Well-posedness of monotone semilinear SPDEs with semimartingale noise
We prove existence and uniqueness of strong solutions for a class of semilinear stochastic evolution equations driven by general Hilbert space-valued semimartingales, with drift equal to the sum of a linear maximal monotone operator in variational form and of the superposition operator associated to a random time-dependent monotone function defined on the whole real line. Such a function is only assumed to satisfy a very mild symmetry-like condition, but its rate of growth towards infinity can be arbitrary. Moreover, the noise is of multiplicative type and can be path-dependent. The solution is obtained via a priori estimates on solutions to regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, and ensuing compactness properties. A key role is played by an infinite-dimensional Doob-type inequality due to Métivier and Pellaumail.
Let us consider semilinear stochastic evolution equations of the type
in , where is a smooth bounded domain of . Here is linear coercive maximal monotone operator on , is a random time-dependent maximal monotone graph everywhere defined on the real line, is a Hilbert space-valued semimartingale, and the coefficient satisfies a suitable Lipschitz continuity assumption (precise hypotheses on the data are given in §2 below).
Our main result is the existence and uniqueness of a strong solution to (1.1) (in the sense of Definition 3.1 below), and its continuous dependence on the initial datum in a suitable topology. We thus extend in several directions the well-posedness results obtained in  (see also ), where we considered the case with non-random time-independent and driving noise given by a Wiener process. We also cover the case where the coefficient , while still satisfying suitable predictability properties, is path-dependent. Equations with random time-dependent drift terms appear naturally, for instance, in the context of stochastic control problems.
The approach to the problem derives from the one adopted in , but the much more general character of the equations considered here gives rise to several non-trivial difficulties that need different ideas to be successfully overcome. For instance, in op. cit we considered the family of regularized equations obtained replacing with its Yosida approximation , which could be treated by the classical variation theory by Pardoux, Krylov and Rozovskiĭ (see [9, 20]). Such an approach does not work with general semimartingale noise, so we need to regularize also the operator , thus obtaining approximating equations with bounded coefficients that admits classical solutions in . In contrast to the case treated in , one also needs to determine sufficient conditions on such that is at least a progressively measurable function (see §2 below for detail on this technical issue). Interpreting such approximating equations either as “genuine” stochastic equations or as deterministic evolution equations with random coefficients, we obtain a priori estimates for their solutions in various topologies. In contrast to the case where is Wiener noise, hence a square-integrable process, if is a general semimartingale we cannot expect to obtain estimates in spaces of processes with finite moments. To overcome this problem, we first suppose that satisfies extra integrability conditions that are removed in a second step. A fundamental tool is an infinite-dimensional maximal inequality for stochastic integrals with respect to semimartingales due to Métivier and Pellaumail (see [18, 19]). These a priori estimates imply enough compactness to pass to the limit in the regularized equations, thus solving an auxiliary version of (1.1) with additive noise, i.e. where does not depend on . The assumption that is everywhere defined plays here a crucial role, as it allows to use weak compactness techniques in spaces. In order to treat the general case with multiplicative noise, a simple fixed-point argument as in  is no longer sufficient. We proceed instead as follows: using localization techniques, we first show the existence of strong solutions on closed stochastic intervals. This technique allows us also to remove the extra integrability assumption on . Uniqueness of solutions on closed stochastic intervals implies that such local solutions form a directed system, so that it is natural to construct a maximal solution. Finally, the linear growth of is shown to imply that the maximal solution can be extended to any compact time interval. One can also show that the solution depends continuously on the initial datum in the sense of the topology of uniform (in time) convergence in probability.
Several auxiliary results are needed to carry out the program outlined above, some of which may be interesting in their own right. For instance, we prove a general version of Itô’s formula for the square of the -norm in a variational setting with possibly singular terms. This can be seen as an extension of the classical formulas by Pardoux, Krylov, and Rozovskiĭ [9, 20], as well as by Krylov and Győngy , at least in the case where the variational triple is Hilbertian. We shall investigate in more detail Itô-type formulas in (generalized) variational settings in a work in preparation. We also give a characterization of weakly càdlàg processes in terms of essential boundedness (in time) and a weak càdlàg property in a larger space, extending the classical result on weak continuity for vector-valued functions by Strauss (see ).
While there is a sizable literature on extensions and refinements of the variational approach to SPDEs with Wiener noise (see, e.g.,  and references therein), equations driven by more general processes have received comparably less attention. Probably the first contribution in this direction is , where the well-posedness result of  is extended to the case where the driving noise is a quasi left-continuous locally square-integrable martingale, although under a rather restrictive growth assumption on the (nonlinear) drift term. In particular, semilinear equations such as (1.1) can be treated with this approach only if is Lipschitz-continuous. More recently, nonlinear equations in the variational setting driven by compensated Poisson random measures have been treated, also under relaxed monotonicity conditions, in . Semilinear equations with drift , as in (1.1), can be treated within this framework under polynomial growth assumptions on that depend on the dimension of the domain (cf.  for a discussion of this issue). Multivalued stochastic equations with possibly càdlàg additive noise have been studied also in , under a linear growth condition on the drift, so that semilinear equations such as (1.1) can be treated only if has at most linear growth. Our results largely generalize also those obtained in [13, 14] by semigroup methods, where grows polynomially and the equation is driven by a Wiener process and a compensated Poisson random measure (one should note, however, that needs not admit a variational formulation).
