A combinatorial in�nitesimal representation of L�vy processes Sergio Albeverio, Frederik S. Herzberg ∗ ∗ Abteilung Stochastik, IAM, Universit�t Bonn
L�vy processes: de�nition De�nition 1 . Consider a probability space (Ω , A , P ) and let d ∈ N . A stochastic process x : Ω × R + → R d is called a L�vy process if and only if it is pinned to zero and has stationary and independent increments , i.e. 1. x 0 = 0 on Ω , 2. x t − x s is independent of F s = σ ( x u : u ≤ s ) for all t > s , 3. the law of x t − x s equals the law of x t − s for all t ≥ s , and 4. P -almost all paths of ( x t ) t ∈ R + are right-continuous with left limits (c�dl�g). 1
Translation-invariant Markovian semigroups There is a one-to-one correspondence between • L�vy processes on the space D [0 , + ∞ ) of c�dl�g paths in R d • Markovian semigroups ( p t ) t ∈ R + on R d that are 1. continuous (i.e. t �→ p t is continuous with respect to the vague topology) and 2. (space-)translation invariant (in the sense that p t f ( · + z ) = p t f for all z ∈ R d , t ≥ 0 and any nonnegative Lebesgue-Borel measurable f : R d → R ). This bijection is given by ( x · , (Ω , A , P )) ( P x t ) t ≥ 0 , � � ( p t ) t ≥ 0 ( p J ) J ∈ R + < ℵ 0 (Ionescu-Tulcea-Kolmogorov). 2
L�vy-Khintchine formula Theorem 1 (L�vy-Khintchine formula, cf. e.g. Revuz and Yor [5], Sato [6]). Consider a Markovian semigroup ( p t ) t ∈ R + on R d , d ∈ N . ( p t ) t ∈ R + is continuous and translation invariant if and only if the in�nitesimal generator ℓ of ( p t ) t ∈ R + can be written as d d ℓ : f �→ 1 � � 2 ∂ i ∂ j f + σ i,j γ i ∂ i f 2 i,j =1 i =1 � + R d ( f ( · + y ) − f )) ν ( dy ) d × d is a symmetric d × d -matrix where σ ∈ R + with nonnegative entries , γ ∈ R d , and ν is a Radon measure on R d satisfying 1. ν { 0 } = 0 , � 2. B 1 (0) | y | 2 ν ( dy ) < + ∞ , � � 3. ν < + ∞ , ∁ B 1 (0) ∁ B 1 (0) denoting the complement of the unit ball in R d centered at 0 . 3
Overview of this paper: theory In this paper we shall, using results of Lindstr�m's [4], construct a particularly simple internal analogue of ℓ for a given positive in�nitesimal h > 0 . This will be an internal operator L such that for all test functions f ∈ C ∞ 0 ( R d ) one has ∀ x ∈ ∗ R d ∀ x ∈ R d ( x ≈ x ⇒ L ∗ f ( x ) ≈ ℓf ( x )) and in addition, the internal translation-invariant Markovian semigroup P = ( P t ) t ∈ h · ∗ N 0 generated by L shall be proven to be the internal convolution of • a weighted multiple of Anderson's random walk as well as • the superposition of hyper�nitely many independent stochastic jumps , corresponding to the di�usion and jump (or: L�vy measure) parts of the initial L�vy process, respectively. L is said to generate the reduced lifting . 4
Application: Towards a weak notion of completeness for L�vy markets (1) Let d = 1 and let Λ denote the internal L�vy measure of the reduced lifting of a given L�vy process x · , that is: the set of pairs of possible jump sizes and intensities. Suppose, the L�vy measure ν of x · is concentrated on R > 0 (�all risks are insured against�). One can show that there is a reduced lifting of x · that, at each time, is the independent sum of • a weighted multiple of Anderson's random walk (the lifting of the di�usion part), and • the superposition of m ∈ ∗ N \ N independent stochastic jumps , each of which is greater than rh , occurring at a probability given by the internal L�vy measure Λ which is derived from ∗ ν . 5
Application: Towards a weak notion of completeness for L�vy markets (2) Thus, the Markov kernel P h generating the internal Markov chain of the reduced lifting of x · can be decomposed according to P h = Q (0) · · · Q ( m ) wherein • Q (0) : f �→ � � � � 1 1 for · − σh + (1 − p 0 ) f · + σh p 0 f 2 2 some σ > 0 , p 0 ∈ (0 , 1) , and • for all i ∈ { 1 , . . . , m } , Q ( i ) : f �→ (1 − p i ) f ( · ) + p i f ( · + α i ) , α i > rh , i ∈ { 1 , . . . , m } , being the jumps of the reduced lifting , counted with multiplicity. This reduces the 2 m +1 -nomial market model P h to a binomial one (uniquely up to permutations of { α i } i ). 6
