Approximations for solutions of Lévy-type stochastic differential equations
aa r X i v : . [ m a t h . P R ] D ec Approximations for solutions of Lévy-type stochasticdifferential equations ∗ Michał Barski
Faculty of Mathematics and Computer Science, University of Leipzig, GermanyFaculty of Mathematics, Cardinal Stefan Wyszy´nski University in Warsaw, Poland
Abstract
The problem of the construction of strong approximations with a given order of con-vergence for jump-diffusion equations is studied. General approximation schemes are con-structed for Lévy type stochastic differential equation. In particular, the paper generalizes theresults from [5] and [2]. The Euler and the Milstein schemes are shown for finite and infiniteLévy measure.
Key words : strong approximations, Lévy-type differential equations, Itô-Taylorexpansion, discrete approximating schemes
AMS Subject Classification : 60H10, 60G57.
The problem of approximation construction for solution of stochastic differential equationis widely studied throughout many papers. The authors’ attention is focused mainly on theequation of the form: Y t = Y + Z t f ( Y s − ) dZs, (1.1)where Y is a random variable with known distribution, f -some regular function and Z -adriving process. There are many approximation methods for the solution of (1.1) dependingon the driving process and the optimality criteria imposed on the approximating error. Thecase when Z is a Wiener process the problem is comprehensively studied in the book [5], forjump diffusion case see, for instance, [3], [4]. In [5] various schemes for the so called weakand strong approximations are presented, in particular their dependence on the mesh of thepartition of the interval [0 , T ] . Denoting by ¯ Y the approximation, the optimality criteria forweak solutions have a form: E [ g ( Y T ) − g ( ¯ Y T )] −→ min , where g is some regular function,while for strong solutions: E sup t | Y t − ¯ Y t | −→ min . The schemes use the increments oftime, increments of the Wiener process and, for higher order of convergence, some normallydistributed random variables correlated with the increments of the Wiener process. Thusfor practical implementation we have to generate normally distributed, correlated randomvariables. ∗ Research supported by Polish KBN Grant P03A 034 29 „Stochastic evolution equations driven by Lévy noise” he simplest approximating scheme for the equation (1 . is the Euler scheme which hasthe following structure: ¯ Y = Y , ¯ Y ( i +1) Tn = f ( ¯ Y iTn )( Z ( i +1) Tn − Z iTn ) , where { iTn , i = 0 , , ..., n } is a partition of the interval [0 , T ] . In the case of the Wienerdriving process it is easy to construct. However, for a general Lévy driving process it is nolonger so simple. This is because of the difficulty of practical construction of the incrementsof Z when the Lévy measure is infinite, i.e. when the measure of a unit ball is infinite. Ifthe increments can not be simulated, then they themselves have to be approximated in somesense and the accuracy of such construction should be studied. This way of approximating ispresented for example in [9] and [7]. The main idea in these papers is to reduce the problemby replacing increments of Z by suitable increments of the compound Poisson process, whichcan be practically simulated. It should be pointed out that our approach is more general sincea significant majority of papers consider approximation problem using different modificationsof the Euler scheme.In this paper we work with a stochastic differential equation of the form: Y t = Y + Z t b ( Y s − ) ds + Z t σ ( Y s − ) dW s + Z t Z | x | < F ( Y s − , x ) ˜ N ( ds, dx )+ Z t Z | x |≥ G ( Y s − , x ) N ( ds, dx ) , where b, σ, F, G are some regular functions, W - a standard Wiener process and N, ˜ N - aPoisson random measure and its compensated measure respectively. We focus on the strongapproximations, i.e. the error is measured by E sup t | Y t − ¯ Y t | . The strong approximation isof order γ if E sup t | Y t − ¯ Y t | ≤ δ γ , where δ is the mesh of partition of the interval [0 , T ] .Our aim is to construct the strong approximation for a previously fixed number γ > . Theidea is to apply the Itô formula to the process Y many times, i.e. to the process Y and then tothe coefficients in its expansion. The approximation is built of some of the coefficients whichare chosen appropriately. The main result is Theorem 4.1 providing the description of theapproximation. This theorem is a generalization of the results from [5] for diffusion processesand [2] for diffusion processes with jumps generated by a standard Poisson process. For γ = we obtain the Euler approximation but we can also built approximations of higher order. Theapproximation given by Theorem 4.1 has one limitation - in case when the Lévy measureof a unit ball is infinite, some ingredients are hard to simulate. This difficulty concerns thepossibility of simulating integrals with respect to the compensated Poisson measure on unitballs. This problem hasn’t appeared in [5] or [2] since there were no jumps or were equal to only. To overcome this problem we modify the approximation by replacing all unit balls with ε - discs which are obtained by cutting ε - balls from unit balls. This procedure causes thatthe error depends not only on δ but on ε as well. Theorem 5.3 provides the error description.It is a sum of δ γ and some function of ε which tends to zero when ε −→ . The speed ofconvergence of this function depends on the behavior of the Lévy measure near . Concluding,if the Lévy measure is finite then the approximation is given by Theorem 4.1, if it is not - byTheorem 5.3, but then the error depends on ε also. Note that in the first case we are able toconstruct strong approximations of higher order than the Euler scheme.The paper is organized as follows, in Section 2 we present known facts concerning Lévy-type stochastic differential equation and describe the procedure of solution expansion withthe use of the Itô formula. Section 3 contains precise formulation of the problem whichis being successively solved in Section 4. This section consists of three preceding lemmaswhich are used in the main Theorem 4.1. In this section we adopt some ideas and estimationfrom [5] to the present jump-diffusion settings. Section 5 is devoted to the modification of he approximation in the case where the Lévy measure is infinite. Section 6 consists of twoexamples of strong approximations schemes for γ = and γ = 1 , i.e. the Euler and Milsteinschemes. Let (Ω , F t ; t ∈ [0 , T ] , P ) be a probability space with filtration generated by two independentprocesses: a standard Wiener process W and a random Poisson measure N . The Poissonrandom measure defined on R + × ( R \{ } ) is assumed to have the intensity measure ν whichis a Lévy measure. By ˜ N we denote the compensated Poisson random measure. Since wewill consider stochastic integrals of different types, the class of integrands should be specified.While the integrals with respect to time and the Poisson measure are well understood, the classof integrands with respect to W and ˜ N should be made precise. Definition 2.1