The remaining text is organized as follows: in §2 we fix the notation, collect all standing assumptions, and discuss some notable consequences thereof that are going to be used extensively. The definition of strong solution, both in the global and the local sense, and the statement of the main well-posedness result are given in §3. In §4 we recall some elements of the above-mentioned approach by Métivier and Pellaumail to stochastic integration with respect to semimartingales in Hilbert space, centered around a fundamental stopped Doob-type inequality. We also prove an extension to the càdlàg case of a classical criterion for weak continuity of vector-valued function due to Strauss, as well as a slight generalization of a classical criterion for uniform integrability by de la Vallé-Poussin. In §5 we prove an Itô-type formula for the square of the norm of a process that can be decomposed into the sum of a stochastic integral with respect to a (Hilbert-space-valued) semimartingale and of a Lebesgue integral of a singular drift term. This result is an essential tool to obtain, in §6, an auxiliary well-posedness result for a version of (1.1) with additive noise. Finally, the proof of the main result is presented in §7.
Acknowledgment. Large part of the work for this paper was done during several stays of the first-named author at the Interdisziplinäres Zentrum für Komplexe Systeme (IZKS), Universität Bonn, Germany, as guest of Prof. S. Albeverio. His kind hospitality and the excellent working conditions at IZKS are gratefully acknowledged.
2 Assumptions and first consequences
For any Banach spaces and , we shall denote the Banach space of continuous linear operators from to by , if endowed with the operator norm, and by , if endowed with the strong operator topology (i.e. with the topology of simple convergence). If , we shall just write in place of . The usual Lebesgue-Bochner spaces of -valued function on a measure space will be denoted by , , where is endowed with the (metrizable) topology of convergence in measure. The set of continuous functions and of weakly continuous functions on with values in will be denoted by and , respectively. Analogously, the symbols and stand for the corresponding spaces of càdlàg functions. A function will be called strongly measurable if it is the limit in the norm topology of of a sequence of elementary functions.
We shall denote by a smooth bounded domain of , and by the Hilbert space with its usual scalar product and norm .
All random elements will be defined on a fixed probability space endowed with a filtration satisfying the “usual assumptions” of right-continuity and completeness. Identities and inequalities between random variables will always be meant to hold -almost surely, unless otherwise stated. Two (measurable) processes will be declared equal if they are indistinguishable. By we shall denote a fixed semimartingale taking values in a (fixed) separable Hilbert space . The standard notation and terminology of stochastic calculus for semimartingales will be used (see, e.g., ).
The following hypotheses will be in force throughout the paper. An arbitrary but fixed terminal time will be denoted by .
Assumption (A). We assume that , where is a separable Hilbert space densely, continuously and compactly embedded in , and that there exist a constant such that
We denote by the part of in , i.e. the operator , with domain . Furthermore, we assume that there exists a sequence of linear injective operators on such that, for every ,
is sub-Markovian, i.e., if with a.e. in , then a.e. in ;
is ultracontractive, i.e .
Moreover, denoting the restriction of to by the same symbol, we assume that
for every and it can be extended to a continuous linear operator on , still denoted by the same symbol;
converges to the identity in , with , as .
Throughout the work, we shall denote by a Hilbert space continuously embedded in and dense in . Thanks to the assumptions on such a space always exists, for instance setting , with an arbitrary (but fixed) natural number.
Assumption (J). Let be a function satisfying the following conditions:
is progressively measurable for all ;
is convex and continuous for every , with ;
uniformly with respect to .
For every , the maximal monotone graph is defined as the subdifferential of , i.e. if and only if
Seeing the maximal monotone graph as a multivalued map, we assume that
is everywhere defined for every ;
is bounded on bounded sets uniformly with respect to .
The forthcoming assumptions on the coefficient are formulated in terms of control processes for semimartingales, whose definition is given in §4.1 below.
Assumption (B). Let be a map satisfying the following conditions:
is a strongly predictable -valued process for every adapted càdlàg -valued process ;
for every stopping time , and for every adapted processes càdlàg -valued processes , ,
for every control process of there exists an increasing, nonnegative, right-continuous, adapted process such that, for every and every adapted càdlàg -valued processes , , one has
Assumptions (a) and (b) are immediately satisfied if for all , where is strongly measurable with respect to the product -algebra of the predictable -algebra and of the Borel -algebra of . A more refined criterion can be found in [19, §§6.2–6.4].
