Lifting Lévy processes to hyperfinite random walks

Bull. Sci. math. 130 (2006) 697–706 www.elsevier.com/locate/bulsci

Lifting Lévy processes to hyperfinite random walks Sergio Albeverio a , Frederik S. Herzberg b,∗ a Abteilung für Stochastik, Institut für Angewandte Mathematik der Universität Bonn, Wegelerstraße 6, D-53115 Bonn,

Germany; BiBoS (Bielefeld-Bonn-Stochastics); SFB 611; IZKS; CERFIM (Locarno); Acc. Arch. (USI, Mendrisio) b Abteilung für Stochastik, Institut für Angewandte Mathematik der Universität Bonn, Wegelerstraße 6, D-53115 Bonn, Germany Received 20 January 2006 Available online 9 March 2006

Abstract An internal lifting for an arbitrary measurable Lévy process is constructed. This lifting reflects our intuitive notion of a process which is the infinitesimal sum of its infinitesimal increments, those in turn being independent from and closely related to each other – for short, the process can be regarded as some kind of random walk (where the step size generically will vary). The proof uses the existence of càdlàg modifications of Lévy processes and certain features of hyperfinite adapted probability spaces, commonly known as the “model theory of stochastic processes”. © 2006 Elsevier Masson SAS. All rights reserved. MSC: primary 28E05, 60J30; secondary 03H05, 60H05 Keywords: Lévy processes; Liftings; Saturated models

1. Introduction Lévy processes have attracted increasing interest in recent years, partially as suitable generalisations of the Brownian motion for purposes of mathematical modelling, but also because they are a promising object of investigation in their own right: There are various classical representation theorems for Lévy processes, such as the Lévy–Khintchine formula, and e.g. local times, exit/entry times and infinitesimal generators have been studied extensively (cf. [4,15] for compilations of classical and recent results on Lévy processes). In spite of this, there has been, * Corresponding author.

E-mail address: [email protected] (F.S. Herzberg). 0007-4497/$ – see front matter © 2006 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.bulsci.2006.02.001

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until very recently, little effort to develop a nonstandard theory of Lévy processes. The crucial step in this enterprise is of course the construction of ‘canonical’ well-behaved liftings for Lévy processes. Thus motivated, we will show that every measurable Lévy process on a hyperfinite adapted probability space has a reasonably regular lifting that resembles a hyperfinite random walk, in having stationary independent infinitesimal increments in a weak sense that will be made precise. This readily yields a result on all measurable Lévy processes (on arbitrary adapted probability spaces) because of the universality of hyperfinite adapted spaces (which in turn is a result of Keisler’s [9]). The model theory of stochastic processes, systematically developed by Fajardo and Keisler [6] on the basis of work by Hoover and Keisler [7,9] is moreover useful to pursue the ℵ1 -saturation argument required for the construction of the mentioned lifting for a measurable Lévy process on a hyperfinite adapted space. The regularity of the lifting can be ensured by resorting to Stroyan and Bayod’s lifting theorems for càdlàg processes, i.e. stochastic processes whose paths are almost surely right-continuous with left limits, while setting up the ℵ1 -saturation argument. Considering the large number of applications that Anderson’s related result on the Brownian motion [3] has entailed, one might hope to improve one’s understanding of Lévy processes by regarding them as hyperfinite random walks as suggested in this talk. In particular, one might expect applications to local-time formulae and stochastic differential equations (such as Perkins’ results on Brownian local time and Hoover and Perkins’ investigations into stochastic differential equations [8,13,14]. Some time after our first manuscript on this topic appeared as a preprint [2], Tom Lindstrøm completed a paper [10], in which the notion of a “hyperfinite Lévy process”, which is a significantly less technical concept than the one that would have to be introduced to find a name for our nonstandard representation of measurable Lévy process in Theorem 2.2 was defined and a oneto-one correspondence with standard Lévy processes proven – his work uses the jump-diffusion decomposition of Lévy processes. Unfortunately it is not immediate clear how to achieve regularity results on Lindstrøm’s lifting beyond them being right liftings, therefore the decision of whether to employ Lindstrøm’s lifting or ours will depend on the specific application one has in mind. 2. Main results and proofs: Nonstandard representation of an arbitrary measurable Lévy process Now we are coming to the crucial part of this paper. Probably the most important one is Theorem 2.2, since it allows for internal representations of stochastic integrals with respect to measurable Lévy processes, as we will see in the last paragraph of this article. However, as the result can be generalised, we shall proceed from the general to the special case and give the proof of Theorem 2.2 after the proof of Theorem 2.1. Additionally, these theorems also provide mathematical formulations of our informal statement in the introduction. For this whole chapter, we would like to emphasise that it is possible to apply the same techniques that we are using to processes in arbitrary separable Banach spaces instead of R. 2.1. The case of a general increment construction Definition 2.1. Let x be a Lévy process, adapted to an adapted probability space Ω, starting at zero, i.e. x0 = 0 a.s. An increment construction in distribution is a sequence ((x (0,n) , . . . , x (n,n) ):

