ON OPTIMAL UNIFORM APPROXIMATION OF L´EVY PROCESSES ON BANACH SPACES WITH FINITE VARIATION PROCESSES

. For a general c`adl`ag L´evy process X on a separable Banach space V we estimate values of inf c ≥ 0 (cid:8) ψ ( c ) + inf Y ∈A X ( c ) E TV( Y, [0 , T ]) (cid:9) , where A X ( c ) is the family of processes on V adapted to the natural ﬁltration of X , a

By A X (c) we denote the family of V -valued processes Y t , t ≥ 0, adapted to the natural filtration of X and such that Y − X ∞,[0,T ] ≤ c a.s. Also that is for ω ∈ Ω, TV(Y (ω), [0, T ]) is the total variation of the trajectory Y (ω) on the interval [0, T ].
In this paper we deal with the following optimisation problem. Given are T > 0 and a non-decreasing function ψ : [0, +∞) → [0, +∞) calculate (or estimate up to universal constants) (1.1) Thus we want to find a process with possibly small total variation, adapted to the natural filtration of X and whose trajectories a.s. uniformly approximate trajectories of X with given accuracy c, but the worse accuracy of the approximation is, the bigger penalty ψ(c) we pay. Interestingly, V X (ψ) may be finite even if the total variation of X is infinite. Lower and upper bounds for V X (ψ) are given in Theorem 2.6. For finite-dimensional processes finiteness of these bounds is equivalent with E |X 1 − X 0 | < +∞, see for example Theorem 25.3 of [11]. This type of optimization problems appear naturally in several situations. For example, in financial models with small proportional transaction costs where X is the process representing optimal investment strategy on (frictionless) market without transaction costs, while Y is the approximation of X and the total variation of Y is proportional to the transactions costs of the implementation of the strategy Y , see for example [6,7]. This type of optimization problems have no unified, algorithmic solution since the generator of the total variation functional is not well defined. Moreover, we deal with very general Lévy processes taking their values in general Banach spaces.
Another problems when approximation by functions with possible small total variation is used, is image denoising, see for example [4]. Here however one deals with more complicated domains (usually R d ) and different norms measuring the quality of the approximation (like L 1 ).
Using well known results of the renewal theory, some ad-hoc reasoning, results obtained for the functional called truncated variation and assuming that ψ grows no faster than some polynomial, we will be able to estimate (1.1) up to universal constants, depending on ψ and some simpler quantities describing of the process X (like its Lévy measure and laws of its exit times from the balls centered at X 0 ). Together with the estimates we will provide the construction of the process Z uniformly approximating X, for which these estimates hold.
From the triangle inequality we immediately get (1. 2) The quantity in the last line of (1.2) is called the truncated variation of X. In the case when V = R, from the results of Remark 15 in [10] it is possible to prove that for any c > 0 there exists a process X c ∈ A X (c) such that X − X c ∞,[0,T ] ≤ c and which means that if ψ grows no faster than some polynomial (and no slower than some increasing linear function) then inf c>0 ψ (c) + ETV 2c (X, [0, T ]) and V X (ψ) are comparable up to universal constants depending on ψ only. For a general Banach space-valued Lévy process, using similar construction as in the proof of Theorem 1 from [9], we get that there exists a process From this, assuming that there exists a constant K ψ such that for any thus again we see that both quantities: inf c>0 ψ (c) + ETV 2c (X, [0, T ]) and V X (ψ) are comparable up to universal constants (depending on ψ only). Since X has càdlàg trajectories, the construction of the process Y c appearing in (1.3) and (1.4) simplifies to the following one. First, we define stopping times τ c 0 = 0 and for n = 1, 2, . . .
Remark 1.1. The construction in the proof of Theorem 1 in [9] rather uses timesτ c n defined in the following wayτ c n = inf t >τ c n−1 : which may be not stopping times, but it is straightforward to verify that for the times defined by (1.5) and Y c defined by (1.6) the estimates (1.4) hold as well (see the proof of [9], Thm. 1).
In what follows, we will use the presented construction to obtain more straightforward estimates of V X (ψ) in terms of the characteristics of the process X.
This paper is organised as follows. In the next section we prove useful estimates of ETV(Y c , [0, T ]), where Y c is the process defined by equation (1.6), and then prove two universal estimates of V X (ψ) (Thms. 2.5 and 2.6) expressed in terms of simpler functionals of X. In the last, third section, we apply obtained estimates in three specific cases, namely when: (1) X is a Brownian motion with drift on R, (2) X is a standard Brownian motion on R d and (3) X is a symmetric α-stable process (α ∈ (1, 2)) on R.

