.Notice by the Markov property of W that, given W 1 , γ 1 and sgn(W u ) are independent for any u > 1.Thus, on where Φ is the function defined in (2.4).Therefore, on [g t (Y ) > 1] On the other hand, the conditional law of W 1 given γ 1 = s is (see Exercise XII.3.8 in [14]) Using this density, one can directly show that Hence, we arrive at which yields the claimed independence.
Since μ is adapted to F Y by definition, we deduce that the filtered Azéma martingale of the first kind is independent of γ.This is in stark contrast to Azéma's martingale, µ, which is a function of the process γ.

Filtered Azéma martingale of the second kind
We now return to study the solutions of equation (1.5) and the associated projection of W . Recall that the equation (1.5) is the following SDE: where L(Y ) is the symmetric local time of Y at 0. The right local time of Y at 0 will be denoted with (Y ).We will write L and instead of L(Y ) and (Y ), respectively, when no confusion arises.L t (X) = lim  where β := • 0 sgn(W gs(X) )dB s , the first equality is due to Lemma 2.1 and the second is due to the fact that support of the measure dL(X) is contained in the zero set of X.This shows that sgn(W gt(X) )X t is a weak solution to (3.1).By working backwards one can also see that sgn(W g(Y ) )Y is a weak solution to (1.6).Since there is a one-to-one correspondence between Y and sgn(W g(Y ) )Y , we obtain the uniqueness in law of the solutions to (3.1) from the analogous property of the solutions to (1.6).Again, since the solutions to (1.6) ECP 17 (2012), paper 62. Page 3/13 ecp.ejpecp.orgFiltered Azéma martingales solutions, we obtain

Remark 2 . 3 .
If we set Z t = sgn(W gt(Y ) )Y t and thereby note that g(Z) = g(Y ), we obtain via balayage formula Z t = t 0 sgn(W gs(Z) )dB s + αt.

Theorem 2 . 5 .
Let p(t, y − x) be the transition density of a standard Brownian motion and set

Thus, Y satisfies Y t = t 0
sgn(W gs(X) )dB s + α t 0 sgn(W gs(X) )dL s (X) = β t + α t 0 sgn(W s )dL s (X) = β t + α t 0 sgn(W s )dL s (Y ), [14]ere is a unique weak solution to(3.1).Moreover, Y sgn(W g(Y ) ) |Y | is a reflecting Brownian motion.The symmetric and nonsymmetric local times, and L, respectively, of Y at 0 are related by Suppose X is the skew Brownian motion that solves(1.6).As observed in Introduction, this SDE has a unique solution.Next let Y t = sgn(W gt(X) )X t .Observe that Y is a continuous semimartingale in view of Lemma 2.1 and [X, X] t = [Y, Y ] t = t.Moreover, L(X) = L(Y ).Indeed, (see Exercise VI.1.25in[14]) iProof.