Feynman-Kac formula for Lévy processes
with discontinuous killing rate
Technische Universität München, Center for Mathematics
The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to compute option prices in Lévy models by solving partial integro differential equations have been developed. In order to provide a solid mathematical foundation for these methods, we derive a Feynman-Kac representation of variational solutions to partial integro differential equations that characterize conditional expectations of functionals of killed time-inhomogeneous Lévy processes. We allow for a wide range of underlying stochastic processes, comprising processes with Brownian part as well as a broad class of pure jump processes such as generalized hyperbolic, multivariate normal inverse Gaussian, tempered stable, and -semi stable Lévy processes. By virtue of our mild regularity assumptions as to the killing rate and the initial condition of the partial integro differential equation, our results provide a rigorous basis for numerous applications, in financial mathematics, probability theory and physics. We reencounter the original ideas of Feynman and Kac, but now revealing the normal inverse Gaussian process in its role connecting the relativistic Schrödinger equation to stochastic processes. In Regard to finance we suggest a flexible class of employee options. We implement a Galerkin scheme to solve the attendant pricing equation numerically and illustrate the effect of a killing rate.
Key words and phrases:
Time-inhomogeneous Lévy process, killing rate, Feynman-Kac representation, weak solution, variational solution, parabolic evolution equation, partial integro differential equation, pseudo differential equation, nonlocal operator, fractional Laplace operator, Sobolev-Slobodeckii spaces, option pricing, Laplace transform of occupation time, relativistic Schrödinger equation, employee option, Galerkin method
2000 Mathematics Subject Classification:35S10, 60G51, 60-08, 47G20, 47G30
Feynman-Kac formulas play a distinguished role in probability theory and functional analysis. Ever since their birth in 1949, Feynman-Kac-type formulas have been a constant source of fascinating insights in a wide range of disciplines. They originate in the description of particle diffusion by connecting Schrödinger’s equation and the heat equation to the Brownian motion, see (34). A type of Feynman-Kac formula also figures at the beginning of modern mathematical finance: In their seminal article of 1973, Black and Scholes derived their Nobel Prize-winning option pricing formula by expressing the price as a solution to a partial differential equation, thereby rediscovering Feynman and Kac’s deep link.
The fundamental contribution of Feynman-Kac formulas is to link stochastic processes to solutions of deterministic partial differential equations. Thus they also establish a connection between probability theory and numerical analysis, two disciplines that have evolved largely separately. Although both enjoy great success, transfer between them has remained only incidental. This may very well be the reason for applications of Feynman-Kac still appearing so surprisingly fresh. In computational finance, they enable the development of option pricing methods by solving deterministic evolution equations. These have proven to be highly efficient, particularly when compared to Monte Carlo simulation. Thus, like other deterministic methods, they come into play whenever efficiency is essential and the complexity of the pricing problem is not too high. This is the case for recurring tasks, such as calibration and real-time pricing, and over the last few decades has given rise to extensive research in computing option prices by solving partial differential equations. The challenge to extend these methods to price options in advanced jump models has furthermore inspired researchers in recent years to develop highly efficient and widely applicable algorithms, see for instance (14), Hilber, Reich, Schwab and Winter (2009), Hilber, Reichmann, Schwab and Winter (2013), Salmi, Toivanen and Sydow (2014) and (28).
In this article we derive a Feynman-Kac-type formula so as to provide a solid mathematical basis for fast option pricing in time-inhomogeneous Lévy models using partial integro differential equations (PIDEs). While large parts of the literature focus on numerical aspects of these pricing methods, only little is known about the precise link between the related deterministic equations and the corresponding conditional expectations representing option prices. Our main question therefore is: Under which conditions is there a Feynman-Kac formula linking option prices given by conditional expectations with solutions to evolution equations?