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n ∈ N) of adapted processes where for each n the components of the nth part of the sequence have the same finite-dimensional distribution such that one has n  x (i,n) ≡0 x ∀n ∈ N i=1

and whenever m divides n, n

∀k  m ∀t ∈ [0, 1]

km 

(i,n)

xt

=

i=1

as well as for all n ∈ N, t  0,

k 

(i,m)

xt

,

i=1

(0,n) xt

= 0 a.s. and

∀n ∈ N ∀i  n x (i,n) ≡0 x·/n One can easily think of simple examples other than the usual increment construction. Assume for this purpose that x is a stochastic integral with the integrating process being a Brownian motion b. Choose some Lebesgue-measurable J ⊂ [0, +∞) and define ct := bt χJ (t) − bt (1 − χJ (t)) for all t  0. Now define another process y exactly the same way as x, except that b will be replaced by c as integrating process. Then the usual increments of y, defined by ∀i ∈ {1, . . . , n} ∀t ∈ [0, 1]

(i,n)

yt

:= yti/n − yt (i−1)/n ,

form a general increment construction for x. Theorem 2.1. Let x˜ be an arbitrary adapted measurable Lévy process on an adapted probability space (Γ, (Ct )t∈[0,1] , P ) and (Ω, (Ft )t∈[0,1] , L(μ)) be a hyperfinite adapted probability space of mesh H !. Assume P -a.s. x˜0 = 0 and that the family ((x˜ (0,n) , . . . , x˜ (n,n) ): n ∈ N) is a generalised increment construction. Then we can find: a process x ≡ x˜ on Ω with a lifting X¯ : Ω × Th → ∗ R on Ω × Th for some hyperfinite infinite h  H and a H1 ! -càdlàg lifting X on (1,h!) (h!,h!) , . . . , X1 and an internal bijective adapted autoΩ × T; internal random variables X1 ~ (h!) morphism χ : Ω ↔ Ω with the property ∀i, j  h! ∀e, g ∈ Q ∃M, N ∈ ∗ N \ N       1 1 1 1 (j,h!) (i,h!) = L(μ) X1 ∈ e − ,g + ∈ e − ,g + L(μ) X1 M M N N such that:

  m ∈ Ω × Th , for L(μΩ×Th )-a.e. ω, h!   m  m (i,h!) (h!) X1 ◦ χ (ω) = x ω, h! i=1

and h!

∀m ∈ N ∀k  m

km  i=1

X1(i,h!) =

k 

x1(i,m) .

i=1

Using this notation, we also obtain the following weak independence assertion:

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∀a, b ∈ Q

I

    1 1 (j,h!) ∗ X1 μ ∈ aj − , bj + m m j ∈I   

1 1 (j,h!) ∗ μ X1 ∈ a j − , bj + ≈ m m j ∈I

for every finite I ∈ P ({1, . . . , h!}) and some hyperfinite infinite m which depends on I . Proof. The existence of x ≡ x˜ is a consequence of the universality of hyperfinite adapted spaces as introduced in several works by Fajardo, Hoover and Keisler [6,7,9], e.g. [6,9, Adapted Universality Theorem], and the existence of a t := H1 ! -càdlàg-lifting X for x follows from investigations by Stroyan and Bayod [16, Theorem 5.3.23]. Let n ∈ N. Exploiting the homogeneity of adapted hyperfinite spaces and the well-known (i,n!) for i ∈ lifting theorems for random variables [11], we have internal random variables X1 {1, . . . , n} such that the approximations   m m   m (i,n!) (i,n!) , X1 ◦ χ (n!) ≈ x1 ◦ χ (n!) = xm/n! = lim Xl ·, L(μ)-a.s. ∀m  n! l→∞ n! i=1