Notation (1) Fix an in�nite hyper�nite number H , and let H · ∗ Z d be the lattice of mesh size ¯ L := 1 H . For an arbitrary N ∈ ∗ N \ N , we set η := 1 [ − N, N ] d ∩ ¯ L =: L , thereby ensuring that L is hyper�nite . By ρ := ρ L we shall denote the L - rounding operation ρ L , de�ned by ρ L : x �→ (sup { y i ≤ x i : y ∈ L } ∨ − N ) d i =1 , Owing to the particular shape of L as a discrete set, the supremum in each component is even a maximum and ρ : ∗ R d → L . 7
Notation (2) Let 0 < h ≈ 0 and de�ne, for α ∈ L and λ ∈ ∗ R > 0 , �rstly the internal in�nitesimal generator for a hyper�nite Poisson process L ( α ) : f �→ f ( · + α ) − f h and the corresponding internal Markov kernel P ( α ) := f + hL ( α ) = f + h ( f ( · + α ) − f ) . h h Varying the intensity, we also set P ( α,λ ) := f + hλL ( α ) h . h These kernels generate internal Markov chains via � � ◦ ( t/h ) P ( α,λ ) P ( α,λ ) , P ( α ) := P ( α, 1) ∀ t ∈ h ∗ N 0 := . t t t h The modi�ed internal in�nitesimal generator for in�nitesimal t ∈ h ∗ N 0 is de�ned by : f �→ P ( α ) f − f L ( α ) t . t t 8
Existence of a twice S -continuous S -in�nitesimal generator for superpositions The (modi�ed) internal in�nitesimal generator has a standard part : Lemma 1 . Let f : R d → R continuous with compact support, and consider a Radon measure ν , whether �nite or in�nite, on R d . Then for any two hyper�nite real numbers y ≈ x and all 0 ≈ t ∈ h ∗ N , one has: � � ∗ ν ◦ ρ − 1 � L ( α ) f ( y ) ( dα ) t L � �� � α ∈ L L ( α ) = P f ( y ) · ∗ ν [ ρ − 1 { α } ] t � � ∗ ν ◦ ρ − 1 � L ( α ) ≈ h f ( x ) ( dα ) . L � �� � α ∈ L L ( α ) = P f ( x ) · ∗ ν [ ρ − 1 { α } ] h Proof idea. A combination of elementary estimates yields the result for �nite ν ; the general result will follow by truncation and monotone convergence. 9
Notation: composition of hyper�nite kernels Let AB for two hyper�nite translation-invariant kernels A , B on ∗ R d denote the translation-invariant kernel obtained by convolving the two associated measures : If � � � � A : f �→ p i f ( · − α i ) , B : f �→ q j f · − β j , i j then we de�ne the product of A and B as � � � � AB : f �→ · − γ p i q j f . i,j n o γ ∈ { α i } i ∪ β j αi + βj = γ j Then AB = BA is again a hyper�nite translation-invariant kernel and we can de�ne the product � A ∈ A A for a hyper�nite set A of hyper�nite translation-invariant kernels recursively in the internal cardinality of A . Analogously, powers of hyper�nite translation-invariant kernels can be de�ned. 10
Superposition of hyper�nite Poisson processes (1) Lemma 2 . Consider an internal hyper�nite family { x i } i<M ⊆ L of vectors in ∗ R d and an internal hyper�nite family of positive hyperreal numbers ( λ i ) i<M such that: 1. C 0 := � 1 ≤| x i | λ i as well as | x j |≤ 1 λ j | x j | 2 are �nite; C 1 := � 2. Setting C 2 := � | x i |≤ 1 λ i and C 3 := � | x j | < 1 λ j | x j | , one has √ √ h ≈ C 3 h ≈ C 2 · h ≈ 0 . N (These requirements may be read as regularity conditions on the measure A �→ � i λ i χ A ( x i ) ) De�ne, for t ∈ h · ∗ N , Q ( i ) := P ( x i ,λ i ) ∀ i < M . t t 11
Then for all f ∈ C 2 � � with a �nite R d , R C 2 ( R d ) -norm, there exists an R = R ( f ) ∈ ∗ R with hR ≈ 0 such that for all k < M , � � � � � M − 1 Q ( j ) � � � h − id � Q ( i ) h − id � j = i +1 � � ≤ R. f � � h h � � � � i ≥ k Moreover, this R ( f ) can be chosen to be a 1 -homogeneous function in f by setting 2 + 4 C 0 C 2 � � R ( f ) := 4 C 0 sup R d | f | R d | f ′′ | . + ( NC 3 + C 0 C 1 + 4 C 2 C 1 ) sup Proof idea for Lemma 2. Apply the transfer principle to Taylor's Theorem for functions in C 2 � � . R d , R 12
Superposition of hyper�nite Poisson processes (2) Remark 1 . The conditions imposed in assumption (2) of Lemma 2 can be viewed as conditions on the internal measure Λ on L = ¯ L ∩ [ − N, N ] d , induced by ( λ i ) i and ( x i ) i via A �→ � i<M λ i χ A ( x i ) . They are exactly the regularity properties of Lindstr�m's hyper�nite L�vy measure (as constructed in the proof in his hyper�nite representation theorem for standard L�vy processes [4, Theorem 9.1]). 13
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