Optimal uniform approximation of L evy processes on Banach spaces - - PowerPoint PPT Presentation

Optimal uniform approximation of L´ evy processes on Banach spaces with finite variation processes

Rafa l Marcin Lochowski

Szko la G l´

• wna Handlowa (Warsaw School of Economics)

Probablity and Analysis 2017

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 1 / 17

Optimisation problem, setting

Xt, t ≥ 0, is a c` adl` ag L´ evy process attaining its values in a Banach space V (i.e. a process with a.s. c` adl` ag paths and independent and stationary increments). AX is a family of V -valued processes Yt, t ≥ 0, adapted to the natural filtration of X. |·| denotes the norm in V and for T > 0 and two processes Y , Z : Ω × T → V , where T is an index set such that [0, T] ⊂ T , we denote Y − Z∞,[0,T] := sup

0≤t≤T

|Yt − Zt| and TV(Y , [0, T]) := sup

n

sup

0≤t0<t1<···<tn≤T n

• i=1
• Yti − Yti−1
• .

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 2 / 17

Optimisation problem, formulation

We will deal with the following optimisation problem. Given are T, θ > 0 and non-decreasing function ψ : [0, +∞) → [0, +∞) calculate (or estimate up to universal constants) VX (ψ, θ) := E inf

Y ∈AX

• ψ
• X − Y ∞,[0,T]
• + θ · TV(Y , [0, T])
• .

(1) To make the problem non-trivial we assume that E |X1| < +∞.

Remark

This type of optimisation problems appears naturally in several stituations. However, it has no unified, algorithmic solution since the generator of the total variation functional is not well defined. Moreover, we deal with very general L´ evy processes attaining their values in general Banach spaces.

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 3 / 17

Optimisation problem - first observations

From the triangle inequality we immediately get that X − Y ∞,[0,T] ≤ c/2 ⇒ |Yt − Ys| ≥ max {|Xt − Xs| − c, 0} for any 0 ≤ s ≤ t ≤ T. Thus TV(Y , [0, T]) ≥ TVc(X, [0, T]) := sup

n

sup

0≤t0<t1<···<tn≤T n

• i=1

max {|Xt − Xs| − c, 0} . (2) The quantity on the right side of (2) is called the truncated variation of X.

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 4 / 17

Optimisation problem - alternative statement

From the results of [LochowskiMilosSPA], [LochowskiGhomrasniMMAS] it is possible to prove that for any c > 0 there exists a process X c ∈ AX such that X − X c∞,[0,T] ≤ c/2 and TVc(X, [0, T]) ≤ TV(X c, [0, T]) ≤ TVc(X, [0, T]) + c, (3) thus in the case V = R we have the estimate inf

c>0

• ψ

c 2

• + θ · ETVc(X, [0, T])
• ≤ E inf

Y ∈AX

• ψ
• X − Y ∞,[0,T]
• + θ · TV(Y , [0, T])
• ≤ inf

c>0

• ψ

c 2

• + θ · ETVc(X, [0, T]) + θc
• ,

which means that if ψ (x) grows no faster than some polynomial and no slower than a linear function then both quantities infc>0 {ψ (c/2) + θ · ETVc(X, [0, T])} and VX (ψ, θ) are comparable up to universal constants depending on θ and ψ only.

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 5 / 17

Optimisation problem - some results for real Brownian motion with drift

Unfortunately, the quantity TVc(X, [0, T]) is still not easy one to calculate/estimate. In [LochowskiBPAS] the following estimates of ETVc(X, [0, T]) were given for standard Brownian motion with drift Wt = Bt + µt : for T such that √ T ≥ χ (c, µ) , 1 264 1 c + |µ|

• T ≤ ETV c (W , [0; T]) ≤ 64

1 c + |µ|

• T;

for T such that c − |µ| T ≤ √ T < χ (c, µ) , 1 747

• 2

√ T + |µ| T − c

• ≤ ETV c (W , [0; T]) ≤ 340
• 2

√ T + |µ| T − c

• and for T such that

√ T < c − |µ| T, 1 227T 3/2 e−(c−|µ|T)2/(2T) (c − |µ| T)2 ≤ ETV c (W , [0; T]) ≤ 493·T 3/2 e−(c−|µ|T)2/(2T) (c − |µ| T)2 .