i=1

1 l -forward

hold, where Xl denotes the average of the t-interpolation X t of X. This implies the L(μ)-stochastic convergence of the Xl , since L(μ)-almost sure convergence always implies L(μ)-stochastic convergence. Keeping this in mind, we define “exceptional sets” by 

m 1  m   (i,n!) − X1 ◦ χ (n!)  Fl (k, n) := Xl ·, k n! mn!

i=1

for all k, n ∈ N. Recalling that l → Xl is internal, the sets Fl (k, n) are internal for every k, n ∈ N. On the other hand, we may apply results from Albeverio, Fenstad, Høegh-Krohn and Lind strøm [1] to the Hausdorff space R with its topological basis { ni=1 ]ai , bi [: n ∈ N, a, b ∈ Qn }, (1,n!) (n!,n!) and using the independence of the random variables x1 , . . . , x1 we conclude: ∀I ∈ P ({1, . . . , n!}) ∀a, b ∈ QI     1 1 (j,n!) ∗ X1 lim μ ∈ aj − , bj + m→∞ m m j ∈I     1 1 (j,n!) ∗ X1 ∈ a j − , bj + = inf L(μ) m∈N m m j ∈I     1 1 (j,n!) ∗ X1 ∈ a j − , bj + = L(μ) m m j ∈I m∈N      (j,n!)  (j,n!)   −1 X1 x1 ∈ st [aj , bj ] = L(μ) ∈ [aj , bj ] = L(μ) =

j ∈I

=



j ∈I

j ∈I

j ∈I

 (j,n!) 

 (j,n!)  L(μ) x1 ∈ [aj , bj ] = L(μ) X1 ∈ st−1 [aj , bj ] j ∈I

    1 1 (j,n!) ∗ X1 L(μ) ∈ a j − , bj + m m m∈N

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   1 1 (j,n!) ∗ = inf L(μ) X1 ∈ a j − , bj + m∈N m m j ∈I   

1 1 |I |<ℵ0 (j,n!) ∗ . = lim μ X1 ∈ a j − , bj + m→∞ m m

j ∈I

Therefore, for every I ∈ P ({1, . . . , n!}), a, b ∈ QI and k ∈ N there exists an m0 ∈ N, satisfying for m  m0 the formula     (j,n!) ∗ 1 1 μ X1 ∈ aj − , bj + m m j ∈I

−

 

 (j,n!) 1 1 1  , μ X1 ∈ ∗ a j − , bj + m m k

(1)

j ∈I

whose length urges us to introduce the abbreviations ∀I ∈ P ({1, . . . , n!}) ∀j ∈ I ∀a, b ∈ QI    (j,n!) 1 1 ∗ X1 , G(n!, I, a, b, m) := ∈ a j − , bj + m m j ∈I    1 1 (j,n!) ∗ H (n!, j, a, b, m) := X1 . ∈ aj − , bj + m m Note that this convention shall also apply to infinite m ∈ ∗ N and that the sets G(n!, I, a, b, m), H (n!, j, a, b, m) are internal for finite |I | and arbitrary a, b, j, m, n. Of course, we aim at performing an ℵ1 -saturation argument. The observations we just made suffice to convince ourselves that the sets ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Bk := k  n  H,

n ∈ ∗N :