In order to further specify the problem, we focus on time-inhomogeneous Lévy models and options whose path dependency may be expressed by a killing rate. In this setting with the Kolmogorov operator of a time-inhomogeneous Lévy process, killing rate (or potential) , source and initial condition , the Kolmogorov equation is of the form
Adopting a heuristic approach, one would typically assume that equation (1) has a classical solution . If this solution is sufficiently regular to allow for an application of Itô’s formula and moreover satisfies an appropriate integrability condition, the following Feynman-Kac-type representation
follows by standard arguments and taking conditional expectations, see equations (76) and (77) on page 77 for a detailed derivation. Then, the conditional expectation (2) can be obtained by solving Kolmogorov equation (1) by means of a deterministic numerical scheme. Such an argumentation hinges on a strong regularity assumption on the solution and thus implicitly on the data of the equation, , , and . We have to realize, however, that this constitutes a serious restriction on the applicability of such a heuristic approach.
To do justice to the complexities of financial applications, we pay special attention to identifying appropriate conditions for the validity of equation (2) for financial applications. Often, discontinuous killing rates constitute a natural choice, as we will show in several detailed examples in Section 5. In particular indicator functions as killing rates turn out to be key to a wide variety of applications, both in mathematical finance and in probability theory. As one typical application we propose and study a flexible family of employee options in Section 5.1 and illustrate the numerical effect of such killing rates in Section 6. The fundamental role of killing rates of indicator type is killing the process outside a specified domain, which makes them attractive for applications. Moreover, they are closely related to occupation times and exit times of stochastic processes as we outline in Sections 5.3 and 5.4. We furthermore find that discontinuous killing rates form a common root of exit probabilities of stochastic processes and the distribution of supremum processes. As such they apply to the prices of path-dependent options like those of barrier, lookback, and American type. In view of these considerations, which are both of a theoretical and applied nature, we will also want to allow for non-smooth and even discontinuous killing rates in Kolmogorov equation (1).
Discontinuities in the killing rate result in non-smoothness of the solution of Kolmogorov equation (1). In particular, one cannot expect . Assume and in (1), then implies , which obviously is a contradiction. Hence, for our purposes, the assumption that Itô’s formula can be applied to the solution is futile. Neither is it reasonable to assume that equation (1) has a classical solution. Let us emphasize that such irregularity is not only inherent in equation (1) if the killing rate is discontinuous, but also a typical feature of Kolmogorov equations for other path-dependent option prices. Prominent examples are boundary value problems related to barrier options in Lévy models as well as free boundary value problems for American option prices. In each of these cases, the use of a generalized solution concept is called for.
Among the possible generalizations of classical solutions of partial differential equations, we find that viscosity and weak solutions are the ones that are most commonly discussed. Viscosity solutions directly abstract from pointwise solutions by introducing comparison functions that are sufficiently regular, while the root of weak solutions is the problem formulation in a Hilbert space. Conceptually, both have their advantages. From a numerical perspective, viscosity solutions relate to finite difference schemes, whereas weak solutions are the theoretical foundation of Galerkin methods, a rich class of versatile numerical methods to solve partial differential equations. Relying on their elegant Hilbert space formulation, Galerkin methods by their very construction lead to convergent schemes as well as to a lucid error analysis. They furthermore distinguish themselves by their enormous flexibility towards problem types as well as compression techniques. Both theory and implementation of Galerkin methods have experienced a tremendous advancement over the past fifty years. They have become indispensable for today’s technological developments in such diverse areas as aeronautical, biomechanical, and automotive engineering.
In mathematical finance, Galerkin pricing algorithms have been developed for various applications, even for basket options in jump models. Furthermore, numerical experiments and error estimates have confirmed their efficiency both in theory as well as in practice. See (26), and e.g. Matache, von Petersdorff and Schwab (2004), Matache, Schwab and Wihler (2005), von Petersdorff and Schwab (2004). We present the implementation of a related Galerkin method to price call options adjusted with a killing rate in Section 6. Furthermore, Galerkin-based model reduction techniques have a great potential in financal applications, see Cont, Lantos and Pironneau (2011), (39), and (41), Haasdonk, Salomon and Wohlmuth (2012) and (23).