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 6 / 17

The problem gets even worse in the Banach space setting

The problem gets even worse for more general L´ evy processes. Moreover, in the Banach space setting (even in R2) the estimate (3) is no longer valid. Fortunately, we have the following (not difficut to obtain) result:

Theorem (Banach space estimate)

For any c > 0 and any regulated process X there exists a process Y c ∈ AX such that X − Y c∞,[0,T] ≤ c/2 and ETVc(X, [0, T]) ≤ ETV(Y c, [0, T]) ≤ inf

λ>1 λ · ETV(λ−1)·c/(2λ)(X, [0, T]) .

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 7 / 17

Banach space estimate

From the Banach space estimate theorem and assuming that for any a ≥ 0, ψ (2a) ≤ Kψ · ψ (a) , we get inf

c>0

• ψ

c 2

• + θ · ETVc(X, [0, T])
• ≤ inf

c>0

• ψ

c 2

• + θ · ETV(Y c, [0, T])
• ≤ max
• K 2

ψ, 2

• inf

c>0

• ψ

c 2

• + θ · ETVc(X, [0, T])
• (4)

thus again we see that both quantities infc>0 c

2 + θ · ETVc(X, [0, T])

• and VX (ψ, θ) are comparable up to universal constants (depending on ψ
• nly).

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 8 / 17

Banach space estimate - construction

In the case when X has c` adl` ag trajectories, the construction of the process Y c simplifies to the following one. First, we define stopping times τ c

0 = 0

and for n = 1, 2, . . . τ c

n =

• inf
• t > τ c

n−1 :

• Xτ c

n−1 − Xt

• > c

2

• if τ c

n−1 < +∞;

+∞ if τ c

n−1 = +∞

and then we define Y c

t = +∞

• n=0

Xτ c

n 1[τ c n ;τ c n+1) (t) . Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 9 / 17

Banach space estimate expressed in simple quantities

Theorem

Let ψ : [0, +∞) → [0, +∞) be a non-decreasing function such that for a ≥ 0, ψ (2a) ≤ Kψ · ψ (a) . For any T, θ > 0 the following estimates hold: E inf

Y ∈AX

• ψ
• X − Y ∞,[0,T]
• + θ · TV(Y , [0, T])
• ≤ inf

c>0

• ψ

c 2

• + θ · e E
• |Xτ c − X0| 1{τ c≤T}
• 1 − E exp (−τ c/T)
• ,

E inf

Y ∈AX

• ψ
• X − Y ∞,[0,T]
• + θ · TV(Y , [0, T])
• ≥

1 max

• K 2

ψ, 2

inf

c>0

• ψ

c 2

• + θ · e − 1

2e E

• |Xτ c − X0| 1{τ c≤T}
• 1 − E exp (−τ c/T)
• ,

where τ c = inf {t > 0 : |Xt − X0| > c/2} .

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 10 / 17

Some special cases - the case of a Brownian motion with drift

The easiest case is the case of the standard Brownian. Slightly more complicated is the case of the Brownian motion with drift. Using the already presented estimates for the Brownian motion with drift

• r a little bit refined reasoning we get

VX (ψ, θ) = κ2 inf

c>0

               ψ c 2

• + θ · c

1 − Φ c

2 − |µ| T

• /

√ T

• 1 −

2 cosh( cµ

2 ) sinh

• c

2

• 2

T +µ2

• sinh
• c
• 2

T +µ2

• 

              , where κ2 ∈

• e−1

14e max(K 2

ψ,2), 4e

• and Φ (x) = (2π)−1/2 x

−∞ e−t2/2dt.