⎫ ⎪ ⎪ ⎪ internal automorphic ⎪ ⎪ ∃χ (n!) : Ω → Ω ∃l0  k∀l  l0 ⎪ ⎪ ⎪ (1,n!) (n!,n!) ⎪ Ω n! ⎪ ∃(X1 , . . . , X1 ) ∈ (∗ R ) ⎪ ⎪ ⎪ ⎪ ⎪ ∃M, N ∈ ∗ N ∩ {·  k} ∀i, j  n! ⎪   ⎪ ⎞⎪ ⎛ ⎪ μ e − 1  X (i,n!)  e + 1 ⎪ ⎪ r s 1 1 M M ⎪ ⎪ ∀r, s  k  ,   ⎪ ⎟ ⎜ k (j,n!) 1 1 ⎪ − μ er −  X  e + ⎪ ⎟ ⎜ s ⎬ 1 N N ⎟ ⎜ 1 ⎟ ⎜ μ(Fl (k, n))  k , ⎟⎪ ⎜ ⎟⎪ ⎜ ∀m  n ∀  m ⎪ ⎟⎪ ⎜ ⎪ ⎟⎪ ⎜ n!     ⎪  (i,n!) (i,m)  ⎪ ⎟ ⎜ 1 1 m ⎪  μ  X − x , ⎪ ⎟ ⎜ i=1 1 i=1 1 k k ⎪ ⎪ ⎟ ⎜ 1 ⎪ (n!) ⎪ ⎟ ⎜ ∀A ∈ A μ(χ (A)) − μ(A)  k , ⎪ ⎟⎪ ⎜ ⎪ ⎪ ⎟ ⎜ ∀I ∈ P ({1, . . . , n!}) ∃m  k ∀l  k ⎪ ⎟⎪ ⎜ ⎪ ⎪ ⎟ ⎜ μ(G(n!, I, ∗ Z(l, |I |), m)) ⎪ ⎪ ⎠ ⎝ ⎪ 1 ⎪   ⎭ − j ∈I μ(H (n!, j, ∗ Z(l, |I |), m)) k

are nonempty for any k ∈ N and internal. Furthermore, (Ck )k∈N is decreasing and ℵ1 -saturation gives us an infinite hyperfinite h ∈ ∗ k∈N Ck . So we find an l0 ∈ N \ N, such that for all l  l0

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L(μ)-a.s. ∀m  h! m 



X1(i,h!)

i=1

◦χ

(h!)

m ≈ Xl ·, h!

 !

m 1 h! + l



=l·

m s= h! ,step H1 !

" 1 X(·, s) . H!

(2)

Now we shall show that m  m X1(i,h!) ◦ χ (h!) → h! i=1

is indeed an internal stochastic process with a time index set   m Th = : m ∈ ∗ N0 , m  h! h! such that

 m ∈ Ω × Th for L(μΩ×Th )-f.a. ω, h!     m  m m = x ω, . X1(i,h!) ◦ χ (h!) ω, h! h! 

(3)

i=1

In order to achieve this, choose a lifting X¯ : Ω × Th → ∗ R on the hyperfinite adapted space # H ! $T Ω × Th = Ω0 T × Th = Ω0 h! h × Th , and obtain for L(μΩ×Th )-a.e. (ω, t) ∈ Ω × Th

¯ X(ω, t) = x(ω, t).

Following Stroyan’s and Bayod’s theory of liftings for processes with paths in D[0, 1] [16, Lemma 5.3.15, Theorem 5.3.23], we know for L(μ)-a.e. ω ∈ Ω ∀t ∈ [0, 1] # $ x(ω, t) = stk X(ω, ·) (t) = lim Xl (ω, t). l→∞

But the projective inverse images of sets with L(μ) = L(μΩ )-measure zero have L(μΩ×Th )measure zero. This is why the S-continuity of liml→∞ Xl (Xl was proven to be S-equicontinuous in [16, Lemma 5.3.15]) lets us readily conclude: for L(μΩ×Th )-a.e. (ω, t) ∈ Ω × Th ¯ lim Xl (ω, t) = lim Xl (ω, t) = x(ω, t) = X(ω, t).

l→∞

l→∞

So Xl goes L(μΩ×Th )-a.s. – hence also L(μΩ×Th )-stochastically – to X¯ as l tends to infinity. So     ¯ 1 1 ∀k, n ∈ N Mk,n :=  ∈ ∗ N: ∀l   μΩ×Th |Xl − X| n k defines a system of nonempty and internally defined – that is, internal – sets. This system (Mk,n )k,n∈N has the finite intersection property, expressing that every finite intersection of the sets Mk1 ,n1 , . . . , Mkm ,nm is nonempty (this is true because Mmaxi ki ,maxi ni is a nonempty subset

