Linear filtering of systems with memory and application to finance

. We study the linear ﬁltering problem for systems driven by continuous Gaussian processes V (1) and V (2) with memory described by two parameters. The processes V ( j ) have the virtue that they possess stationary increments and simple semimartingale representations simultaneously. It allows for straightforward parameter estimations. After giving the semimartingale representations of V ( j ) by innovation theory, we derive Kalman–Bucy-type ﬁltering equations for the systems. We apply the result to the optimal portfolio problem for an investor with partial observations. We illustrate the tractability of the ﬁltering algorithm by numerical implementations


Introduction
Let T be a positive constant.In this paper, we use the following Gaussian process V = (V t , t ∈ [0, T ]) with stationary increments as the driving noise process: (s−u) dW u ds, 0 ≤ t ≤ T, (1.1) where p and q are real constants such that 0 < q < ∞, −q < p < ∞, (1.2) and (W t , t ∈ R) is a one-dimensional Brownian motion satisfying W 0 = 0.The parameters p and q describe the memory of V .In the simplest case p = 0, V is reduced to the Brownian motion, i.e., V t = W t , 0 ≤ t ≤ T .
It should be noticed that (1.1) is not a semimartingale representation of V with respect to the natural filtration Using the innovation theory as described in Liptser and Shiryayev [21] and a result in Anh et al. [2], we show (Theorem 2.1) that there exists a one-dimensional Brownian motion B = (B t , t ∈ [0, T ]), called the innovation process, satisfying σ(B s , s ∈ [0, t]) = σ(V s , s ∈ [0, t]), 0 ≤ t ≤ T, and V t = B t − t 0 s 0 l(s, u)dB u ds, 0 ≤ t ≤ T, (1.3) where l(t, s) is a Volterra kernel given by l(t, s) = pe −(p+q)(t−s) 1 − 2pq (2q + p) 2 e 2qs − p 2 , 0 ≤ s ≤ t ≤ T. (1.4)With respect to the natural filtration F B = (F B t , t ∈ [0, T ]) of B, which is equal to F V , (1.3) gives the semimartingale representation of V .Thus the process V has the virtue that it possesses the property of a stationary increment process with memory and a simple semimartingale representation simultaneously.We know no other process with this kind of properties.The two properties of V become a great advantage, for example, in its parameter estimation.
In Anh and Inoue [1], Anh et al. [2] and Inoue et al. [10], the process V is used as the driving process for a financial market model with memory.The empirical study for S&P 500 data in Anh et al. [3] shows that the model captures reasonably well the memory effect when the market is stable.The work in these references suggests that the process V can serve as an alternative to Brownian motion when the random disturbance exhibits dependence between different observations.
In this paper, we are concerned with the filtering problem of the twodimensional partially observable process ((X t , Y t ), t ∈ [0, T ]) governed by the following linear system of equations: t , 0 ≤ t ≤ T, X 0 = ρ, dY t = µX t dt + dV (2) t , 0 ≤ t ≤ T, Y 0 = 0.
The basic filtering problem for the linear model (1.5) with memory is to calculate the conditional expectation E[X t |F Y t ], called the (optimal) filter, where ) is the natural filtration of the observation process Y .In the classical Kalman-Bucy theory (see Kalman [12], Kalman and Bucy [13], Bucy and Joseph [4], Davis [5] and Liptser and Shiryayev [21]), Brownian motion is used as the driving noise.Attempts have been made to resolve the filtering problem of dynamical systems with memory by replacing Brownian motion by other processes.In Kleptsyna et al. [16,17,18] and others, fractional Brownian motion was used.Notice that fractional Brownian motion is not a semimartingale.In the discretetime setting, autoregressive processes are used as driving noise (see, e.g., Kalman [12], Bucy and Joseph [4] and Jazwinski [11]).Our model may be regarded as a continuous-time analogue of the latter since it is shown in Anh and Inoue [1] that V is governed by a continuous-time AR(∞)-type equation (see Section 7).
The Kalman-Bucy filter is a computationally tractable scheme for the optimal filter of a Markovian system.We aim to derive a similar effective filtering algorithm for the system (1.5) which has memory.However, rather than considering (1.5) itself, we start with a general continuous Gaussian process X = (X t , t ∈ [0, T ]) as the state process and as the observation process, where µ(•) is a deterministic function and ) is a process which is independent of X and given by (1.1).Using (1.3) and (1.4), we derive explicit Volterra integral equations for the optimal filter (Theorem 3.1).In the special case (1.5), the integral equations are reduced to differential equations, which give an extension to Kalman-Bucy filtering equations (Theorem 3.4).Due to the non-Markovianness of the formulation (1.5), it is expected that the resulting filtering equations would appear in the integral equation form (cf. Kleptsyna et al. [16]).The fact that good Kalman-Bucy-type differential equations can be obtained here is due to the special properties of (1.5).This interesting result does not seem to hold for any other formulation where memory is inherent.We apply the results to an optimal portfolio problem in a partially observable financial market model.More precisely, we consider a stock price model that is driven by the process V = (V t , t ∈ [0, T ]) given by (1.1).Assuming that the investor can observe the stock price but not the drift process, we discuss the portfolio optimization problem of maximizing the expected logarithmic utility from terminal wealth.To solve this problem, we make use of our results on filtering to reduce the problem to the case where the drift process is adapted to the observation process.We then use the martingale methods (cf.Karatzas and Shreve [14]) to work out the explicit formula for the optimal portfolio (Theorem 4.1).This paper is organized as follows.In Section 2, we prove the semimartingale representation (1.3) with (1.4) for V .Section 3 is the main body of this paper.We derive closed form equations for the optimal filter.In Section 4, we apply the results to finance.In Section 5, we illustrate the advantage of V in parameter estimation.Some numerical results on Monte Carlo simulation are presented.In Section 6, we numerically compare the performance of our filter with the Kalman-Bucy filter in the presence of memory effect.Finally, a possible extension of this work is discussed in Section 7.