Feynman-Kac representations for viscosity solutions with application to option pricing in Lévy models have been derived in (13) and (14). Results linking jump processes with Brownian part to variational solutions had already been proven earlier in (6). However, in order to cover some of the most relevant financial models, we have to consider pure jump processes, i.e. processes without a Brownian component, as well. Pure jump Lévy models have been shown to fit market data with high accuracy and have enjoyed considerable popularity, see for instance (15), (44), (12). Moreover, statistical analysis of high-frequency data supports the choice of pure jump models, see (1).
We realize that pure jump processes differ significantly from processes with a Brownian part. The Brownian component translates to a second order derivative in the Kolmogorov operator, while the pure jump part corresponds to an integro differential operator of a lower order of differentiation. Accordingly, the second order derivative is only present in Kolmogorov operators of processes with a Brownian component. As a consequence, the solution to the Kolmogorov equation of a pure jump Lévy process does not lie in the Sobolev space , the space of quadratic integrable functions with a square integrable weak derivative. Therefore we need a more general solution space. In order to make an appropriate choice, recall that Lévy processes are nicely characterized through the Lévy-Khinchine formula by the Fourier transform of their distribution or, equivalently, by the symbol. Moreover, the symbol is typically available in terms of an explicit parametric function and as such is the key quantity to parametric Lévy models. For a wide range of processes, the asymptotic behaviour of the symbol ensures that the solution of the Kolmogorov equation belongs to a Sobolev-Slobodeckii space, i.e. it has a derivative of fractional order. Even more, parabolicity with respect to Sobolev-Slobodeckii spaces of the Kolmogorov equations related to Lévy processes has been characterized in terms of growth conditions on the symbol in (21).
So as to allow for typical initial conditions, such as the payout function of a call option in logarithmic variables and the Heaviside step function that relates to distribution functions, we base our analysis more generally on exponentially weighted Sobolev-Slobodeckii spaces. We therefore generalize the characterization of parabolicity to time-inhomogeneous Lévy processes and to exponentially weighted Sobolev-Slobodeckii spaces. In (16) existence and uniqueness of weak solutions in exponentially weighted Sobolev-Slobodeckii spaces of Kolmogorov equations related to time-inhomogeneous Lévy processes and a Feynman-Kac formula has been established. Here, we generalize these results to solutions of Kolmogorov equations related to time-inhomogeneous Lévy processes with possibly discontinuous killing rates. Technically, the present setting is more difficult since the Fourier transform of the solution is not explicitly available and, moreover, the solution is not sufficiently regular for an application of Itô’s formula.
The fruitful relation between pseudo differential operators (PDOs) and Markov processes via their symbols has already been extensively used to establish existence of stochastic processes, see for instance the monographs of Jacob from (2001), (2002) and (2005). For a short overview on the different approaches to construct Feller processes and the use of pseudo differential calculus in this context see Chapter III in the monograph of Böttcher, Schilling and Wang (2013). Let us observe that our question is of a different nature: We establish a Feynman-Kac-type representation of the form (2), while existence of the stochastic processes involved, and the conditional expectation, are known. An interesting feature of our approach is that we do need not impose growth conditions on the (higher-order) derivatives of the symbol as in the standard symbolic calculus. Our approach is more closely related to (27), where a class of martingale problems is solved tracing back the existence of the processes to parabolicity of the Kolmogorov equations with respect to anisitropic Sobolev-Slobodeckii spaces. Compared to the setting in (27), we restrict ourselves to isotropic spaces and constant coefficients, but, more generally, allow for exponentially weighted spaces and possibly discontinuous killing rates.
To comprise all of the requirements, we state our research question more precisely as follows: Under which conditions on the time-inhomogeneous Lévy process , the possibly discontinuous killing rate , the source and initial condition is there a unique weak solution in an exponentially weighted Sobolev-Slobodeckii space of Kolmogorov equation (1) that allows for a stochastic representation of form (2)?