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of the intersection). So the more general formulation of the ℵ1 -saturation principle grants that the whole system (Mk,n )k,n∈N has a nonempty intersection. So we get an l1 ∈ ∗ N \ N such that for all l  l1     ¯  1 ≈ μΩ×Th |Xl − X| ¯  1 ≈ 0, ∀n ∈ N L(μΩ×Th ) |Xl − X| n n and therefore

  % ¯ = L(μΩ×Th ) ¯ 1 L(μΩ×Th ){Xl ≈ X} |Xl − X| = 0, n n∈N

holds. This proves for every l  l1 : for L(μΩ×Th )-f.a. (ω, t) ∈ Ω × Th

¯ Xl (ω, t) = X(ω, t) = x(ω, t).

Referring to (2), we obtain as a consequence of the latter equation that for every l  l2 := max{l0 , l1 } the formula   m ∈ Ω × Th for L(μΩ×Th )-f.a. ω, h!       m  m m m (i,h!) (h!) ¯ X1 ◦ χ (ω) = Xl ω, = X ω, = x ω, h! h! h! i=1

holds. This completes the proof of (3).

2

2.2. The case of the usual increment construction Theorem 2.2. Let x˜ be an arbitrary adapted measurable Lévy process on an adapted probability space (Γ, (Ct )t∈[0,1] , P ) and (Ω, (Ft )t∈[0,1] , L(μ)) be a hyperfinite adapted probability space of mesh H !. Assume P -a.s. x˜0 = 0. Then we can find: a process x ≡ x˜ on Ω with a lifting X¯ : Ω × Th → ∗ R on Ω × Th for some hyperfinite infinite h  H and a H1 ! -càdlàg lifting X on Ω × T; internal random variables X1(1,h!) , . . . , X1(h!,h!) with the property ∀i, j  h! ∀e, g ∈ Q ∃M, N ∈ ∗ N \ N       1 1 1 1 (j,h!) (i,h!) = L(μ) X1 ∈ e − ,g + ∈ e − ,g + L(μ) X1 M M N N such that:

  m ∈ Ω × Th for L(μΩ×Th )-a.e. ω, h!

m  i=1

  m . X1(i,h!) (ω) = x ω, h!

Using this notation, we also obtain the following weak independence assertion:     1 1 (j,h!) ∗ X1 ∀a, b ∈ QI μ ∈ a j − , bj + m m j ∈I   

1 1 (j,h!) ∗ μ X1 ∈ aj − , bj + ≈ m m j ∈I

for every finite I ∈ P ({1, . . . , h!}) and some hyperfinite infinite m which depends on I .

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Proof. Define (i,n)

:= x˜ti/n − x˜t (i−1)/n

(i,n)

:= xti/n − xt (i−1)/n .

∀i ∈ {1, . . . , n} ∀t ∈ [0, 1]

x˜t

∀i ∈ {1, . . . , n} ∀t ∈ [0, 1]

xt

and By the stationarity of the increments of the process x˜ we gain – for arbitrary choice of n ∈ N, 0 = t0 < · · · < tn as well as B1 , . . . , Bn ∈ B(R) – the assertion ' & n ' & n {x˜tk i/n − x˜tk (i−1)/n ∈ Bk } = P {x˜tk /n ∈ Bk } , P k=1

k=1

implying that the finite-dimensional distributions of the processes x˜ (i,n) for i ∈ {1, . . . , n} are given by the convolution semigroup (Px˜t/n )t∈[0,1] and the start distribution δ0 (via the Ionescu– Tulcea–Kolmogorov construction of the projective limit). Observe furthermore that n 

x˜ (i,n) = x˜

i=1

and n 

x (i,n) = x.