A mapping g : Ω × [0 , T ] −→ R is integrable with respect to W if it ispredictable and satisfies the integrability condition: E R T g ( s ) ds < ∞ . Definition 2.2
Let E be a subset of R . A mapping g : Ω × [0 , T ] × E −→ R is integrablewith respect to ˜ N if it is predictable and satisfies the integrability condition: E R T R E g ( s, x ) ν ( dx ) ds < ∞ . In these classes of integrands both integrals are square-integrable martingales and the follow-ing isometric formulas hold: E (cid:16) Z T g ( s ) dW s (cid:17) = E Z T g ( s ) ds E (cid:16) Z T Z E g ( s, x ) ˜ N ( ds, dx ) (cid:17) = E Z T Z E g ( s, x ) ν ( dx ) ds. Throughout all the paper we will work with a stochastic differential equation of the form: Y t = Y + Z t b ( Y s − ) ds + Z t σ ( Y s − ) dW s + Z t Z B F ( Y s − , x ) ˜ N ( ds, dx )+ Z t Z B ′ G ( Y s − , x ) N ( ds, dx ) , (2.2)where t ∈ [0 , T ] , B = { x : | x | < } , B ′ = { x : | x | ≥ } . For simplicity the initial conditionis assumed to be deterministic, i.e. Y ∈ R . Coefficients b : R −→ R , σ : R −→ R , F : R × R −→ R , G : R × R −→ R are measurable and satisfy the following conditions. (A1) Lipschitz condition : there exists a constant K > such that: | b ( y ) − b ( y ) | + | σ ( y ) − σ ( y ) | + Z B | F ( y , x ) − F ( y , x ) | ν ( dx )+ Z B ′ | G ( y , x ) − G ( y , x ) | ν ( dx ) ≤ K | y − y | ∀ y , y ∈ R . (A2) Growth condition : there exists a constant K > such that: | b ( y ) | + | σ ( y ) | + Z B | F ( y, x ) | ν ( dx )+ Z B ′ | G ( y, x ) | ν ( dx ) ≤ K | y | ∀ y ∈ R . heorem 2.3 Under assumptions ( A1 ) and ( A2 ) there exists a unique, adapted, càdlàg solu-tion of (2.2). Moreover, the solution satisfies: E | Y t | ≤ C (1 + Y ) ∀ t ∈ [0 , T ] , (2.3) where C ≥ . Theorem 2.3 is a consequence of Theorem . . , Theorem . . and Corollary . . in [1],where the Lipschitz and the growth conditions are imposed on the coefficients b, σ, F only and G ( · , x ) is assumed to be continuous. The estimation (2 . itself is a consequence of Corollary . . in [1] and the proof of Theorem . . , where the inequality: E | Y t | ≤ C ( t )(1 + Y ) ∀ t ∈ [0 , T ] , (2.4)is shown for equation (2.2) but without the term R t R B ′ G ( Y s − , x ) N ( ds, dx ) . Under assump-tions ( A1 ) , ( A2 ) the same estimation can be obtained for (2.2) with the use of similar ar-guments. Moreover, C ( · ) is a continuous function and as such it is bounded on the interval [0 , T ] and thus (2 . holds.In the sequel the proposition below will be used and for the reader’s convenience weprovide the proof. Proposition 2.4
Under assumptions ( A1 ) and ( A2 ) the solution Y of (2 . satisfies the esti-mation: E sup ≤ s ≤ T | Y s | ≤ C (1 + Y ) for some constant C ≥ . Proof:
We write the solution in the form: Y s = Y + Z s b ( Y u − ) du + Z s σ ( Y u − ) dW u + Z s Z B F ( Y u − , x ) ˜ N ( du, dx )+ Z s Z B ′ G ( Y u − , x ) ˜ N ( du, dx ) + Z s Z B ′ G ( Y u − , x ) ν ( dx ) du, and thus: Y s ≤ (cid:18) Y + (cid:16) Z s b ( Y u − ) du (cid:17) + (cid:16) Z s σ ( Y u − ) dW u (cid:17) + (cid:16) Z s Z B F ( Y u − , x ) ˜ N ( du, dx ) (cid:17) + (cid:16) Z s Z B ′ G ( Y u − , x ) ˜ N ( du, dx ) (cid:17) + (cid:16) Z s Z B ′ G ( Y u − , x ) ν ( dx ) du (cid:17) (cid:19) . Using the Doob and Schwarz inequalities as well as isometric formulas for stochastic integralswe obtain: E sup ≤ s ≤ T Y s ≤ (cid:18) Y + T E Z T b ( Y u − ) du + 4 E Z T σ ( Y u − ) du + 4 E Z T Z B F ( Y u − , x ) ν ( dx ) du + 4 E Z T Z B ′ G ( Y u − , x ) ν ( dx ) du + T ν ( B ′ ) E Z T Z B ′ G ( Y u − , x ) ν ( dx ) du (cid:19) . Using assumption ( A2 ) we obtain: E sup ≤ s ≤ T Y s ≤ (cid:18) Y + K ( T + 12 + T ν ( B ′ )) Z T (1 + E Y u ) du (cid:19) . y (2.3) we have: E sup ≤ s ≤ T Y s ≤ (cid:18) Y + K ( T + 12 + T ν ( B ′ )) Z T (cid:16) C (1 + Y ) (cid:17) du (cid:19) , and finally we have the desired estimation: E sup ≤ s ≤ T Y s ≤ C (1 + Y ) . (cid:3) For the process Y being a solution of (2.2) and for a real function f of class C we have thefollowing form of the Itô formula: f ( Y t ) = f ( Y ) + Z t f ′ ( Y s − ) b ( Y s − ) ds + Z t f ′ ( Y s − ) σ ( Y s − ) dW s + 12 Z t f ′′ ( Y s − ) σ ( Y s − ) ds + Z t Z B ′ (cid:8) f ( Y s − + G ( Y s − , x )) − f ( Y s − ) (cid:9) N ( ds, dx )+ Z t Z B (cid:8) f ( Y s − + F ( Y s − , x )) − f ( Y s − ) (cid:9) ˜ N ( ds, dx ) (2.5) + Z t Z B n f ( Y s − + F ( Y s − , x )) − f ( Y s − ) − F ( Y s − , x ) f ′ ( Y s − ) o ν ( dx ) ds. Introducing the following operators: L f ( y ) := f ′ ( y ) b ( y ) + 12 f ′′ ( y ) σ ( y ) + Z B { f ( y + F ( y, x )) − f ( y ) − F ( y, x ) f ′ ( y ) } ν ( dx ) L f ( y ) := f ′ ( y ) σ ( y ) L f ( y, x ) := f ( y + F ( y, x )) − f ( y ) L f ( y, x ) := f ( y + G ( y, x )) − f ( y ) , we can write (2.5) in the operator form: f ( Y t ) = f ( Y ) + Z t L f ( Y s − ) ds + Z t L f ( Y s − ) dW s + Z t Z B L f ( Y s − , x ) ˜ N ( ds, dx ) + Z t Z B ′ L f ( Y s − , x ) N ( ds, dx ) . We would like to apply the Itô formula not only to the function f , but to the coefficientfunctions: L f , L f , L f , L f or in general to any function which is smooth enough as well.Since functions L f and L f depend on two arguments ( x, y ) , we admit the following rules f acting operators on the multiargument real function g ( y, x , x , ..., x l ) : L g ( y, x , ..., x l ) := ∂∂y g ( y, x , ..., x l ) b ( y ) + 12 ∂ ∂y g ( y, x , ..., x l ) σ ( y )+ Z B n g ( y + F ( y, x ) , x , ..., x l ) − g ( y, x , ..., x l ) − F ( y, x ) ∂∂y g ( y, x , ..., x l ) o ν ( dx ) L g ( y, x , ..., x l ) := ∂∂y g ( y, x , ..., x l ) σ ( y ) L g ( y, x , ..., x l , x l +1 ) := g ( y + F ( y, x l +1 ) , x , ..., x l ) − g ( y, x , ..., x l ) L g ( y, x , ..., x l , x l +1 ) := g ( y + G ( y, x l +1 ) , x , ..., x l ) − g ( y, x , ..., x l ) . To describe the higher order Itô expansion of f we will use the notion of multiindices and multiple stochastic integrals . A multiindex α = ( α , α , ..., α l ( α )) is a finite sequence ofelements such that α i ∈ { , , , } for i = 1 , , ..., l ( α ) . The number of all elements equal toa) will be denoted by s ( α ) ,b) will be denoted by w ( α ) ,c) will be denoted by ˜ n ( α ) ,d) will be denoted by n ( α ) .The length l ( α ) of α is thus given as l ( α ) = s ( α ) + w ( α ) + ˜ n ( α ) + n ( α ) . For the sake ofconvenience we also define k ( α ) := ˜ n ( α ) + n ( α ) . For technical reasons we also considerthe empty index denoted by v with length , i.e. l ( v ) = 0 . For a given multiindex α =( α , α , ..., α l ( α ) ) let us define: α − = ( α , α , ..., α l ( α ) − ) − α = ( α , ..., α l ( α ) ) . Definition 2.5