To answer our research question, we introduce in the next section the necessary notation and concepts. We use this framework first to characterize parabolicity of the Kolmogorov equation in terms of properties of the symbol in Theorem 3.3. Prepared thus, we formulate our main result, the Feynman-Kac-type representation of the weak solution of Kolmogorov equation (1) in Theorem 3.4. In Section 4 we find that it is a wide and interesting class of stochastic processes that fall within the scope of this result. Analysing its applications in Section 5 leads us from typical financial problems further to the characterization of purely probabilistic objects and finally back to the original quantum mechanical ideas of Feynman and Kac—yet in a relativistic guise. Exploiting the advantages of Theorem 3.4 further, we return to its practical realization and implement a Galerkin scheme to solve Kolmogorov equation (1) in Section 6. We find that thanks to Theorem 3.4 the solutions obtained thus correspond to option prices. With the numerical implementation at hand, we visualize and discuss the effect of killing rates of indicator type. Section 7 presents a robustness result for weak solutions that is required in our proof of Theorem 3.4 in Section 8. In this last section we also identify desirable regularity properties of the solutions to the Kolmogorov equation. Appendix A provides two technical lemmata for the symbol and the operator, and Appendix B concludes with the proof of Theorem 3.3.
2. Preliminaries and notation
In order to present the main result of the present article, we first introduce the underlying stochastic processes, the Kolmogorov equation with killing rate, its weak formulation as well as the solution spaces of our choice. We denote by the set of smooth real-valued functions with compact support in and let
be the Fourier transform of and be its inverse.
Let a stochastic basis be given and let be an -valued time-inhomogeneous Lévy process with characteristics . That is has independent increments and for fixed its characteristic function is given by
where, for every and , the symbol of the process is defined as
Here, for every , is a symmetric, positive semi-definite -matrix, , and is a Lévy measure, i.e. a positive Borel measure on with and . Moreover, is a truncation function i.e. such that with in a neighbourhood of . We assume the maps , and to be Borel-measurable with, for every ,
where is a norm on the vector space formed by the -matrices.
The Kolmogorov operator of the process is given by
for every , where denotes the -th component of the truncation function . By some elementary manipulations we obtain
which shows us that the Kolmogorov operator is a pseudo differential operator with symbol .
Following the classical way to define solution spaces of parabolic evolution equations, we introduce a Gelfand triplet , which consists of a pair of separable Hilbert spaces and and the dual space of such that there exists a continuous embedding from into . We then denote by the space of weakly measurable functions with and by the derivative of with respect to time in the distributional sense. The Sobolev space
will serve as solution space for equation (1). For a more detailed introduction to the space , which relies on the Bochner integral, we refer to Section 24.2 in (47). More information on Gelfand triplets can be found for instance in Section 17.1 in (47).
Usually, variational equations of a similar type as the heat equation are formulated with respect to Sobolev spaces, and thus are based on both and . Since we include pure jump processes in our analysis, operator (7) may be of fractional order. We therefore work with Sobolev-Slobodeckii spaces, which formalize the notion of a derivative of fractional order. Turning to a typical financial problem, we express the price of a call option as solution to a Kolmogorov equation of type (1). We then obtain and , while the initial condition is given by . We now have to realize that the initial condition and we cannot use an -based approach. The exponentially dampened function, , though belongs to for every . Thus, in order to incorporate initial conditions that typically arise in financial problems, we allow for an exponential weight. We further increase the class of function spaces by a domain splitting argument, see Remark 5.2 on page 5.2.
To make these considerations formally precise, we define the exponentially weighted Sobolev-Slobodeckii space with index and weight as the completion of with respect to the norm given by
Observe that this is a separable Hilbert space. For the space coincides with the Sobolev-Slobodeckii space as it is defined e.g. in (47). For the space coincides with the weighted space of square integrable functions . Furthermore, we denote the dual space of by .
Let be a family of bilinear forms that are measurable in with associated linear operators given by
and whose related symbols are such that
We close the section with the weak formulation of Kolmogorov equation (1).
Let and , measurable and bounded, and . Then is a weak solution of Kolmogorov equation (1), if for almost every ,
and converges to for in the norm of .
3. Main results
Equipped with the necessary notation and concepts, we now focus on our main purpose, providing a Feynman-Kac formula linking weak solutions of PIDEs with killing rates to conditional expectations.