Driving noise process with memory
Let T ∈ (0, ∞), and let (Ω, F, P) be a complete probability space.For a process (A t , t ∈ [0, T ]), we denote by Let V = (V t , t ∈ [0, T ]) be the process given by (1.1).The process V is a continuous Gaussian process with stationary increments.The aim of this section is to prove (1.3) with (1.4).
To find a good representation of the process α, we recall the following result from Anh et al. [ Proof.We have k(t, s) = a(t)b(s) for 0 ≤ s ≤ t ≤ T , where, for t ∈ [0, T ], Fix s ∈ [0, T ] and define x(t) = x s (t) and λ = λ s by Then, from (2.3) we obtain The solution x is given by We have By the change of variable x(u) = (2q + p) 2 e 2qu − p 2 , we obtain x(s) Thus or (1.4), as desired.

Filtering equations
3.1.General result.We consider a general two-dimensional centered continuous Gaussian process ((X t , U t ), t ∈ [0, T ]).The process X = (X t , t ∈ [0, T ]) represents the state process, while U = (U t , t ∈ [0, T ]) is another process which is related to the dynamics of X.
In this subsection, (B t , t ∈ [0, T ]) is not an innovation process but just a one-dimensional Brownian motion that is independent of (X, U ).Let l(t, s) be an arbitrary Volterra-type bounded measurable function on [0, T ] 2 (i.e., l(t, s) = 0 for s > t).Though the function given by (1.4) satisfies this assumption, (1.4) itself is not assumed in this subsection.We define the processes α = (α t , t ∈ [0, T ]) and V = (V t , t ∈ [0, T ]) by (2.4) and (2.2), respectively.Thus, in particular, α is not assumed to be given by a conditional expectation of the type (2.1), and V is not necessarily a stationary increment process.We consider the observation where * denotes the transposition.Notice that Γ AC is R d×d -valued.Specifically, we consider and define the error matrix P (t, s) ∈ R 3×3 by The next theorem gives an answer to the filtering problem for the partially observable process ((X t , Y t ), t ∈ [0, T ]).It turns out that this will be useful in the filtering problem for (1.5) for example.
and the error matrix P (t, s) is the unique solution to the following matrix Riccati-type integral equation such that P (t, t) is symmetric and nonnegative definite for 0 ≤ t ≤ T : where Proof.Since (X, U ) is independent of B, (X, U, α, Y ) forms a Gaussian system.We have Thus we can define the innovation process I = (I t , t ∈ [0, T ]) by which is a Brownian motion satisfying F Y = F I (cf.Liptser and Shiryayev [21,Theorem 7.16]).Notice that I can be written as By Corollary to Theorem 7.16 in [21], there exists an R 3 where | • | denotes the Euclidean norm.Now let g(t) = (g 1 (t), g 2 (t), g 3 (t)) be an arbitrary bounded measurable row-vector function on [0, T ].Then, for t ∈ [0, T ], From this, (3.3), (3.4) and the fact that (X, U ) and B are