A set of multiindices A is called a hierarchical set if ∀ α ∈ A : l ( α ) < ∞ and α ∈ A\{ v } = ⇒ − α ∈ A . A set of multiindices B ( A ) , where A is a hierarchical set, is called a remainder set of A if ∀ α ∈ B ( A ) α / ∈ A and − α ∈ A . Assume that g ( s, x , x , ..., x l ) is a regular stochastic process, i.e. such that all the stochasticintegrals written below exist in the sense of Definitions 2.1 and 2.2. Let ρ and τ be fixedpoints in the interval [0 , T ] s.t. ρ ≤ τ . A multiple stochastic integral on the interval [ ρ, τ ] with respect to any multiindex α s.t. k ( α ) ≤ l is defined by the induction procedure. First,we define the integral with respect to the empty index: I v [ g ] τρ ( x , ..., x l ) = g ( τ, x , ..., x l ) . Now, assume that I α − [ g ] τρ ( x , x , ..., x k ) depends on k parameters, where ≤ k ≤ l . Thenwe define the multiple integral as follows: ) if α l ( α ) = 0 then I α [ g ] τρ ( x , ..., x k ) = Z τρ I α − [ g ] s − ρ ( x , ..., x k ) ds,
2) if α l ( α ) = 1 then I α [ g ] τρ ( x , ..., x k ) = Z τρ I α − [ g ] s − ρ ( x , ..., x k ) dW s ,
3) if α l ( α ) = 2 and k ≥ then I α [ g ] τρ ( x , ..., x k − ) = Z τρ Z B I α − [ g ] s − ρ ( x , ..., x k ) ˜ N ( ds, dx k ) ,
4) if α l ( α ) = 3 and k ≥ then I α [ g ] τρ ( x , ..., x k − ) = Z τρ Z B ′ I α − [ g ] s − ρ ( x , ..., x k ) N ( ds, dx k ) . Let us notice that it follows from the description above that I α [ g ] depends on l − k ( α ) param-eters, i.e. I α [ g ] τρ = I α [ g ] τρ ( x , x , ..., x l − k ( α ) ) . Example
Let g = g ( s, x , x , x ) . Then: I (1) [ g ] τρ ( x , x , x ) = Z τρ g ( s − , x , x , x ) dW s ,I (213) [ g ] t ( x ) = t Z Z B ′ s − Z s − Z Z B g ( s − , x , x , x ) ˜ N ( ds , dx ) dW s N ( ds , dx ) . The processes which serve as integrands in multiple integrals in the expansion of f ( Y ) willbe obtained with the use of coefficient functions f α , where α is a multiindex. We define thecoefficient function with respect to any multiindex α by the induction procedure: f v ( y ) = f ( y ) ,f α ( y, x , ..., x k ( α ) ) = L α h f − α ( y, x , ..., x k ( − α ) ) i ( y, x , ..., x k ( α ) ) . Example
For a given function f = f ( y ) we get: f (10) ( y ) = L L f,f (2013) ( y, x , x ) = L L L L f. For simplicity we omit here the dependence on arguments on the right hand side.Notice, that the coefficient function f α = f α ( y, x , ..., x k ( α ) ) depends on k ( α ) parameters,i.e. on x , x , ..., x k ( α ) . However, the multiple integral I α [ f α ] τρ = I α [ f α ( y, x , ..., x k ( α ) )] τρ does not depend on any parameter.We have the following analogue of Theorem 5.5.1 in [5] which is also called the Itô - Taylorexpansion. It is a consequence of the Itô formula and definitions of the hierarchical andremainder sets. heorem 2.6 For any hierarchical set A and a smooth function f we have the followingrepresentation: f ( Y τ ) = X α ∈A I α [ f α ( Y ρ , x , ..., x k ( α ) )] τρ + X α ∈B ( A ) I α [ f α ( Y •− , x , ..., x k ( α ) )] τρ , (2.6) assuming that all the integrals above exist. Notice that the first sum in (2.6) consists of all integrals for which the integrands do not de-pend on time while the second sum contains all integrals with the integrands dependent ontime. Since we are interested in the approximation of the process Y itself, to the end of thepaper we will consider the identity function only, i.e. f ( y ) = y .In the sequel we use two auxiliary lemmas. Lemma 2.7 (The Gronwall lemma)
Let g, h : [0 , T ] −→ R be integrable and satisfy: ≤ g ( t ) ≤ h ( t ) + L Z t g ( s ) ds for t ∈ [0 , T ] and L > . Then: g ( t ) ≤ h ( t ) + L Z t e L ( t − s ) h ( s ) ds for t ∈ [0 , T ] . Lemma 2.8
Let g be a càdlàg function on the interval [0 , T ] . Then for any ( ρ, τ ] ⊆ [0 , T ] wehave: sup s ∈ ( ρ,τ ] g ( s − ) ≤ sup s ∈ ( ρ,τ ] g ( s ) . Proof:
Let ( s n ) n =1 , ,... be a sequence such that s n ∈ ( ρ, τ ] for n = 1 , , ... satisfying g ( s n − ) −→ sup s ∈ ( ρ,τ ] g ( s − ) := K . Since g is càdlàg, for any ε > there exits a sequence ( s εn ) n =1 , ,... such that s εn ∈ ( ρ, τ ] for n = 1 , , ... and satisfies: g ( s εn ) ≥ g ( s n − ) − ε for n = 1 , , ... and thus: lim n −→∞ g ( s εn ) ≥ K − ε. Letting ε −→ we obtain sup s ∈ ( ρ,τ ] g ( s ) ≥ K. (cid:3) Our approximation of the process Y , which is the solution of (2.2), will be based on a fixedpartition τ < τ < ... < τ n = T of the interval [0 , T ] . For the sake of simplicity all the partition points are assumed to benon-random. The diameter of this partition is assumed to be smaller than δ , i.e. max i =0 , ,...,n − ( τ i +1 − τ i ) < δ . The approximation denoted by Y δ is obtained from the firstsum of multiple integrals in the Itô-Taylor expansion (2.6). The procedure can be described as ollows. Starting from the known value Y δ , which can be equal to Y , we calculate the value Y δt for t ∈ (0 , τ ] using the first sum in (2.6). Using value Y δτ we repeat the procedure for t ∈ ( τ , τ ] and so on. Denoting n t = max { k : τ k ≤ t } we define process Y δ as: Y δt = X α ∈A I α [ f α ( Y δτ nt , x , ..., x k ( α ) )] tτ nt . (3.7)The motivation for the form of the approximation is justified by the possibility of practicalcalculation multiple integrals for which integrands does not depend on time (at least for loworder integrals). In fact, in the case of integrals with respect to the compensated Poissonmeasure additional difficulty occurs which is related to the property of Lévy measure. It isdiscussed in Section 5. We focus on the problem of finding a strong approximation of order γ > , i.e. such that E sup t ∈ [0 ,T ] | Y t − Y δt | ≤ Cδ γ (3.8)for some constant C > . The rate of convergence γ is fixed and in practical application it isthe multiplicity of , i.e. γ = , , , ... .Thus our goal can be summarized as follows: for a fixed γ > find a hierarchical set A suchthat the approximation Y δ defined by (3.7) satisfies (3.8). Before formulating the main theorem let us introduce the following notation. For any multiin-dex α s.t. k ( α ) > we denote by β ( α ) a multiindex which is obtained from α by deleting allthe coordinates equal to or . Then the sets B αi for i = 1 , , ..., k ( α ) are defined as follows B αi := ( B if β ( α ) k ( α )+1 − i = 2 B ′ if β ( α ) k ( α )+1 − i = 3 . Recall that B is a unit ball and B ′ its complement. The following result is a generalization ofTheorem . . in [5] and Theorem in [2]. Theorem 4.1