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 11 / 17

Some special cases - the case of a standard Brownian motion on Rn

VX (ψ, θ) = κ3 inf

c>0

     ψ c 2

• + θ · c

2 1 −

21−ν Γ(ν+1)

+∞

k=1 jν−1

ν,k

Jν+1(jν,k)e−2j2

ν,kT/c2

1 −

(c/2)ν Γ(ν+1)(2T)ν/2Iν

• c√

1/(2T)

• 

    , where κ3 ∈

• e−1

2e max(K 2

ψ,2), e

• , Jν denotes the Bessel function of the first

kind, Jν (y) = y 2 ν +∞

• m=0

(−1)m m!Γ (m + ν + 1) y 2 2m and 0 < jν,1 < jν,2 < . . . denote all positive zeros of the function Jν, and Iν denotes the modified Bessel function Iν (y) = y 2 ν +∞

• m=0

1 m!Γ (m + ν + 1) y 2 2m .

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 12 / 17

The case of a standard Brownian motion on Rn - remarks

The formula presented on the previous slide is really hard to apply. However, if we fix n or allow to depend the accuracy of our estimate on n then one may apply the results of Grzegorz Serafin proved his recent paper Exit times densities of the Bessel process, Proc. Amer. Math. Soc., to appear, DOI: 10.1090/proc/13419.

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 13 / 17

Special case - real, discontinuous L´ evy processes

In this case we have discontinuities and thus we need to take into account

• vershoots while estimating

E

• |Xτ c − X0| 1{τ c≤T}
• .

We have the following Quintuple Law at First passage of Doney and Kyprianou: P

• τ c − ¯

Gτ c− ∈ dt, ¯ Gτ c− ∈ ds, Xτ c − c ∈ du, x − Xτ c− ∈ dv, x − ¯ Xτ c− ∈ dy

• = U(ds, x − dy) ˆ

U(dt, dv − y)Π(du + v), where ¯ Gτ c− is the time of the last maximum prior to first passage, τ c − ¯ Gτ c− is the length of the excursion making first passage, Xτ c − c is the overshoot at first passage, x − Xτ c− is the undershoot at first passage, x − ¯ Xτ c− is the undershoot of the last maximum at first passage.

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 14 / 17

Special case - real, discontinuous L´ evy processes, cont.

Finally, U and ˆ U are bivariate potential measures of the ascending ladder process (L−1, H) and the decending ladder process (ˆ L−1, ˆ H) respectively: U(ds, dx) = ∞ dtP

• L−1 ∈ ds, H ∈ dx
• ˆ

U(ds, dx) = ∞ dtP

• ˆ

L−1 ∈ ds, ˆ H ∈ dx

• .

Unfortunately, these quantities are computable for very special families of processes (even in the α-stable case).

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 15 / 17

Some references

Doney, R.A. and Kyprianou, A.E. Overshoots and undershoots of Lvy

• processes. Ann. Appl. Probab. 16, 91-106.

[LochowskiMilosSPA] On truncated variation, upward truncated variation and downward truncated variation for diffusions. Stochastic Process. Appl., 123(2):446-474, 2013. [LochowskiGhomrasniMMAS] The play operator, the truncated variation and the generalisation of the Jordan decomposition. Math. Methods Appl. Sci., 38(3):403-419, 2015. [LochowskiBPAS] On truncated variation of Brownian motion with drift.

• Bull. Pol. Acad. Sci. Math., 56(3-4):267-281, 2008.
• R. M.
• Lochowski. Riemann-Stieltjes integrals driven by irregular signals in

Banach spaces and rate-independent characteristics of their irregularity. Preprint arXiv:1602.02269v1, 2016.

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 16 / 17

Thank you for your attention!

Rafa l Marcin Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 17 / 17