Following a classical way to prove existence and uniqueness of weak solutions of a parabolic equation, we verify continuity and a Gårding inequality of its bilinear form. We specify the notion of parabolicity accordingly and adapt it to our framework:
Let be an operator associated with bilinear form .
We say , respectively , is parabolic with respect to , if there exist constants , such that uniformly for all and all ,
|(Gårding inequality (Gård-))|
For , we say that the parabolicity of , respectively , (with respect to ) is uniform in , if for all the mapping is càdlàg and there exist constants , such that uniformly for all , all and inequalities (Cont-) and (Gård-) are satisfied.
As highlighted in equation (8), the Kolmogorov operator of a time-inhomogeneous Lévy process is a pseudo differential operator. Its symbol is explicitly known for various classes and in general is characterized by the exponent of the Lévy-Khinchine representation. Therefore, we express our main assumptions in terms of the symbol of the process. For Lévy processes with symbols , it has been shown in (21), Theorem 3.1, that the corresponding bilinear form is parabolic with respect to , if and only if constants and exist such that for every ,
We generalize this growth condition so as to render it suitable for the setting of time-inhomogeneous Lévy processes and weighted Sobolev-Slobodeckii spaces. We find that an extension of the bilinear form to weighted Sobolev-Slobodeckii spaces corresponds to a shift of the symbol in the complex plane. Symbols can be extended to complex domains if the appropriate exponential moment condition is satisfied. Let be a time-inhomogeneous Lévy process. First notice that is infinitely divisible with Lévy measure for every , as has been shown by (17), Lemma 1. Theorem 25.17 in (43) now implies that, for all ,
is equivalent to the exponential moment condition and
We therefore formulate the conditions in terms of an exponential moment condition on the process and growth conditions on the symbol extended to a complex domain. It turns out that this complex domain can conveniently be chosen as a tensorized complex strip. More precisely, for weight , let
From Theorem 25.17 in (43), we also know that the complex set on which is definable is convex. Lemma 2.1 (c) in (16) shows for the present setting that the map has a continuous extension to the complex domain that is analytic in the interior .
We will derive the main results related to the following set of conditions.
For weight and index , let be a symbol with extension to and, if available, let denote the time-inhomogeneous Lévy process with symbol .
For every ,
(Exponential moment condition )
There exists a constant such that uniformly for all and ,
(Continuity condition (Cont-))
There exist constants , and such that uniformly for all and ,
(Gårding condition (Gård-))
For every fixed and the mapping is càdlàg.
We say that has Sobolev index uniformly in , if has an extension to that satisfies (A2)–(A4). If is the symbol of process , we also say has Sobolev index uniformly in .
Conditions (A1)–(A4) are satisfied for a large set of processes, for instance for tempered stable and normal inverse Gaussian processes as well as their time-inhomogeneous extensions. In Section 4 we look in detail at the verification of the conditions.
Notice that for we have . Thus (A1) is trivially satisfied and (A2)–(A3) simplify accordingly. This case corresponds to the case of Sobolev-Slobodeckii spaces without weighting and is covered by the following results. If, moreover, the symbol is constant in time, (A4) is trivially satisfied and (A1)–(A4) reduce to (14) and (15). Conditions (A1)–(A3) were introduced in (16) to show existence and uniqueness of weak solutions of the related Kolmogorov equation (without killing rate) along with a Feynman-Kac-type formula with application to European option prices in time-inhomogeneous Lévy models. We additionally require (A4), which only imposes a mild technical restriction.
Our framework defined, let us now state our main results. We first show in Theorem 3.3 the equivalence between parabolicity with respect to weighted Sobolev-Slobodeckii spaces and growth conditions (A2) and (A3), thereby generalizing the result for Sobolev-Slobodeckii spaces and conditions (14) and (15) to the present setting. The characterization of (uniform) parabolicity in terms of conditions on the symbol is interesting in its own right. It is, moreover, one of the key steps in our proof of Theorem 3.4 below, which establishes a Feynman-Kac-type representation.