Let us assume that coefficients in equation (2.2) satisfy conditions (A1) , (A2) .Let Y δ be the approximation of the form (3.7), for the solution Y of (2.2), constructed withthe use of the hierarchical set A γ , where: A γ := (cid:26) l ( α ) + s ( α ) ≤ γ or l ( α ) = s ( α ) = γ + 12 (cid:27) . (4.9) Moreover, assume that coefficient functions f α satisfy: (A3) for any α ∈ A γ holds: Z B α Z B α ... Z B αk ( α ) | f α ( y , x , ..., x k ( α ) ) − f α ( y , x , ..., x k ( α ) ) | ν ( dx k ( α ) ) ...ν ( dx ) ≤ K α | y − y | , (A4) for any α ∈ A γ ∪ B ( A γ ) holds: Z B α Z B α ... Z B αk ( α ) | f α ( y, x , x , ..., x k ( α ) ) | ν ( dx k ( α ) ) ...ν ( dx ) ≤ L α (1 + y ) , here K α , L α are some constants.Then for δ ∈ (0 , the inequality: E sup s ∈ [0 ,T ] | Y s − Y δs | ≤ E ( γ, T ) | Y − Y δ | + E ( γ, T, Y ) δ γ holds. The proof is presented at the end of this section. First we present three auxiliary lemmas anda proposition.
Lemma 4.2
Let ρ, τ be two fixed points in the interval [0 , T ] s. t. ρ < τ , τ − ρ < δ . If all theintegrals below exist then we have: E sup s ∈ ( ρ,τ ] n Z sρ g ( u, x , x , ..., x l ) du o ≤ δ E n sup u ∈ ( ρ,τ ] g ( u, x , x , ..., x l ) o , (4.10) E sup s ∈ ( ρ,τ ] n Z sρ g ( u, x , x , ..., x l ) du o ≤ δ Z τρ E (cid:8) g ( u, x , x , ..., x l ) (cid:9) du, (4.11) E sup s ∈ ( ρ,τ ] n Z sρ g ( u, x , x , ..., x l ) dW u o ≤ δ E n sup u ∈ ( ρ,τ ] g ( u, x , x , ..., x l ) o , (4.12) E sup s ∈ ( ρ,τ ] n Z sρ g ( u, x , x , ..., x l ) dW u o ≤ Z τρ E (cid:8) g ( u, x , x , ..., x l ) (cid:9) du, (4.13) E sup s ∈ ( ρ,τ ] n Z sρ Z B g ( u, x , ..., x l ) ˜ N ( du, dx l ) o ≤ δ Z B E n sup u ∈ ( ρ,τ ] g ( u, x , ..., x l ) o ν ( dx l ) , (4.14) E sup s ∈ ( ρ,τ ] n Z sρ Z B g ( u, x , ..., x l ) ˜ N ( du, dx l ) o ≤ Z τρ Z B E (cid:8) g ( u, x , x , ..., x l ) (cid:9) ν ( dx l ) du, (4.15) E sup s ∈ ( ρ,τ ] n Z sρ Z B ′ g ( u, x , ..., x l ) N ( du, dx l ) o ≤ δ (4 + δν ( B ′ )) Z B ′ E n sup u ∈ ( ρ,τ ] g ( u, x , ..., x l ) o ν ( dx l ) , (4.16) E sup s ∈ ( ρ,τ ] n Z sρ Z B ′ g ( u, x , ..., x l ) N ( du, dx l ) o ≤ δν ( B ′ )) Z τρ Z B ′ E (cid:8) g ( u, x , ..., x l ) (cid:9) ν ( dx l ) du. (4.17) ote, that due to Lemma 2.8, the lemma above remains true if we replace the upper limit " s "in the left hand side integrals with ” s − ” . Proof:
All these inequalities are proved with the use of the Schwarz and Doob inequalities,the isometric formula for stochastic integrals and Fubini’s theorem.(4.11) E sup s ∈ ( ρ,τ ] n Z sρ g ( u, x , x , ..., x l ) du o ≤ E sup s ∈ ( ρ,τ ] δ n Z sρ g ( u, x , x , ..., x l ) du o ≤ δ E n Z τρ g ( u, x , x , ..., x l ) du o = δ Z τρ E (cid:8) g ( u, x , x , ..., x l ) (cid:9) du (4.10) E sup s ∈ ( ρ,τ ] n Z sρ g ( u, x , x , ..., x l ) du o ≤ δ Z τρ E (cid:8) g ( u, x , x , ..., x l ) (cid:9) du ≤ δ E n sup u ∈ ( ρ,τ ] g ( u, x , x , ..., x l ) o (4.13) E sup s ∈ ( ρ,τ ] n Z sρ g ( u, x , x , ..., x l ) dW u o ≤ s ∈ ( ρ,τ ] E n Z sρ g ( u, x , x , ..., x l ) dW u o = 4 sup s ∈ ( ρ,τ ] E n Z sρ g ( u, x , x , ..., x l ) du o ≤ E n Z τρ g ( u, x , x , ..., x l ) du o = 4 Z τρ E n g ( u, x , x , ..., x l ) o du (4.12) E sup s ∈ ( ρ,τ ] n Z sρ g ( u, x , x , ..., x l ) dW u o ≤ E n Z τρ g ( u, x , x , ..., x l ) du o ≤ δ E n sup u ∈ ( ρ,τ ] g ( u, x , x , ..., x l ) o (4.15) E sup s ∈ ( ρ,τ ] n Z sρ Z B g ( u, x , ..., x l ) ˜ N ( du, dx l ) o ≤ s ∈ ( ρ,τ ] E n Z sρ Z B g ( u, x , ..., x l ) ˜ N ( du, dx l ) o = 4 sup s ∈ ( ρ,τ ] E n Z sρ Z B g ( u, x , ..., x l ) ν ( dx l ) du o ≤ Z τρ Z B E (cid:8) g ( u, x , x , ..., x l ) (cid:9) ν ( dx l ) du (4.14) E sup s ∈ ( ρ,τ ] n Z sρ Z B g ( u, x , ..., x l ) ˜ N ( du, dx l ) o ≤ E n Z τρ Z B g ( u, x , x , ..., x l ) ν ( dx l ) du o ≤ E n δ sup u ∈ ( ρ,τ ] Z B g ( u, x , x , ..., x l ) ν ( dx l ) o ≤ δ E n Z B sup u ∈ ( ρ,τ ] g ( u, x , x , ..., x l ) ν ( dx l ) o = 4 δ Z B E n sup u ∈ ( ρ,τ ] g ( u, x , ..., x l ) o ν ( dx l ) E sup s ∈ ( ρ,τ ] n Z sρ Z B ′ g ( u, x , ..., x l ) ˜ N ( du, dx l ) o = E sup s ∈ ( ρ,τ ] n Z sρ Z B ′ g ( u, x , ..., x l ) N ( du, dx l ) + Z sρ Z B ′ g ( u, x , ..., x l ) ν ( dx l ) du o ≤ n E sup s ∈ ( ρ,τ ] n s Z ρ Z B ′ g ( u, x , ..., x l ) ˜ N ( du, dx l ) o + E sup s ∈ ( ρ,τ ] n s Z ρ Z B ′ g ( u, x , ..., x l ) ν ( dx l ) du o o (4.18)The first component is bounded by analogous expression as in (4.15). For the second we havethe following inequalities: E sup s ∈ ( ρ,τ ] n Z sρ Z B ′ g ( u, x , ..., x l ) ν ( dx l ) du o ≤ E sup s ∈ ( ρ,τ ] n Z sρ Z B ′ ν ( dx l ) du · Z sρ Z B ′ g ( u, x , ..., x l ) ν ( dx l ) du o ≤ δν ( B ′ ) Z τρ Z B ′ E (cid:8) g ( u, x , ..., x l ) (cid:9) ν ( dx l ) du. As a consequence we obtain: E sup s ∈ ( ρ,τ ] n s Z ρ Z B ′ g ( u, x , ..., x l ) N ( du, dx l ) o ≤ δν ( B ′ )) τ Z ρ Z B ′ E (cid:8) g ( u, x , ..., x l ) (cid:9) ν ( dx l ) du. (4.16)For the second term in (4 . we have the following inequalities: E sup s ∈ ( ρ,τ ] n Z sρ Z B ′ g ( u, x , ..., x l ) ν ( dx l ) du o ≤ δν ( B ′ ) E n Z τρ Z B ′ g ( u, x , ..., x l ) ν ( dx l ) du o ≤ δ ν ( B ′ ) E n sup u ∈ ( ρ,τ ] Z B ′ g ( u, x , ..., x l ) ν ( dx l ) o ≤ δ ν ( B ′ ) E n Z B ′ sup u ∈ ( ρ,τ ] g ( u, x , ..., x l ) ν ( dx l ) o = δ ν ( B ′ ) Z B ′ E n sup u ∈ ( ρ,τ ] g ( u, x , ..., x l ) o ν ( dx l ) . Taking into account (4.18), the inequality above and (4 . we obtain: E sup s ∈ ( ρ,τ ] n s Z ρ Z B ′ g ( u, x , ..., x l ) N ( du, dx l ) o ≤ δ (4 + δν ( B ′ )) Z B ′ E n sup u ∈ ( ρ,τ ] g ( u, x , ..., x l ) o ν ( dx l ) . (cid:3) Lemma 4.3
Let ρ, τ be two fixed points in the interval [0 , T ] s.t. ρ < τ , τ − ρ < δ and α = v be a fixed multiindex. If all the integrals below exist for the process g = g ( u, x , ..., x l ) , where ≥ k ( α ) , then we have: E { sup s ∈ ( ρ,τ ] I α [ g ] sρ ( x , ..., x l − k ( α ) ) } ≤ δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) ·· Z τρ E Z B α Z B α ... Z B αk ( α ) g ( u, x , x , ...x l ) ν ( dx l ) ν ( dx l − ) ... ν ( dx l − k ( α )+1 ) du. (4.19) Proof:
We will apply the induction procedure with respect to the length of α . If l ( α ) = 1 then(4.19) follows from inequalities (4.11), (4.13), (4.15), (4.17) in Lemma 4.2 applied to α = 0 , α = 1 , α = 2 , α = 3 respectively.Now assume that (4 . is true for α − and let us show that it is also true for α . We willconsider several cases. a ) α l ( α ) = 0 ; In this case k ( α − ) = k ( α ) and B α − i = B αi for i = 1 , , ..., k ( α ) . By (4.10),Lemma 2.8 and the inductive assumption we have: E { sup s ∈ ( ρ,τ ] I α [ g ] sρ ( x , ..., x l − k ( α ) ) } = E n sup s ∈ ( ρ,τ ] Z sρ I α − [ g ] u − ρ ( x , ..., x l − k ( α − ) ) du o ≤ δ E n sup u ∈ ( ρ,τ ] I α − [ g ] u − ρ ( x , ..., x l − k ( α − ) ) o ≤ δ δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z τρ E Z B α − Z B α − ... Z B α − k ( α − ) g ( u, x , x , ..., x l ) ν ( dx l ) ν ( dx l − ) ... ν ( dx l − k ( α − )+1 ) du = δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) ·· Z τρ E Z B α Z B α ... Z B αk ( α ) g ( u, x , x , ..., x l ) ν ( dx l ) ν ( dx l − ) ... ν ( dx l − k ( α )+1 ) du. ) α l ( α ) = 1 ; In this case k ( α − ) = k ( α ) and B α − i = B αi for i = 1 , , ..., k ( α ) . By (4.12),Lemma 2.8 and the inductive assumption we have: E { sup s ∈ ( ρ,τ ] I α [ g ] sρ } ( x , ..., x l − k ( α ) ) = E n sup s ∈ ( ρ,τ ] Z sρ I α − [ g ] u − ρ ( x , ..., x l − k ( α − ) ) dW u o ≤ δ E n sup u ∈ ( ρ,τ ] I α − [ g ] u − ρ ( x , ..., x l − k ( α − ) ) o ≤ δ δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z τρ E Z B α − Z B α − ... Z B α − k ( α − ) g ( u, x , x , ...x l ) ν ( dx l ) ν ( dx l − ) ... ν ( dx l − k ( α − )+1 ) du = δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) ·· Z τρ E Z B α Z B α ... Z B αk ( α ) g ( u, x , x , ...x l ) ν ( dx l ) ν ( dx l − ) ... ν ( dx l − k ( α )+1 ) du.c ) α l ( α ) = 2 ; By (4.14), Lemma 2.8 and the inductive assumption we have: E { sup s ∈ ( ρ,τ ] I α [ g ] sρ ( x , ..., x l − k ( α ) ) } = E n sup s ∈ ( ρ,τ ] Z sρ Z B I α − [ g ] u − ρ ( x , ..., x l − k ( α )+1 ) ˜ N ( du, dx l − k ( α )+1 ) o ≤ δ Z B E n sup u ∈ ( ρ,τ ] I α − [ g ] u − ρ ( x , ..., x l − k ( α )+1 ) o ν ( dx l − k ( α )+1 ) ≤ δ δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z B Z τρ E Z B α − Z B α − ... Z B α − k ( α − ) g ( u, x , x , ...x l ) ν ( dx l ) ν ( dx l − ) ... ν ( dx l − k ( α − )+1 ) du ν ( dx l − k ( α )+1 )= δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) ·· Z τρ E Z B α Z B α ... Z B αk ( α ) g ( u, x , x , ...x l ) ν ( dx l ) ν ( dx l − ) ... ν ( dx l − k ( α )+1 ) du. ) α l ( α ) = 3 ; By (4.16), Lemma 2.8 and the inductive assumption we have: E { sup s ∈ ( ρ,τ ] I α [ g ] sρ ( x , ..., x l − k ( α ) ) } = E n sup s ∈ ( ρ,τ ] Z sρ Z B ′ I α − [ g ] u − ρ ( x , ..., x l − k ( α )+1 ) N ( du, dx l − k ( α )+1 ) o ≤ δ + δ ν ( B ′ )) Z B ′ E n sup u ∈ ( ρ,τ ] I α − [ g ] u − ρ ( x , ..., x l − k ( α )+1 ) o ν ( dx l − k ( α )+1 ) ≤ δ + δ ν ( B ′ )) δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z B ′ Z τρ E Z B α − Z B α − ... Z B α − k ( α − ) g ( u, x , x , ...x l ) ν ( dx l ) ν ( dx l − ) ... ν ( dx l − k ( α − )+1 ) du ν ( dx l − k ( α )+1 )= δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) ·· Z τρ E Z B α Z B α ... Z B αk ( α ) g ( u, x , x , ...x l ) ν ( dx l ) ν ( dx l − ) ... ν ( dx l − k ( α )+1 ) du. (cid:3) For any multiindex α = v and a process g = g ( s, x , ..., x k ( α ) ) we define two auxiliaryfunctionals: F αt [ g ] := E sup s ∈ [0 ,t ] n s − X i =0 I α [ g ] τ i +1 τ i + I α [ g ] sτ ns ! , (4.20) G αρ,τ [ g ] := E sup s ∈ [ ρ,τ ] Z B α Z B α ... Z B αk ( α ) g ( s, x , ..., x k ( α ) ) ν ( dx k ( α ) ) ...ν ( dx ) . (4.21) Lemma 4.4
For any multiindex α = ν and a process g s.t. G α ,t [ g ] < ∞ we have thefollowing inequality: F αt [ g ] ≤ t δ l ( α )+ s ( α ) − R t G α ,u [ g ] du if l ( α ) = s ( α ) C ( α, t ) δ l ( α )+ s ( α ) − R t G α ,u [ g ] du if l ( α ) = s ( α ) . Proof:
We consider several cases:a) l ( α ) = s ( α ) ,b) { w ( α ) > or ˜ n ( α ) > } andb1) α l ( α ) = 0 ,b2) α l ( α ) = 1 ,b3) α l ( α ) = 2 ,b4) α l ( α ) = 3 ,c) n ( α ) > and w ( α ) = ˜ n ( α ) = 0 . ) l ( α ) = s ( α ) By the Schwarz inequality and Lemma 4.3 we have: F αt [ g ] = E sup s ∈ [0 ,t ] (cid:16) Z s I α − [ g ] u − τ nu du (cid:17) ≤ E sup s ∈ [0 ,t ] h s Z s I α − [ g ] u − τ nu du i ≤ t Z t E (cid:16) I α − [ g ] u − τ nu (cid:17) du ≤ t Z t E sup s ∈ ( τ nu ,u ] (cid:16) I α − [ g ] s − τ nu (cid:17) du ≤ t Z t δ l ( α − )+ s ( α − ) − Z uτ nu E g ( s ) ds du ≤ t Z t δ l ( α − )+ s ( α − ) − Z uτ nu E sup w ∈ [ τ nu ,s ] g ( w ) ds du ≤ t Z t δ l ( α − )+ s ( α − ) − Z uτ nu G ατ nu ,s [ g ] ds du ≤ t Z t δ l ( α − )+ s ( α − ) − δG ατ nu ,u [ g ] du ≤ tδ l ( α )+ s ( α ) − Z t G α ,u [ g ] du. b1) { w ( α ) > or ˜ n ( α ) > } and α l ( α ) = 0 The following inequality holds: F αt [ g ] ≤ E sup s ∈ [0 ,t ] n s − X i =0 I α [ g ] τ i +1 τ i ! + 2 E sup s ∈ [0 ,t ] (cid:16) I α [ g ] sτ ns (cid:17) . Notice that the process P n s − i =0 I α [ g ] τ i +1 τ i is a martingale because it contains integral withrespect to the Wiener process or with respect to the compensated Poisson measure. First letus consider the first sum. E sup s ∈ [0 ,t ] n s − X i =0 I α [ g ] τ i +1 τ i ! ≤ s ∈ [0 ,t ] E n s − X i =0 I α [ g ] τ i +1 τ i ! = 4 sup s ∈ [0 ,t ] E n s − X i =0 I α [ g ] τ i +1 τ i + I α [ g ] τ ns τ ns − ! = 4 sup s ∈ [0 ,t ] E n s − X i =0 I α [ g ] τ i +1 τ i ! + 2 n s − X i =0 I α [ g ] τ i +1 τ i I α [ g ] τ ns τ ns − + I α [ g ] τ ns τ ns − ≤ s ∈ [0 ,t ] E n s − X i =0 I α [ g ] τ i +1 τ i ! + E sup u ∈ ( τ ns − ,τ ns ] I α [ g ] uτ ns − ≤ s ∈ [0 ,t ] n E n s − X i =0 I α [ g ] τ i +1 τ i ! + δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) ·· Z τ ns τ ns − E Z B α Z B α ... Z B αk ( α ) g ( u, x , x , ...x k ( α ) ) ν ( dx k ( α ) ) ν ( dx k ( α ) − ) ... ν ( dx ) du o s ∈ [0 ,t ] ( E n s − X i =0 I α [ g ] τ i +1 τ i ! + δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) ·· Z τ ns τ ns − G ατ ns − ,u du ) ≤ s ∈ [0 ,t ] ( E n s − X i =0 I α [ g ] τ i +1 τ i ! + δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) ·· Z τ ns − τ ns − G ατ ns − ,u du + Z τ ns τ ns − G ατ ns − ,u du ! ) ≤ s ∈ [0 ,t ] ( δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) ·· Z τ τ G ατ ,u du + Z τ τ G ατ ,u du + ... + Z τ ns τ ns − G ατ ns − ,u du ! ) ≤ δ l ( α )+ s ( α ) − w ( α )+˜ n ( α )+1 n δν ( B ′ )) o n ( α ) Z t G α ,u du. For the second sum we have the following inequalities: E sup s ∈ [0 ,t ] (cid:16) I α [ g ] sτ ns (cid:17) ≤ δ E sup s ∈ [0 ,t ] Z sτ ns I α − [ g ] u − τ ns du ≤ δ E sup s ∈ [0 ,t ] Z sτ ns sup w ∈ ( τ ns ,u ] I α − [ g ] w − τ ns du ≤ δ E Z t sup w ∈ ( τ nu ,u ] I α − [ g ] w − τ nu du = δ Z t E sup w ∈ ( τ nu ,u ] I α − [ g ] w − τ nu du ≤ δ Z t δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z uτ nu E Z B α Z B α ... Z B αk ( α ) g ( w, x , x , ...x k ( α ) ) ν ( dx k ( α ) ) ν ( dx k ( α ) − ) ... ν ( dx ) dw du ≤ δ δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) Z t Z uτ nu G ατ nu ,w [ g ] dw du ≤ δ δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) Z t G ατ nu ,u [ g ] du = δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) Z t G α ,u [ g ] du. Finally we obtain: F αt [ g ] ≤ · w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) δ l ( α )+ s ( α ) − Z t G α ,u [ g ] du.
