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´ owna 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 X t , 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). A X is a family of V -valued processes Y t , 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 | Y t − Z t | 0 ≤ t ≤ T and n � � . � � TV( Y , [0 , T ]) := sup sup � Y t i − Y t i − 1 n 0 ≤ t 0 < t 1 < ··· < t n ≤ T i =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) � � � � V X ( ψ, θ ) := E inf ψ � X − Y � ∞ , [0 , T ] + θ · TV( Y , [0 , T ]) . (1) Y ∈A X To make the problem non-trivial we assume that E | X 1 | < + ∞ . 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 ⇒ | Y t − Y s | ≥ max {| X t − X s | − c , 0 } for any 0 ≤ s ≤ t ≤ T . Thus TV( Y , [0 , T ]) ≥ TV c ( X , [0 , T ]) n � := sup sup max {| X t − X s | − c , 0 } . (2) n 0 ≤ t 0 < t 1 < ··· < t n ≤ T i =1 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 ∈ A X such that � X − X c � ∞ , [0 , T ] ≤ c / 2 and TV c ( X , [0 , T ]) ≤ TV( X c , [0 , T ]) ≤ TV c ( X , [0 , T ]) + c , (3) thus in the case V = R we have the estimate � c � � � + θ · E TV c ( X , [0 , T ]) inf ψ 2 c > 0 � � � � ≤ E inf ψ � X − Y � ∞ , [0 , T ] + θ · TV( Y , [0 , T ]) Y ∈A X � c � � � + θ · E TV c ( X , [0 , T ]) + θ c ≤ inf ψ , 2 c > 0 which means that if ψ ( x ) grows no faster than some polynomial and no slower than a linear function then both quantities inf c > 0 { ψ ( c / 2) + θ · E TV c ( X , [0 , T ]) } and V X ( 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 TV c ( X , [0 , T ]) is still not easy one to calculate/estimate. In [LochowskiBPAS] the following estimates of E TV c ( X , [0 , T ]) were given for standard Brownian motion with drift W t = B t + µ t : √ for T such that T ≥ χ ( c , µ ) , 1 � 1 � � 1 � T ≤ E TV c ( W , [0; T ]) ≤ 64 c + | µ | c + | µ | T ; 264 √ for T such that c − | µ | T ≤ T < χ ( c , µ ) , √ √ 1 � � ≤ E TV c ( W , [0; T ]) ≤ 340 � � 2 T + | µ | T − c 2 T + | µ | T − c 747 √ and for T such that T < c − | µ | T , 227 T 3 / 2 e − ( c −| µ | T ) 2 / (2 T ) ≤ E TV c ( W , [0; T ]) ≤ 493 · T 3 / 2 e − ( c −| µ | T ) 2 / (2 T ) 1 . ( c − | µ | T ) 2 ( 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 R 2 ) 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 ∈ A X such that � X − Y c � ∞ , [0 , T ] ≤ c / 2 and E TV c ( X , [0 , T ]) ≤ E TV ( Y c , [0 , T ]) λ> 1 λ · E TV ( λ − 1) · c / (2 λ ) ( X , [0 , T ]) . ≤ inf 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 (2 a ) ≤ K ψ · ψ ( a ) , we get � c � � � + θ · E TV c ( X , [0 , T ]) inf ψ 2 c > 0 � c � � � + θ · E TV( Y c , [0 , T ]) ≤ inf ψ 2 c > 0 � c � � � K 2 � � + θ · E TV c ( X , [0 , T ]) ≤ max ψ , 2 inf ψ (4) 2 c > 0 � c 2 + θ · E TV c ( X , [0 , T ]) � thus again we see that both quantities inf c > 0 and V X ( ψ, θ ) are comparable up to universal constants (depending on ψ only). 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 , . . . � � � � � � > c t > τ c if τ c inf n − 1 : � X τ c n − 1 − X t n − 1 < + ∞ ; � � τ c 2 n = if τ c + ∞ n − 1 = + ∞ and then we define + ∞ � Y c t = n +1 ) ( t ) . X τ c n 1 [ τ c n ; τ c n =0 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 (2 a ) ≤ K ψ · ψ ( a ) . For any T , θ > 0 the following estimates hold: � � � � E inf ψ � X − Y � ∞ , [0 , T ] + θ · TV ( Y , [0 , T ]) Y ∈A X � � � | X τ c − X 0 | 1 { τ c ≤ T } � � c + θ · e E � ≤ inf ψ , 1 − E exp ( − τ c / T ) 2 c > 0 � � � � E inf ψ � X − Y � ∞ , [0 , T ] + θ · TV ( Y , [0 , T ]) Y ∈A X � � � | X τ c − X 0 | 1 { τ c ≤ T } � 1 � c + θ · e − 1 E � ≥ � inf ψ , � 1 − E exp ( − τ c / T ) 2 2 e K 2 c > 0 max ψ , 2 where τ c = inf { t > 0 : | X t − X 0 | > 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 or a little bit refined reasoning we get       √  �� c  �   �  1 − Φ 2 − | µ | T /  T   � c  �  V X ( ψ, θ ) = κ 2 inf ψ + θ · c , 2 � � c > 0 � 2 cosh ( c µ 2 2 ) sinh c T + µ 2   2    1 −    � �   �  2  T + µ 2 sinh c   � � and Φ ( x ) = (2 π ) − 1 / 2 � x −∞ e − t 2 / 2 d t . e − 1 where κ 2 ∈ ψ , 2 ) , 4 e 14 e max ( K 2 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 R n j ν − 1   J ν +1 ( j ν, k ) e − 2 j 2 ν, k T / c 2 2 1 − ν � + ∞ ν, k 1 −  k =1  � c + θ · c Γ( ν +1)   � V X ( ψ, θ ) = κ 3 inf ψ , ( c / 2) ν 2 2 c > 0 1 − c √   � �  Γ( ν +1)(2 T ) ν/ 2 I ν  1 / (2 T ) � � e − 1 where κ 3 ∈ ψ , 2 ) , e , J ν denotes the Bessel function of the first 2 e max ( K 2 kind, � ν + ∞ ( − 1) m � y � y � 2 m � J ν ( y ) = 2 m !Γ ( m + ν + 1) 2 m =0 and 0 < j ν, 1 < j ν, 2 < . . . denote all positive zeros of the function J ν , and I ν denotes the modified Bessel function � ν + ∞ � y 1 � y � 2 m � I ν ( y ) = . 2 m !Γ ( m + ν + 1) 2 m =0 Rafa� l Marcin � Lochowski (SGH) Optimal uniform approximation Probablity and Analysis 2017 12 / 17

The case of a standard Brownian motion on R n - 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