For and , let be a time-inhomogeneous Lévy process satisfying exponential moment condition (A1). Then the following two assertions are equivalent.
The Kolmogorov operator of is uniformly parabolic in with respect to .
has Sobolev index uniformly in .
The proof of Theorem 3.3 is given in Appendix B, where, moreover, Theorem B.1 provides a more general version of this result for operators and symbols that are not necessarily related to stochastic processes.
Assume (A1)–(A4). Then Theorem 3.3 shows in particular the parabolicity of the related bilinear form uniformly in time with respect to and . Now the classical existence and uniqueness result, see for instance Theorem 23.A in (48), gives us that Kolmogorov equation (1) has a unique weak solution in the space .
We now turn to the stochastic representation of this solution. For an integrable or nonnegative random variable we denote
where is the factorization of the conditional expectation and the expectation with respect to the probability measure such that .
For and , let be an -valued time-inhomogeneous Lévy process with symbol that satisfies (A1)–(A4). Then
for measurable and bounded, and Kolmogorov equation (1) has a unique weak solution ;
if, additionally, for some with , then, for every and a.e. ,
As we have already seen, part 1 of Theorem 3.4 follows from Theorem 3.3 and the classical existence and uniqueness result for solutions of parabolic equations. Part 2 is considerably more involved and Section 8 is devoted to its proof.
The major benefit of Theorem 3.4 for financial applications is that conditional expectations of form (20), which naturally appear as derivatives and asset prices, are now characterized by weak solutions of PIDEs. Therefore, the prices can be computed by numerically solving an equation of form (1). So as to illustrate the method and the effect of killing rates, we present among others an application to employee options in Section 5 and provide its Galerkin discretization in Section 6.
Theorem 3.4 furthermore shows a specific type of regularity of conditional expectation (20). It is interesting to identify sufficient conditions for Hölder continuity. Theorem 8.2 in Nezza, Palatucci and Valdinoci (2011) provides the appropriate Sobolev embedding result. Thus we obtain as an immediate consequence of Theorem 3.4 the following corollary.
4. Examples of classes of time-inhomogeneous Lévy processes
Let us explore the nature of Conditions (A1)–(A4) and show that they are satisfied for a wide class of processes. Conditions (A1)–(A4) naturally apply to processes that are specified through their symbol. Notice that the symbol is expressed in terms of the characteristics of the process. We exploit this in Proposition 4.7 to establish concrete accessible conditions for real valued time-inhomogeneous pure jump Lévy processes with absolutely continuous Lévy measures, while Proposition 4.3 treats time-inhomogeneous multivariate jump diffusions.
We should, however, realize that Conditions (A1)–(A4) are not satisfied by all Lévy processes. On the one hand, continuity and Gårding condition (A2) and (A3) have implications for the distributional properties of the process:
Fix and , and let be a time-inhomogeneous Lévy process with symbol . If Gårding condition (A3) is satisfied for weight and index , then there exist such that uniformly for all and ,
In particular, (A3) implies for every that the distribution of has a smooth Lebesgue density.
On the other hand, continuity and Gårding condition (A2) and (A3) relate to the path behaviour of the process: If a Lévy process with symbol satisfies (A2) and (A3) for and , then is its Blumenthal-Getoor index, as shown in (21), Theorem 4.1. Hence, every pure jump Lévy process satisfying assumptions (A2) and (A3) has infinite jump activity. On this basis we may for instance conclude that compound Poisson processes do not satisfy (A3).
Variance Gamma processes have Blumenthal-Getoor index and thus do not satisfy both (A2) and (A3), as noticed in part (iv) of Example 4.1 in (21). However, pure jump Lévy processes can be approximated by a sequence of Lévy jump diffusion processes with nonzero Brownian part. This can always be achieved by adding a diffusion part and letting its volatility coefficient tend to zero. Example 4.4 shows that pure jump Lévy processes can be approximated by Lévy processes for which (A1)–(A3) are satisfied for weight and index . (4) provide a sequence with better approximation properties for Monte Carlo techniques, which could be exploited further.