2) { w ( α ) > or ˜ n ( α ) > } and α l ( α ) = 1 By Doob’s inequality, the isometric formula for Wiener integrals and Lemma 4.3 we obtain: F αt [ g ] = E sup s ≤ t (cid:18)Z s I α − [ g ] u − τ nu dW u (cid:19) ≤ s ≤ t E (cid:18)Z s I α − [ g ] u − τ nu dW u (cid:19) = 4 sup s ≤ t E Z s I α − [ g ] u − τ nu du = 4 Z t E (cid:16) I α − [ g ] u − τ nu (cid:17) du ≤ Z t E sup w ∈ ( τ nu ,u ] I α − [ g ] w − τ nu du ≤ Z t δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) Z uτ nu G α − τ nu ,s [ g ] ds du ≤ δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) δ Z t G α − τ nu ,u [ g ] du = δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) Z t G α ,u [ g ] du. b3) { w ( α ) > or ˜ n ( α ) > } and α l ( α ) = 2 By Doob’s inequality, the isometric formula for integrals with respect to the compensatedPoisson measure and Lemma 4.3 we obtain: F αt [ g ] = E sup s ≤ t (cid:18)Z s Z B I α − [ g ] u − τ nu ( x ) ˜ N ( du, dx ) (cid:19) ≤ s ≤ t E (cid:18)Z s Z B I α − [ g ] u − τ nu ( x ) ˜ N ( du, dx ) (cid:19) = 4 sup s ≤ t E (cid:18)Z s Z B I α − [ g ] u − τ nu ( x ) ν ( dx ) du (cid:19) = 4 E (cid:18)Z t Z B I α − [ g ] u − τ nu ( x ) ν ( dx ) du (cid:19) ≤ t Z Z B E sup w ∈ ( τ nu ,u ] I α − [ g ] w − τ nu ( x ) ν ( dx ) du ≤ t Z Z B δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z uτ nu E Z B α − Z B α − ... Z B α − k ( α − ) g ( w, x , x , ...x k ( α ) ) ν ( dx k ( α ) ) ν ( dx k ( α ) − ) ... ν ( dx ) dw ν ( dx ) du ≤ δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z t E Z uτ nu Z B α Z B α ... Z B αk ( α ) g ( w, x , x , ...x k ( α ) ) ν ( dx k ( α ) ) ν ( dx k ( α ) − ) ... ν ( dx ) ν ( dx ) dwdu δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z t E Z uτ nu sup s ∈ [ τ nu ,w ] Z B α Z B α ... Z B αk ( α ) g ( s, x , ...x k ( α ) ) ν ( dx k ( α ) ) ν ( dx k ( α ) − ) ... ν ( dx ) ν ( dx ) dwdu ≤ δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z t E δ sup s ∈ [ τ nu ,u ] Z B α Z B α ... Z B αk ( α ) g ( s, x , x , ...x k ( α ) ) ν ( dx k ( α ) ) ν ( dx k ( α ) − ) ... ν ( dx ) ν ( dx ) du = 4 δ δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) Z t G ατ nu ,u [ g ] du ≤ δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) Z t G α ,u [ g ] du. b4) { w ( α ) > or ˜ n ( α ) > } and α l ( α ) = 3 We have the following inequality: F αt [ g ] = E sup s ≤ t (cid:18)Z s Z B ′ I α − [ g ] u − τ nu ( x ) N ( du, dx ) (cid:19) = E sup s ≤ t (cid:18)Z s Z B ′ I α − [ g ] u − τ nu ( x ) ˜ N ( du, dx ) + Z s Z B ′ I α − [ g ] u − τ nu ( x ) ν ( dx ) du (cid:19) ≤ E sup s ≤ t (cid:18)Z s Z B ′ I α − [ g ] u − τ nu ( x ) ˜ N ( du, dx ) (cid:19) + 2 E sup s ≤ t (cid:18)Z s Z B ′ I α − [ g ] u − τ nu ( x ) ν ( dx ) du (cid:19) . The first term is bounded as in the case (b3). For the second term we have the followinginequalities: E sup s ≤ t (cid:18)Z s Z B ′ I α − [ g ] u − τ nu ( x ) ν ( dx ) du (cid:19) ≤ E sup s ≤ t (cid:18)Z s Z B ′ ν ( dx ) du · Z s Z B ′ I α − [ g ] u − τ nu ( x ) ν ( dx ) du (cid:19) ≤ δν ( B ′ ) t Z Z B ′ E (cid:16) I α − [ g ] u − τ nu ( x ) (cid:17) ν ( dx ) du ≤ δν ( B ′ ) t Z Z B ′ E sup w ∈ ( τ nu ,u ] I α − [ g ] w − τ nu ( x ) ν ( dx ) du δν ( B ′ ) Z t Z B ′ δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z uτ nu E Z B α − Z B α − ... Z B α − k ( α − ) g ( w, x , x , ...x k ( α ) ) ν ( dx k ( α ) ) ν ( dx k ( α ) − ) ... ν ( dx ) dw ν ( dx ) du = δν ( B ′ ) δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) ·· Z t E Z uτ nu Z B α Z B α ... Z B αk ( α ) g ( w, x , x , ...x k ( α ) ) ν ( dx k ( α ) ) ν ( dx k ( α ) − ) ... ν ( dx ) dwdu... and omitting identical operations as in (b3) we obtain: ... ≤ δ ν ( B ′ ) δ l ( α − )+ s ( α − ) − w ( α − )+˜ n ( α − ) n δν ( B ′ )) o n ( α − ) Z t G α ,u [ g ] du = δ l ( α )+ s ( α ) − δν ( B ′ )4 w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) − Z t G α ,u [ g ] du. Finally, for this case we have: F αt [ g ] ≤ δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) Z t G α ,u [ g ] du + 2 δ l ( α )+ s ( α ) − δν ( B ′ )4 w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) − Z t G α ,u [ g ] du ≤ δ l ( α )+ s ( α ) − w ( α )+˜ n ( α ) n δν ( B ′ )) o n ( α ) − n δν ( B ′ ) o Z t G α ,u [ g ] du. c) n ( α ) > and w ( α ) = ˜ n ( α ) = 0 In this case the multiindex α consists of and only. If α l ( α ) = 3 then the desired inequalityfollows from (b4). In opposite case let us denote r ( α ) := max { i : α i = 3 } . For simplicity ofexposition we show the case when r ( α ) = l ( α ) − . The idea for other cases is exactly thesame. We have the following inequality: F αt [ g ] ≤ E sup s ≤ t (cid:18) Z s u − Z τ nu Z B ′ I α −− [ g ] w − τ nu ( x ) ˜ N ( dx , dw ) du (cid:19) +2 E sup s ≤ t (cid:18) Z s u − Z τ nu Z B ′ I α −− [ g ] w − τ nu ( x ) ν ( dx ) dw du (cid:19) . Calculations for the first term in the sum above are covered by (b1). Applying the Schwarzinequality and Lemma 4.3 for the second term we obtain: E sup s ≤ t (cid:18) Z s u − Z τ nu Z B ′ I α −− [ g ] w − τ nu ( x ) ν ( dx ) dw du (cid:19) ≤ t E Z t (cid:18) u − Z τ nu Z B ′ I α −− [ g ] w − τ nu ( x ) ν ( dx ) dw (cid:19) du t Z t E sup s ∈ ( τ nu ,u ] (cid:18) s − Z τ nu Z B ′ I α −− [ g ] w − τ nu ( x ) ν ( dx ) dw (cid:19) du ≤ tδν ( B ′ ) Z t u Z τ nu Z B ′ E I α −− [ g ] w − τ nu ( x ) ν ( dx ) dw du ≤ tδν ( B ′ ) Z t u Z τ nu Z B ′ E sup s ∈ ( τ nu ,w ] I α −− [ g ] s − τ nu ( x ) ν ( dx ) dw du ≤ tδν ( B ′ ) δ l ( α −− )+ s ( α −− ) − n δν ( B ′ )) o n ( α −− ) ·· Z t u Z τ nu Z B ′ w Z τ nu E Z B α −− ... Z B α −− k ( α −− ) g ( s, x , ..., x k ( α ) ) ν ( dx k ( α ) ) ...ν ( dx ) ds ν ( dx ) dw du ≤ tδ l ( α −− )+ s ( α −− ) ν ( B ′ ) n δν ( B ′ )) o n ( α −− ) · Z t u Z τ nu w Z τ nu E Z B α ... Z B αk ( α ) g ( s, x , ..., x k ( α ) ) ν ( dx k ( α ) ) ...ν ( dx ) ds dw du ≤ tδ l ( α −− )+ s ( α −− ) ν ( B ′ ) n δν ( B ′ )) o n ( α −− ) Z t u Z τ nu δG ατ nu ,w dw du ≤ tδ l ( α −− )+ s ( α −− ) ν ( B ′ ) n δν ( B ′ )) o n ( α −− ) Z t δ G ατ nu ,u du ≤ tδ l ( α )+ s ( α ) − ν ( B ′ ) n δν ( B ′ )) o n ( α ) − Z t G α ,u du. Finally we have: F αt [ g ] ≤ δ l ( α )+ s ( α ) − n δν ( B ′ )) o n ( α ) − n tν ( B ′ ) o Z t G α ,u [ g ] du. (cid:3) Proposition 4.5
Let A be any hierarchical set. If for each α ∈ A the condition: Z B α Z B α ... Z B αk ( α ) | f α ( y, x , x , ..., x k ( α ) ) | ν ( dx k ( α ) ) ...ν ( dx ) ≤ L α (1 + y ) (4.22) holds, then the approximation Y δ given by (3.7) satisfies: E sup ≤ s ≤ T | Y δs | ≤ C (1 + | Y δ | ) ∀ t ∈ [0 , T ] , where C ≥ . roof: Due to (3.7) we write the approximation in the following form Y δs = Y δ + X α ∈A\{ v } n s − X i =0 I α [ f α ( Y δτ i , x , ..., x k ( α ) )] τ i +1 τ i + I α [ f α ( Y δτ ns , x , ..., x k ( α ) )] sτ ns ! . By Lemma 4.4 and assumption (4.22) we have the following inequalities: E sup s ≤ t | Y δs | ≤ ♯ ( A ) (cid:26) | Y δ | + X α ∈A E sup ≤ s ≤ T (cid:16) n s − X i =0 I α [ f α ( Y δτ i , x , ..., x k ( α ) )] τ i +1 τ i + I α [ f α ( Y δτ ns , x , ..., x k ( α ) )] sτ ns (cid:17) (cid:27) ≤ ♯ ( A ) (cid:26) | Y δ | + X α ∈A Z t E sup u ∈ [0 ,s ] Z B α Z B α ... Z B αk ( α ) f α ( Y δτ nu , x , ..., x k ( α ) ) ν ( dx k ( α ) ) ...ν ( dx ) ds (cid:27) ≤ ♯ ( A ) (cid:26) | Y δ | + X α ∈A Z t E sup u ∈ [0 ,s ] Z B α Z B α ... Z B αk ( α ) f α ( Y δu , x , ..., x k ( α ) ) ν ( dx k ( α ) ) ...ν ( dx ) ds (cid:27) ≤ ♯ ( A ) (cid:26) | Y δ | + X α ∈A Z t E sup u ∈ [0 ,s ] L α (1 + | Y δu | ) ds (cid:27) ≤ ♯ ( A ) (cid:26) | Y δ | + T X α ∈A L α + X α ∈A L α Z t E sup u ≤ s | Y δu | ds (cid:27) . By applying the Gronwall lemma 2.7 we obtain the required result. (cid:3)
Now we are ready to present the main result’s proof.