i=1

It is clear that we have thereby constructed increment constructions for x˜ and x. Now the proof is almost the same as the proof of Theorem 2.1. We only need to use a particular increment construction (that is, the usual one) to see that the adapted automorphisms χ (n!) are going to be the identity map again, for every n ∈ ∗ N. 2 Remark 2.1. If one had used the SDJ -liftings of Hoover and Perkins [8] instead of the t-lifting in this construction, one would have had to use an external choice function to find elements of m in the previous theorem. Both the hyperfinite time-line with the properties of the time steps h! the lifting theory by Stroyan and Bayod [16] and the lifting theory by Hoover and Perkins [8] essentially rely on a lifting theorem for random variables that take values metric spaces, in this special case the metric spaces are spaces of càdlàg functions with different Polish topologies. Stroyan’s and Bayod’s theory uses the Kolmogorov metric on D[0, 1], Hoover’s and Perkins’ theory employs Skorokhod’s J1 topology on D[0, 1] (for precise definitions of those topologies, the reader is referred to the comprehensive works by Billingsley [5] and Parthasarathy [12]). 3. Conclusions The following definition is basically due to [14], although a slight generalisation, referring to [1, Definition 4.1.1], is given here. Definition 3.1. Let Ω be an internal probability space, h ∈ ∗ N \ N, let x, y : Ω × [0, 1] → R be stochastic processes allowing for liftings X, Y : Ω × Th → ∗ R. Then the internal process     (·  1 − Y (s) (4) X dY : Ω × Th → ∗ R, (ω, t) → X(ω, s) Y s + H! 0

Th s
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705

is called the internal stochastic integral of the process X with respect to the integrating process Y . If well-defined, we put for any t ∈ Th (t

(t x dy :=

0

X dY 0

to be the stochastic integral of x with respect to the integrating process y between 0 and t. Remark 3.1. There are several sufficient conditions for the stochastic integral to be well-defined. They are studied e.g. in [1,8,16], but also in [14], where we find uniform boundedness and predictability of x to be sufficient conditions [14, p. 179]. Now consider a measurable Lévy process y˜ on an adapted space (Γ, C, P ) and the – by the Adapted Universality Theorem – associated process y ≡ y˜ on a hyperfinite adapted probability space Ω with a lifting Y . Furthermore, consider an arbitrary measurable process w˜ with associated process w ≡ w˜ on Ω, allowing for a lifting W , such that (w, y) ≡ (w, ˜ y) ˜ by the saturation of the hyperfinite adapted space Ω [6, Adapted Saturation Theorem]. According to Theorem 2.2, the above equation (4) becomes (t W dY = 0



(sh!,h!)

W (·, s)Y1

◦ χ (h!)

(5)

Th s
(sh!,h!)

for all t ∈ Th , where Y1 plays the role of X1 in Theorem 2.2, and the other notations are chosen as in Theorem 2.2. There is an obvious analogy to Anderson’s results in [3]. (1,h!) (h!,h!) There, the place of Y1 ◦ χ (h!) , . . . , Y1 ◦ χ (h!) would be filled by random variables on T Ω = {−1, +1} h such that ∀ω ∈ Ω ∀s ∈ Th

(sh!,h!)

Y1

◦ χ (h!) (ω) = ω(s).

Our representation formula (5) for the internal stochastic integral of an arbitrary measurable process with respect to a measurable Lévy process suggests nonstandard analysis to be an appropriate method not only for the stochastic analytic investigation of Brownian motions, but also for research on general measurable Lévy processes. Systematically, one now needs to analyse in which way one can modify the nonstandard proofs for local time formulae and theorems on stochastic differential equations in order to obtain results where general measurable Lévy processes are used instead of the Brownian motion. Acknowledgements The second named author gratefully acknowledges funding by the German National Academic Foundation (Studienstiftung des deutschen Volkes) and a PhD-Scholarship of the German Academic Exchange Service (Doktorandenstipendium des Deutschen Akademischen Austauschdienstes). He is also indebted to Professors Terry Lyons and Boris Zilber for helpful comments on the presentation of the material. Furthermore, the authors are grateful for many constructive suggestions by H. Jérôme Keisler that improved the presentation significantly. A previous version of this paper also contained a non-trivial mistake which was brought to the attention of the authors by H. Jérôme Keisler.

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