Estimation of
where stopping times τ c n , n = 0, 1, . . . , are defined by formula (1.5). Lemma 2.1. For the process Y c defined by equation (1.6) the following inequalities hold: Proof. Let us first notice that where stopping times τ c n , n = 0, 1, . . . , are defined by formula (1.5). From this, using independence of increments of A and the strong Markov property we get an upper bound for ETV(Y c , [0, T ]) , which reads We will use the notion of stochastic domination. We say that a real random variable Q stochastically dominates a real random variable P if for any x ∈ R, P (Q ≥ x) ≥ P (R ≥ x) . We denote this by Q P . We have that The domination in (2.3) follows from the strong Markov property of X since, by the stationarity of the increments of X, the variable Taking expectations of both sides of relations (2.2) and (2.3) and adding them we get where we used the fact that for n > σ c (T ), 1 {τ c n−1 ≤T } ≡ 0 Let us denote τ c = τ c 1 . Remark 2.2. The function U c (T ) is an example of a renewal function, a well known object in the renewal theory. Elementary renewal theorem states that where in the case Eτ c = +∞ we set 1/Eτ c = 0.
Remark 2.3. We have the following estimates which are special cases of results obtained by Erickson in [5]: This gives in the case Eτ c = +∞ the proper order of growth of the renewal function.
Sometimes (and this is often the case when one deals with Lévy processes) it is easier to deal with the Laplace transform of τ c than with the function U c (T ).
Proof. Both estimates follow from Lemma 2.1 and elementary estimates of U c (T ). The estimate from above follows from the estimate From (2.6) and (2.7) we get estimate (2.4).
To bound ETV(Y c , [0, T ]) from below for t > 0 we define σ c,0 (t) = 0 and for k = 1, 2, . . . , such that τ c σ c,k−1 (t) < +∞ let σ c,k (t) be the smallest integer such that τ c σ c, Theorem 2.5. Let X t , t ≥ 0, be a Lévy process on a separable Banach space V with the norm |·| and let A X be the class of processes adapted to the natural filtration of X. Let ψ : [0, +∞) → [0, +∞) be a non-decreasing function such that for a ≥ 0, ψ (2a) ≤ K ψ · ψ (a) . For any T > 0 the following estimates hold: In what follows we will estimate E |X τ c − X 0 | 1 {τ c ≤T } to obtain the following theorem.
Theorem 2.6. Let X t , t ≥ 0, be a Lévy process on a separable Banach space V with the norm |·| and let A X be the class of processes adapted to the natural filtration of X. Let ψ : [0, +∞) → [0, +∞) be a non-decreasing function such that for a ≥ 0, ψ (2a) ≤ K ψ · ψ (a) . For any T, > 0 the following estimates hold: and where τ c = inf {t > 0 : |X t − X 0 | ≥ c} and Π is the image of the Lévy measure of the process X under the transformation x → |x|.
Proof. Let ∆X τ c = X τ c − X τ c − denote the jump of the process at the moment τ c and let us notice that by the triangle inequality and the definition of τ c , for τ c < +∞ we have Thus for τ c < +∞ it follows that (2.14) Let now µ be the joint law of (|∆X τ c | , τ c ). For y ∈ (c, +∞) and t ∈ (0, +∞) one has where dt denotes the Lebesgue measure. This observation follows from the fact that for y ∈ (2c, +∞) and t ∈ (0, +∞) the event {|∆X τ c | ∈ [y, y + dy) , τ c ∈ [t, t + dt)} is equal the intersection of two independent events sup 0≤s<t |X s − X 0 | < c and {|X t+dt − X t− | ∈ [y, y + dy)} which follows from (2.14) and the Lévy-Ito decomposition (see [1]). Now, using (2.14) we easily estimate We naturally also have On the other hand, by the definition of τ c , for τ c < +∞ we have To deal with the integral (0,T ] P (τ c ≥ t) dt let us notice that the following estimates hold: Thus, we have the double-sided estimate Finally, let us notice that (by integration by parts)

Examples
In this section we will apply the obtained estimates in three special cases. In the first case the process X will be a real-valued Brownian motion with drift, in the second case it will be a standard Brownian motion on R d , d = 2, 3, . . . , and in the third case it will be a real valued, symmetric α-stable process with α ∈ (1, 2).

Estimates of V X (ψ) in the case when X is a Brownian motion with drift
Let now B be a standard Brownian motion starting from 0 and X t = B t + µt be a (real-valued) Brownian motion with drift µ. From Theorem 2.5 it follows that in order to estimate V X (ψ) it is sufficient to estimate (up to universal constants) the quantity E |X τ c | 1 {τ c ≤T } / 1 − E2 −τ c /T . From the continuity of Brownian paths we immediately get that |X τ c | = c and Now let us consider two cases.
−∞ e −t 2 /2 dt is the cumulative probability function of a standard normal variable. Case 2. c > √ T + |µ| T. In this case we get the following lower bound On the other hand, in both cases we have
Again, by Theorem 2.5 it is sufficient to estimate the ratio and again, by the continuity of X, we get Moreover, recall that the process R defined by is called d-dimensional Bessel process or a Bessel process of order d or a Bessel process with index ν = d/2 − 1.
3.3. Estimates of V X (ψ) in the case when X is a symmetric, real-valued, strictly α-stable motion (α ∈ (1, 2)) Let now X t , t ≥ 0, be a symmetric, real-valued, strictly α-stable motion (α ∈ (1, 2)) such that X 0 = 0. X has the following scaling property: for t ≥ 0 and a > 0 To fix our attention to the process of a given magnitude, we will assume that X 1 has the following characteristic function . Let β α be such that To estimate V X (ψ) we will apply Theorem 2.6. First we need to estimate 1 − E2 −τ c /T = ln 2 T +∞ 0 2 −t/T P (τ c ≥ t) dt. Let us define the function (0, +∞) c → u (c) ∈ (0, +∞) such that P X u(c) ≥ c = 1 3e 5 .