Proof of Theorem 4.1:
We write the solution Y of (2.2) and its approximation Y δ in theforms: Y s = Y + X α ∈A γ \{ v } n s − X i =0 I α [ f α ( Y τ i , x , ..., x k ( α ) )] τ i +1 τ i + I α [ f α ( Y τ ns , x , ..., x k ( α ) )] sτ ns ! (4.23) + X α ∈B ( A γ ) n s − X i =0 I α [ f α ( Y •− , x , ..., x k ( α ) )] τ i +1 τ i + I α [ f α ( Y •− , x , ..., x k ( α ) )] sτ ns ! ,Y δs = Y δ + X α ∈A γ \{ v } n s − X i =0 I α [ f α ( Y δτ i , x , ..., x k ( α ) )] τ i +1 τ i + I α [ f α ( Y δτ ns , x , ..., x k ( α ) )] sτ ns ! . Due to Proposition 2.4 and Proposition 4.5 the error of the approximation Z t := E sup s ≤ t | Y s − Y δs | is finite and satisfies the inequality: Z t ≤ D ( γ ) | Y − Y δ | + X α ∈A γ \{ v } R αt + X α ∈B ( A γ ) U αt , (4.24) here D ( γ ) = ♯ {A γ ∪ B ( A γ ) } ,R αt = E sup s ≤ t (cid:18) n s − X i =0 I α h f α ( Y τ i , x , ..., x k ( α ) ) − f α ( Y δτ i , x , ..., x k ( α ) ) i τ i +1 τ i (4.25) + I α h f α ( Y τ ns , x , ..., x k ( α ) ) − f α ( Y δτ ns , x , ..., x k ( α ) ) i sτ ns (cid:19) ,U αt = E sup s ≤ t (cid:18) n s − X i =0 I α [ f α ( Y •− , x , ..., x k ( α ) )] τ i +1 τ i + I α [ f α ( Y •− , x , ..., x k ( α ) )] sτ ns (cid:19) . (4.26)Let us denote D ( α, T ) := sup t ∈ [0 ,T ] max { t, C ( α, t ) } where C ( α, t ) is a constant fromLemma (4.4). Since δ l ( α )+ s ( α ) − < δ l ( α )+ s ( α ) − < , by Lemma 4.4 and assumption (A3) we have the following inequality for any α ∈ A γ \{ v } : R αt ≤ D ( α, T ) Z t E sup s ≤ u h f α ( Y τ ns , x , ..., x k ( α ) ) − f α ( Y δτ ns , x , ..., x k ( α ) ) i du ≤ D ( α, T ) Z t E sup s ≤ u Z B α Z B α ... Z B αk ( α ) h f α ( Y s , x , ..., x k ( α ) ) − f α ( Y δs , x , ..., x k ( α ) ) i ν ( dx k ( α ) ) ...ν ( dx ) du ≤ D ( α, T ) K α Z t E sup s ≤ u | Y s − Y δs | du = D ( α, T ) K α Z t Z u du. For any α ∈ B ( A γ ) inequality: l ( α ) + s ( α ) − > l ( α ) + s ( α ) − ≥ γ is satisfied. Due tothis fact, assumption (A4) , Proposition 2.4 and Lemma 2.8 we have the following inequalities: U αt ≤ D ( α, T ) δ γ Z t G α ,u h f α ( Y •− , x , ..., x k ( α ) ) i du ≤ D ( α, T ) δ γ Z t E sup s ≤ u Z B α Z B α ... Z B αk ( α ) h f α ( Y s − , x , ..., x k ( α ) ) i ν ( dx k ( α ) ) ...ν ( dx ) du ≤ D ( α, T ) δ γ L α Z t E sup s ≤ u | Y s − | du ≤ D ( α, t ) δ γ L α Z t (1 + C (1 + Y )) du ≤ δ γ D ( α, T ) L α T (1 + C (1 + Y )) . Finally, denoting shorter relevant constants we have: R αt ≤ D ( α, T ) Z t Z u du, U αt ≤ D ( α, T, Y ) δ γ . Coming back to (4.24) we obtain Z t ≤ D ( γ ) | Y − Y δ | + ˜ D ( γ, T ) Z t Z u du + ˜ D ( γ, T, Y ) δ γ , (4.27) here ˜ D ( γ, T ) := D ( γ ) P α ∈A γ \{ v } D ( α, T ) and ˜ D ( γ, T, Y ) := D ( γ ) P α ∈B ( A γ ) D ( α, T, Y ) . Applying the Gronwall lemma 2.7 to (4.27)we obtain: Z t ≤ E ( γ, T ) | Y − Y δ | + E ( γ, T, Y ) δ γ , where: E ( γ, T ) = D ( γ ) e ˜ D ( γ,T ) T , E ( γ, T, Y ) = ˜ D ( γ, Y , T ) e ˜ D ( γ,T ) T . (cid:3) The strong approximation described by Theorem 4.1 can not always be easily constructedin practice even for low order of convergence. In case when ν ( B ) = ∞ the integrals withrespect to the compensated Poisson measure are difficult to obtain even for simple integrands.In this section we formulate alternative theorem which describes approximation with the useof integrals which can be practically derived.For a fixed ε ∈ (0 , we split the unit ball B into the ball B ε with radius ε and the disc D ε = B \ B ε . Our idea is to modify the approximation given by Theorem 4.1 by exchangingall the integrals on unit balls with respect to the compensated Poisson measure for integralson discs D ε .For the use of this section we extend the inductive definition of multiple stochastic integralintroduced in Section 2. To this end for any multiindex α let us define a set of subscripts Π( α ) consisting of vectors π ( α ) = ( π ( α ) , π ( α ) , ..., π ˜ n ( α ) ( α )) of length ˜ n ( α ) with coordinatesequal to or , i.e. π ( α ) ∈ Π( α ) ⇐⇒ ( π i ( α ) = 0 or π i ( α ) = 1 for i = 1 , , ..., ˜ n ( α ) if ˜ n ( α ) > v if ˜ n ( α ) = 0 . The empty subscript v , i.e. the subscript of length zero is introduced for technical reasons.The subscripts for the multiindices α and α − are related in the following way: π ( α − ) = ( π ( α ) if α l ( α ) = 0 , , π ( α ) , π ( α ) , ..., π ˜ n ( α ) − ( α )) if α l ( α ) = 2 . For a process g = g ( s, x , ..., x l ) , a multiindex α s.t. k ( α ) ≤ l and a subscript π ( α ) ∈ Π( α ) we define the multiple integral by the induction procedure.If ˜ n ( α ) = 0 then I εα v [ g ] τρ ( x , ..., x l ) = I α [ g ] τρ ( x , ..., x l ) . Assume that I εα − π ( α − ) [ g ] τρ ( x , x , ..., x k ) depends on k parameters, where k ≤ l . Then:1) if α l ( α ) = 0 then I εα π ( α ) [ g ] τρ ( x , ..., x k ) = Z τρ I α − π ( α − ) [ g ] s − ρ ( x , ..., x k ) ds,
2) if α l ( α ) = 1 then I εα π ( α ) [ g ] τρ ( x , ..., x k ) = Z τρ I α − π ( α − ) [ g ] s − ρ ( x , ..., x k ) dW s , ) if α l ( α ) = 2 and π ˜ n ( α ) ( α ) = 0 and k ≥ then I εα π ( α ) [ g ] τρ ( x , ..., x k − ) = Z τρ Z B ε I α − π ( α − ) [ g ] s − ρ ( x , ..., x k ) ˜ N ( ds, dx k ) ,
4) if α l ( α ) = 2 and π ˜ n ( α ) ( α ) = 1 and k ≥ then I εα π ( α ) [ g ] τρ ( x , ..., x k − ) = Z τρ Z D ε I α − π ( α − ) [ g ] s − ρ ( x , ..., x k ) ˜ N ( ds, dx k ) ,
5) if α l ( α ) = 3 and k ≥ then I εα π ( α ) [ g ] τρ ( x , ..., x k − ) = Z τρ Z B ′ I α − π ( α − ) [ g ] s − ρ ( x , ..., x l − k ) N ( ds, dx k ) . In fact the last integral does not depend on ε , nevertheless, we use this notation for technicalreasons. Example
Assume that g is of the form g ( s, x , x , ) . Then: I (212) (1 , [ g ] t = t Z Z B ε s − Z s − Z Z D ε g ( s − , x , x ) ˜ N ( ds , dx ) dW s ˜ N ( ds , dx ) . For any hierarchical set A let us denote by A a subset of multiindices containing at least oneelement equal to 2, i.e. α ∈ A iff α ∈ A and ˜ n ( α ) > . Remark 5.1
Let ε > . For any α ∈ A and a process g = g ( s, x , x , ..., x k ( α ) ) thefollowing equality holds: I α [ g ] τρ = X π ∈ Π( α ) I εα π ( α ) [ g ] τρ . Remark 5.2
If we replace in the formulas (4.20),(4.21) the unit balls in integrals with respectto the compensated Poisson measure by ε -balls or ε -discs, then Lemma 4.4 remains true. Asa consequence, for a process: Y δ,εs = X α ∈A\A I α [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns + X α ∈A I εα (1 , ,..., [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns we obtain analogous estimation as in Proposition 4.5, i.e. E sup s ≤ t | Y δ,εs | ≤ C (1 + | Y δ,ε | ) ∀ t ∈ [0 , T ] , where C > , assuming that (4.22) is satisfied. Theorem 5.3
Assume that coefficients in equation (2.2) satisfy conditions (A1),(A2) . Let A γ be a hierarchical set given by (4.9) and assume that (A3),(A4) hold. Assume that for any α ∈ A γ there exists a constant L εα such that for every i s.t. α i = 2 holds: Z B α Z B α ... Z B ε ... Z B αk ( α ) | f α ( y, x , x , ..., x k ( α ) ) | ν ( dx k ( α ) ) ...ν ( dx ) ≤ L εα (1 + y ) , (5.28) here B ε is on the position k ( α ) − i + 1 and L εα −→ ε −→ .Then the approximation defined by the formula: Y δ,εs = X α ∈A γ \A γ I α [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns + X α ∈A γ I εα (1 , ,..., [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns satisfies: E sup s ∈ [0 ,T ] | Y s − Y δ,εs | ≤ N ( γ, T ) | Y − Y δ,ε | + N ( γ, T, Y ) δ γ + N ( γ, T, Y δ,ε , ε ) , where N ( γ, T, Y δ,ε , ε ) −→ ε −→ . Proof:
We write the approximation in the form: Y δ,εs = X α ∈A γ I α [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns − X α ∈A γ I α [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns + X α ∈A γ I εα (1 , ,..., [ f α ( Y δ,ετ ns , x , ..., x ˜ n ( α )+ n ( α ) )] sτ ns = Y δ,ε + X α ∈A γ \{ v } n s − X i =0 I α [ f α ( Y δ,ετ i , x , ..., x k ( α ) )] τ i +1 τ i + I α [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns ! − X α ∈A γ n s − X i =0 I α [ f α ( Y δ,ετ i , x , ..., x k ( α ) )] τ i +1 τ i + I α [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns ! + X α ∈A γ n s − X i =0 I εα (1 , ,..., [ f α ( Y δ,ετ i , x , ..., x k ( α ) )] τ i +1 τ i + I εα (1 , ,..., [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns ! . By Remark 5.2 and Proposition 2.4 the error Z t := E sup s ≤ t | Y s − Y δ,εs | is finite. Takinginto account (4.23) we have: Z t ≤ M ( γ ) | Y − Y δ,ε | + X α ∈A γ \{ v } R αt + X α ∈B ( A γ ) U αt + X α ∈A γ S αt , (5.29)where M ( γ ) = ♯ {A γ } + ♯ {B ( A γ ) } + ♯ {A γ } ,R αt is defined by (4.25) with Y δ replaced by Y δ,ε and U αt by (4.26) and S αt = E sup s ≤ t (cid:18) n s − X i =0 (cid:16) I α [ f α ( Y δ,ετ i , x , ..., x k ( α ) )] τ i +1 τ i − I εα (1 , ,..., [ f α ( Y δ,ετ i , x , ..., x k ( α ) )] τ i +1 τ i (cid:17) + I α [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns − I εα (1 , ,..., [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns (cid:19) . ue to Remark 5.1 we have: S αt = E sup s ≤ t n s − X i =0 (cid:18) X π ( α ) ∈ Π( α ) ,π ( α ) =(1 , ,..., I εα π ( α ) [ f α ( Y δ,ετ i , x , ..., x k ( α ) )] τ i +1 τ i (cid:19) + (cid:18) X π ( α ) ∈ Π( α ) ,π ( α ) =(1 , ,..., I εα π ( α ) [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns (cid:19)! ≤ (cid:16) ♯ { Π( α ) } − (cid:17) X π ∈ Π( α ) ,π =(1 , ,..., E sup s ≤ t n s − X i =0 I εα π ( α ) [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] τ i +1 τ i + I εα π ( α ) [ f α ( Y δ,ετ ns , x , ..., x k ( α ) )] sτ ns ! . In the sum above each integral contains at least one integral on ε -ball. Using assumption (5 . and Remark 5.2 we obtain: S αt ≤ (cid:16) ♯ { Π( α ) } − (cid:17) X π ( α ) ∈ Π( α ) ,π =(1 , ,..., C ( α, t ) L εα Z t E sup s ≤ u (1 + | Y δ,εs | ) du ≤ (cid:16) ( ♯ { Π( α ) } − (cid:17) D ( α, T ) L εα T (1 + C (1 + | Y δ,ε | )) =: L εα · D ( α, T, Y δ,ε ) , Coming back to (5 . and using notation of constants from the proof of Theorem . weobtain: Z t ≤ M ( γ ) | Y − Y δ,ε | + M ( γ, T ) Z t Z u du + M ( γ, T, Y ) δ γ + M ( γ, T, Y δ,ε , ε ) , where M ( γ, T ) = M ( γ ) P α ∈A γ \ v D ( α, T ) , M ( γ, T, Y ) = M ( γ ) P α ∈B ( A γ ) D ( α, T, Y ) and M ( γ, T, Y δ,ε , ε ) = M ( γ ) P α ∈A γ L εα · D ( α, t, Y δ,ε ) . Finally, applying the Gronwalllemma 2.7 we obtain: Z t ≤ N ( γ, T ) | Y − Y δ,ε | + N ( γ, T, Y ) δ γ + N ( γ, T, Y δ,ε , ε ) , where N ( γ, T ) = M ( γ ) e M ( γ,T ) T ; N ( γ, T, Y ) = M ( γ, T, Y ) e M ( γ,T ) T ; N ( γ, T, Y δ,ε , ε ) = M ( γ, T, Y δ,ε , ε ) e M ( γ,T ) T = e M ( γ,T ) T M ( γ ) P α ∈A γ L εα · D ( α, t, Y δ,ε ) . (cid:3) We present the Euler ( γ = ) and Milstein ( γ = 1 ) schemes in for linear coefficients, i.e. b ( y ) = by, σ ( y ) = σy, F ( y, x ) = F yp ( x ) , G ( y, x ) = Gyq ( x ) where σ, b, F, G are constants and functions p ( · ) , q ( · ) satisfy integrability conditions: R B p ( x ) ν ( dx ) < ∞ , R B ′ q ( x ) ν ( dx ) < ∞ . Then assumptions (A1),(A2) are satisfied.For finding integrals with respect to the Poisson random measure we use the representation f random measures, see for instance Th. . in [6], applied to a set E s.t. ν ( E ) < ∞ . Therandom measure N ( · , · ) can be represented as N ( t, E ) = X n> [0 ,t ] × E ( η n , ξ n ) , where η n = r + r + ... + r n and { ξ n } , { r n } are mutually independent random variableswith distributions: P ( ξ n ∈ A ) = ν ( A ∩ E ) ν ( E ) , ∀ A ∈ B ( R ) , P ( r n > s ) = e − ν ( E ) s , s ≥ . In the following constructions we assume that ν ( B ) < ∞ and as a consequence that N (( τ i , τ i +1 ] , B ∪ B ′ ) =: K ( i ) < ∞ . Then all the moments of jumps generated by the Poisson random measure N in the interval ( τ i , τ i +1 ] form a sequence: η < η < ... < η K ( i ) . We omit the depen-dence of this sequence on i to simplify notation. For the sake of clarity we use the followingnotation: ¯ η n = min { η k : η k > η n and ξ k ∈ B ′ } ∧ τ i +1 ,η n = min { η k : η k > η n and ξ k ∈ B } ∧ τ i +1 . Condition ν ( B ) < ∞ guaranties that all the formulas below can be practically derived. If itis not satisfied, then we apply Theorem 5.3 by replacing all unit balls in the approximationby ε - discs. In this case K ( i ) and η n are defined with the use of D ε instead of B . Since N (( τ i , τ i +1 ] , D ε ∪ B ′ ) < ∞ the modified approximation can be calculated. We also find thedependence of the approximation error on ε .Notational remark: if the range of indices in the sums below is empty, then the sum is assumedto be zero. The Euler scheme
The hierarchical set and the remainder sets are of the form A = { v, , , , } , B ( A ) = { , , , , , , , , , , , , , , , } . It can be easily checked that con-ditions (A3), (A4) are also satisfied. The approximation has the following form: Y δτ i +1 = Y δτ i + I [ f ( Y δτ i )] τ i +1 τ i + I [ f ( Y δτ i )] τ i +1 τ i + I [ f ( Y δτ i , x )] τ i +1 τ i + I [ f ( Y δτ i , x )] τ i +1 τ i where: I [ f ( Y δτ i )] τ i +1 τ i = τ i +1 Z τ i bY δτ i ds = bY δτ i △ i ,I [ f ( Y δτ i )] τ i +1 τ i = τ i +1 Z τ i σY δτ i dW s = σY δτ i △ Wi ,I [ f ( Y δτ i , x )] τ i +1 τ i = τ i +1 Z τ i Z B F · Y δτ i p ( x ) ˜ N ( ds, dx ) , F Y δτ i K ( i ) X n =1 B ( ξ n ) p ( ξ n ) − △ i Z B p ( x ) ν ( dx ) ,I [ f ( Y δτ i , x )] τ i +1 τ i = τ i +1 Z τ i Z B ′ G · Y δτ i q ( x ) N ( ds, dx ) = GY δτ i K ( i ) X n =1 B ′ ( ξ n ) q ( ξ n ) , △ i = τ i +1 − τ i , △ Wi = W τ i +1 − W τ i . If ν ( B ) = ∞ we apply Theorem . . Notice that condition (5 . is satisfied since Z B ε | f ( y, x ) | ν ( dx ) = F y Z B ε p ( x ) ν ( dx ) −→ ε → , so L εα = R B ε p ( x ) ν ( dx ) . It follows from the proof of Theorem . that: N (cid:16) , T, Y δ,ε , ε (cid:17) = K ( T, Y δ,ε ) Z B ε p ( x ) ν ( dx ) , where K ( T, Y δ,ε ) > . The Milstein scheme
The hierarchical and remainder sets are of the form: A = { v, , , , , , , , , , , , , } , B ( A ) = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } . Assumptions ( A3 ) , ( A4 ) are satisfied. The approximation is of the following form: Y δτ i +1 = Y δτ i + I [ f ( Y δτ i )] τ i +1 τ i + I [ f ( Y δτ i )] τ i +1 τ i + I [ f ( Y δτ i , x )] τ i +1 τ i + I [ f ( Y δτ i , x )] τ i +1 τ i + I [ f ( Y δτ i )] τ i +1 τ i + I [ f ( Y δτ i )] τ i +1 τ i + I [ f ( Y δτ i , x )] τ i +1 τ i + I [ f ( Y δτ i )] τ i +1 τ i + I [ f ( Y δτ i )] τ i +1 τ i + I [ f ( Y δτ i , x )] τ i +1 τ i + I [ f ( Y δτ i )] τ i +1 τ i + I [ f ( Y δτ i )] τ i +1 τ i + I [ f ( Y δτ i , x )] τ i +1 τ i , where I [ f ( Y δτ i )] τ i +1 τ i , I [ f ( Y δτ i )] τ i +1 τ i , I [ f ( Y δτ i , x )] τ i +1 τ i , I [ f ( Y δτ i , x )] τ i +1 τ i are like in the Eu-ler scheme and I [ f ( Y δτ i )] τ i +1 τ i = 12 σ Y δτ i (cid:0) ( △ Wi ) − △ i (cid:1) ,I [ f ( Y δτ i , x )] τ i +1 τ i = GσY τ i Z τ i +1 τ i Z B ′ q ( x )( W s − W τ i ) N ( ds, dx )= GσY τ i K ( i ) X n =1 q ( ξ n )( W η n − W τ i ) B ′ ( ξ n ) , [ f ( Y δτ i , x )] τ i +1 τ i = F σY τ i (cid:18) Z τ i +1 τ i Z B p ( x )( W s − W τ i ) N ( ds, dx ) − Z τ i +1 τ i Z B p ( x )( W s − W τ i ) ν ( dx ) ds (cid:19) = F σY τ i K ( i ) X n =1 p ( ξ n )( W η n − W τ i ) B ( ξ n ) − Z B p ( x ) ν ( dx ) · △ Zi , where △ Zi = R τ i +1 τ i ( W s − W τ i ) ds is a random variable with distribution N (0 , △ i ) , corre-lated with △ Wi , i.e. E ( △ Wi △ Zi ) = △ i . The pair △ Wi , △ Zi can be generated by transforma-tion of two independent random variables U , U with distributions N (0 , in the followingway: △ Wi = U √△ i , △ Zi = △ i ( U + √ U ) , for more details see [5]. I [ f ( Y δτ i , x )] τ i +1 τ i = GσY τ i Z τ i +1 τ i Z s − τ i Z B ′ q ( x ) N ( du, dx ) dW s = GσY τ i K ( i ) X n =1 n X
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