Before continuing the discussion on the validity of Conditions (A1)–(A4) for other classes of processes, we observe the following.
For and , let be the symbol of a Lévy process that satisfies exponential moment condition (A1). By virtue of Lemma A.1 in Appendix A and the continuity of symbols of Lévy processes (as mappings from to ), the validity of continuity condition (A2) for is equivalent to the following asymptotic condition: For every there exist a constant such that for every ,
We devote the remainder of this section to providing sufficient conditions for the validity of Conditions (A1)–(A4) for time-inhomogeneous jump diffusions, for pure jump Lévy processes, and for time-inhomogeneous processes.
4.1. Jump diffusions
For time-inhomogeneous Lévy jump diffusion processes we find that Conditions (A1)–(A4) are satisfied under remarkably weak conditions:
Fix some . Let be a time-inhomogeneous Lévy process with characteristics such that
Then (A1)–(A3) are satisfied for weight and index .
Due to the equivalence of and the exponential moment condition, (23) implies (A1). Observing that
For Lévy jump diffusion processes the conditions simplify considerably:
Example 4.4 (Multivariate Lévy processes with Brownian part).
Fix and let be an -valued Lévy processes with characteristics such that is a positive definite matrix and the Lévy measure satisfies . Then (A1)–(A3) hold for weight and index .
In order to verify those assumptions of Proposition 4.3 that concern the pure jump part of the process, it suffices to consider the pure jump processes separately, as the following Lemma shows.
For , let be two stochastically independent time-inhomogeneous Lévy processes with symbol such that (A1)–(A4) are satisfied for the same weight and the possibly different indices . Then the sum is a time-inhomogeneous Lévy process with symbol , and (A1)–(A4) are satisfied for weight and index .
4.2. Pure jump Lévy processes and operators of fractional order
We now consider a class of multivariate processes, which frequently occurs in finance and whose symbol is explicitly given.
Example 4.6 (Multivariate Normal Inverse Gaussian (NIG) processes).
Let be an -valued NIG-process, i.e. a Lévy process such that for parameters , and symmetric positive definite matrix with . The symbol of is given by
where we denote by the product for . Compare e.g. equation (2.3) in (24).
Assumptions (A1)–(A3) are satisfied for index and every such that for all . This is in particular the case, if
To summarize, if the parameters of satisfy (25), Conditions (A1)–(A4) are satisfied for weight and Sobolev index .
Since pure jump Lévy processes can be defined through a Lévy measure and a constant drift, we are interested in finding conditions on both the Lévy measure and the drift that imply Conditions (A1)–(A4). Let us address this issue for real-valued time-homogeneous pure jump Lévy processes whose Lévy measure is absolutely continuous. For this class we generalize Proposition 4.2 in (21) to time-inhomogenuity and weights . Thereby we obtain explicit conditions on the characteristics that imply Conditions (A1)–(A4).
Fix and , and let be a real-valued time-inhomogeneous Lévy process with characteristics .
is absolutely continuous for every with density , i.e. . Denote the symmetric part by and the antisymmetric part by .
There exist constants and and a function such that uniformly for all ,
If , there exist constants and such that uniformly for all ,
If , then inequality (26) holds for some and, moreover, for every .
Let be a real-valued time-inhomogeneous pure jump Lévy processes with characteristics . Then,
Condition (F1) is equivalent to (A1);
Conditions (F1)–(F3) imply (A1) and (A3);
Conditions (F1)–(F4) imply (A1)–(A3).
Since , we have , and part directly follows from Theorem 25.17 in (43).
We denote , and . Then the elementary identity yields
We notice that there exists a constant such that
Moreover, since , the triangle inequality implies that for every . This shows that Condition (F3) also remains valid when we replace by . Now part follows from inequality (4.16) in the proof of Proposition 4.2 in (21).
Along the same lines as in the proof of part , we observe that
Thus, the validity of Condition (F4) also remains valid when replacing by . Then, Proposition 4.2 in (21) shows the assertion for .
For we have . According to Lemma A.1, and using the notation therein,