independent, we have The SDE (3.1) follows from (3.5) and the representation The equation (3.2) follows from (3.5) and the equality From this, we have Therefore, P (t, s) is at most Let P 1 and P 2 be two solutions of (3.2).We define Q i (t, s) = P i (t, s) + D(t, s) for i = 1, 2. We put P i (t, s) = 0 for s > t and i = 1, 2. Since µ and l are bounded, it follows from the above estimate that there exists a positive constant It follows that From this and (3.2), we obtain Therefore, Gronwall's lemma gives Thus the uniqueness follows.
Remark 1.We consider the case in which α = 0 and the state process X is the Ornstein-Uhlenbeck process satisfying where θ, σ = 0 and (W t , t ∈ [0, T ]) is a one-dimensional Brownian motion that is independent of B. We also assume that µ(•) = µ, a constant.Then By Theorem 3.1, we have where be the P-augmentation of the filtration generated by ((W t , B t ), t ∈ [0, T ]).Then P XX (t, s) is The equations (3.8) and (3.9) are the well-known Kalman-Bucy filtering equations (see Kalman and Bucy [13], Bucy and Joseph [4], Davis [5], Jazwinski [11] and Liptser and Shiryayev [21]).
3.2.Linear systems with memory.We turn to the partially observable system governed by (1.5).
s ds is However, by elementary calculation, we have Thus the lemma follows.
We put, for 0 ≤ t ≤ T , We also put We define the error matrix P (t) ∈ R 3×3 by Here is the solution to the optimal filtering problem for (1.5). Theorem Proof.For 0 ≤ s ≤ t ≤ T , we put Then we have P (t) = P (t, t).We also put, for 0 ≤ s ≤ t ≤ T , By and Lemma 3.3, we have P (t, s) = M (t − s)P (s), where and M (0) is the unit matrix.Thus we obtain From (3.14) and Theorem 3.1 with U = α (1) and α = α (2) , it follows that The SDE (3.12) follows easily from (3.15).
Remark 2. Suppose that, as in Subsection 3.1, the processes α (j) and V (j) , j = 1, 2, are defined by (3.10) with arbitrary Volterra-type bounded measurable functions l j (t, s) and Brownian motions B (j) .If we further assume that l j (t, s), j = 1, 2, are of the form l j (t, s) = e c j (t−s) g j (s), then we can easily extend Theorem 3.4 to this setting.Notice that, in this case, the noise processes V (1) and V (2) are not necessarily stationary increment processes.
Example 3.7.We consider the estimation problem of the value of a signal ρ from the observation process Y = (Y t , t ∈ [0, T ]) given by where V = (V t , t ∈ [0, T ]) and α = (α t , t ∈ [0, T ]) are as in Section 2. We assume that ρ is a Gaussian random variable with mean zero and variance v.This is the special case θ = σ = 0, µ = 1 of the setting of Corollary 3.6.Let r = p + q and l(•) be as above.Let H(t) and a be as in Corollary 3.6 with µ = 1 and θ = 0. We define where The analytical forms of ψ(•), φ(•), ξ(•) and η(•) can be derived.We omit the details.

Application to finance
In this section, we apply the results in the previous section to the problem of expected utility maximization for an investor with partial observations.
Let the processes t , t ∈ [0, T ]), j = 1, 2, be as in Section 3. In particular, V (1) and V (2) are independent.We consider the financial market consisting of a share of the money market with price S 0 t at time t ∈ [0, T ] and a stock with price S t at time t ∈ [0, T ].The stock price process S = (S t , t ∈ [0, T ]) is governed by the stochastic differential equation where s 0 and η are positive constants and U = (U t , t ∈ [0, T ]) is a Gaussian process following The parameters θ, δ and σ are real constants, and ρ is a Gaussian random variable that is independent of (V (1) , V (2) ).For simplicity, we assume that ) be the P-augmentation of the filtration generated by the process ((V t ), t ∈ [0, T ]) and the random variable ρ.Then U is F-adapted but not F S -adapted; recall from Section 2 that F S is the augmented filtration generated by the process S. Suppose that we can observe neither the disturbance process V (2) nor the drift process U but only the price process S. Thus only F S -adapted processes are observable.
We consider the following expected logarithmic utility maximization from terminal wealth: for given initial capital x ∈ (0, ∞), maximize E[log(X x,π T )] over all π ∈ A(x), ( where and The value π t is the dollar amount invested in the stock at time t, whence π t /S t is the number of units of stock held at time t.The process X x,π = (X x,π t , t ∈ [0, T ]) is the wealth process associated with the self-financing portfolio determined uniquely by π.
An analogue of the problem (4.3) for full observations is solved in Anh et al. [2].For related work, see Karatzas and Zhao [15], Lakner [19,20] and the references therein.We solve the problem (4.3) by combining the results above on filtering and the martingale method as described in Karatzas and Shreve [14].
Solving (4.1), we obtain where the process Y = (Y t , t ∈ [0, T ]) is given by From (4.5), we see that F S = F Y .We regard Y as the observation process.As in the previous sections, for a d-dimensional column vector process A = (A t , t ∈ [0, T ]), we write ) be the innovation process associated with Y that is given by The innovation process I is an F Y -Brownian motion satisfying F S = F Y = F I (see, e.g., Theorem 7.16 in Liptser and Shiryayev [21]).Let L = (L t , t ∈ [0, T ]) be the exponential F-martingale given by We find that, for t ∈ [0, T ], (see, e.g., Liptser and Shiryayev [21,Chapter 7]).The process L = ( L t , t ∈ [0, T ]) is an F Y -martingale.For x ∈ (0, ∞) and π ∈ A(x), we see from the Itô formula that the process ( L t X x,π t , t ∈ [0, T ]) is a local F Y -martingale, whence an F Ysupermartingale since X x,π is nonnegative.It follows that, for x ∈ (0, ∞), π ∈ A(x), and y ∈ (0, ∞), where, in the second inequality, we have used The equalities in (4.6) hold if and only if Thus the portfolio process π satisfying (4.7) is optimal.Put Since x/ L 0 = x and we see from (4.4) that the process π 0 satisfies (4.7), whence it is the desired optimal portfolio process.It also satisfies We put t , α (2) We define the error matrix Combining the results above with Theorem 3.4 which describes the dynamics of U and α (2) , we obtain the next theorem.

Parameter estimation
Let V = (V t , t ∈ [0, T ]) be the process given by (1.1).We can estimate the values of the parameters p and q there from given data of V by a least-squares approach (cf.Anh et al. [3]).In fact, since V is a stationary increment process, the variance of V t − V s is a function in t − s.To be precise, where the function U (t) = U (t; p, q) is given by Suppose that for t j = j∆t, j = 1, . . ., N, the value of V t j is v j .For simplicity, we assume that ∆t = 1.An unbiased estimate of U (t j ) is given by where m j is the mean of v i+j − v i 's: Fitting {U (t j ; p, q)} to {u j } by least squares, we obtain the desired estimated values of p and q.
For example, we generate a sample {v 1 , v 2 , . . ., v 1000 } for V with (p, q) = (0.5, 0.3) by a Monte Carlo simulation.We use this data to numerically calculate the values p 0 and q 0 of p and q, respectively, that minimize 30 j=1 (U (t j ; p, q) − u j ) 2 .
It turns out that p 0 = 0.5049 and q 0 = 0.2915.In Figure 5.1, we plot {U (t j ; p, q)} (theoretical), {u j } (sample) and {U (t j ; p 0 , q 0 )} (fitted).It is seen that the fitted curve follows the theoretical curve very well.We extend the approach above to that for the estimation of the parameters p, q, θ and σ in where θ, σ ∈ (0, ∞), the process V = (V t , t ∈ [0, T ]) is given by (1.1) as above, and the initial value ρ is independent of V and satisfies E[ρ 2 ] < ∞.The solution X = (X t , t ∈ [0, T ]) is the following Ornstein-Uhlenbecktype process with memory: Proposition 5.1.We have by a Monte Carlo simulation.We use this data to numerically calculate the values p 0 , q 0 , θ 0 and σ 0 of p, q, θ and σ, respectively, that minimize 30 j=1 (H(t j ; p, q, θ, σ) − h j (θ)) 2 .

Simulation
As we have seen, the process V = (V t , t ∈ [0, T ]) described by (1.1) has both stationary increments and a simple semimartingale representation as Brownian motion does, and it reduces to Brownian motion when p = 0.In this sense, we may see V as a generalized Brownian motion.Since V is non-Markovian unless p = 0, we have now a wide choice for designing models driven by either white or colored noise.
We see that there are clear differences between the two filters in the cases Θ 2 and Θ 4 .We notice that, in these two cases, p 1 is large than the parameters p 2 and q 2 .In Figure 6.1, we compare the graphs of AE(•) for the two filters in the case Θ = Θ 2 .It is that the error of the optimal filter is consistently smaller than that of the Kalman-Bucy filter.

Possible extension
The two-parameter family of processes V described by (1.1) are those with short memory.A natural problem is to extend the results in the present paper to a more general setting where, as in Anh and Inoue [1], the driving noise process V = (V t , t ∈ R) is a continuous Gaussian process with stationary increments satisfying V 0 = 0 and one of the following continuous-time AR(∞)-type equations: where W = (W t , t ∈ R) is a Brownian motion and dV t /dt and dW t /dt are the derivatives of V and W , respectively, in the random distribution sense.The kernel a(•) is a nonnegative decreasing function that satisfies some suitable conditions.More precisely, using the notation of Anh and Inoue [1], we assume that a(•) satisfies (S2) for (7.1), or either (S1) or (L)  t ∈ R, (7.4) for (7.2).For example, if a(t) = pe −qt for t > 0 with p, q > 0 in (7.1), then the kernel c(•) in (7.3) is given by c(t) = pe −(p+q)t for t > 0, and (7.3) reduces to (1.1) for these p and q.
For the stationary increment process V in the short memory case (7.1) with (S2) or (7.2) with (S1), we can still derive a representation of the form (2.2) with (2.4), in which the kernel l(t, s) is given by an infinite series made up of c(•) and a(•).In the long memory case (7.2) with (L), it is also possible to derive the same type of representation for V if we assume the additional condition a(t) ∼ t −(p+1) (t)p, t → ∞, (7.5) or c(t) ∼ t −(1−p) (t) • sin(pπ) π , t→ ∞, (7.6)where 0 < p < 1/2 and (•) is a slowly varying function at infinity.Notice that (7.5) and (7.6) are equivalent (see Lemma 2.8 in Anh and Inoue [1]).It is expected that results analogous to those of the present paper hold, especially, for V with long memory, using the representation (2.2) with (2.4) thus obtained.This work will be reported elsewhere.

Figure 5 . 1 .
Figure 5.1.Plots of the function v(t), the sample data and the fitted function v 0 (t).

Figure 5 . 2 .
Figure 5.2.Plots of the function h(t), the sample data h(t) with estimated θ and the fitted function h 0 (t).

Figure 6 . 1 .
Figure 6.1.Plots of AE(•) for the optimal and Kalman-Bucy filters with noise parameter Θ = Θ 2 The matrix P (t, t) is clearly symmetric and nonnegative definite.Finally, the uniqueness of the solution to (3.2) follows from Proposition 3.2 below.