Finally, the initial datum is an -valued -measurable random variable.
2.3 On assumptions (A) and (J)
Assumptions (A) and (J) have important consequences that will be extensively used in the sequel. The most important ones are collected in this subsection.
The hypotheses on and ensure that is a Hilbertian variational triple and that the operator is maximal monotone from to . Moreover, as it follows by coercivity, linearity, and monotonicity, is bijective form to . However, in applications it is often necessary to consider only the weaker coercivity on
with a constant. This case can be included in our analysis by considering the operator instead of .
The hypotheses on are met by large classes of differential operators (second order symmetric and non-symmetric divergence-form operators, as well as the fractional Laplacian, for example) – see, e.g.,  for a detailed list of concrete examples.
The standard example of a family of operators that can be shown to satisfy conditions (a)–(d) above for large classes of operators is , with sufficiently large. We refer again to, e.g.,  for a discussion of this issue. Moreover, note that for to belong to it suffices that the commutator can be continuously extended to a linear bounded operator from . In fact, this allows to extend to a linear bounded operator on as follows: for any , by surjectivity of one has , with . Setting , in order to check that this is well defined it is sufficient to prove that if is such that , then . Let be such that . Then , hence , and . Since has already been defined on , we have . Finally, we have
so that is also bounded.
The Banach-Steinhaus theorem implies that the sequence of linear operators is bounded in , , and , i.e.
The continuity property of the adjoint family established next plays an important role in the proof of the Itô-type formula for the square of the -norm in §5.
The sequence of adjoint operators is contained in and converges to the identity in .
By the continuity of in one has, for every , ,
hence converges weakly to in for every . Furthermore, for any and , one has
where . Since is densely and continuously embedded in , this readily implies that and
Since is reflexive, for any sequence , there exist and a subsequence , possibly depending on , and such that converges weakly in to as . Since is compactly embedded in , converges strongly to in . Recalling that converges weakly to as , hence that so does , we infer that , i.e., converges strongly to in . By a standard result of classical analysis, this yields the convergence of to in , that is, along the original sequence, which is independent of . The result can finally be extended to by a density argument: let be a sequence converging to in . The triangle inequality yields
from which one easily concludes. ∎
In general, the adjunction map for linear bounded operators on a Hilbert space is continuous with respect to the uniform and the weak operator topology, but not with respect to the strong operator topology. The previous lemma thus identifies a (very!) special subset of linear bounded operators for which the adjuntion map is continuous also with respect to the strong operator topology.
Let us now discuss some consequences of assumption (J). For every , let denote the convex conjugate of , defined as
The measurability and continuity hypotheses on imply that and are normal integrands, or, equivalently, that their epigraphs are progressively Effros-measurable (see, e.g., [8, 22]). More precisely, let us recall that, given a function , its epigraph at is given by
The progressive Effros-measurability of the epigraph of is then defined as the progressive measurability of the set
for every open .
Moreover, if is a normal integrand, then is also progressively Effros-measurable (see op. cit), which in turn implies that the the resolvent and the Yosida approximation of , both real-valued functions on , are measurable with respect to the product of the progressive -algebra and the Borel -algebra (see, e,g., [12, Proposition 3.12]).
Assumption (c) can be interpreted by saying that, for any fixed , the rates of growth of at plus and minus infinity are comparable. For instance, this is satisfied if is even for every .
Assumption (d) implies that is superlinear at infinity, uniformly with respect to , i.e. that
Lastly, taking in the definition of as subdifferential of , assumption (e) implies that, for all , for all , that is, is bounded on bounded sets uniformly over .
The above measurability conditions are obviously satisfied if is non-random and time-independent, i.e. if is an everywhere defined maximal monotone graph in . Moreover, in this case the convex function such that and is uniquely determined, and implies that is superlinear at infinity.
The boundedness assumption (e) is the natural generalization of the analogous ones commonly used for time-dependent maximal monotone graphs (see, e.g., [1, p. 4]).
Note that all the conditions assumed to hold for every could have been assumed for almost every instead. Indeed, in such a case, if has measure and all hypotheses hold outside , then one can consider the restriction of to the complement of instead of .
3 Main result
The concept of solution we are going to work with is as follows. We recall that is an arbitrary but fixed time horizon.
Let be a stopping time. A strong solution on to (1.1) is a pair , where is an adapted càdlàg -valued process and is an adapted -valued process, such that
and -a.s., with a.e. in ;
is integrable with respect to ;
one has, as an identity in ,
A strong solution on will simply be called a strong solution.
The main results of the paper are collected in the following theorem. These ensure that (1.1) admits a strong solution, which is unique within a natural class of processes, and depends continuously on the initial datum.
Equation (1.1) admits a strong solution , with optional, and it is the only one such that
Moreover, the solution map is continuous from to , where is endowed with topology generated by the supremum norm.
4 Preliminaries and auxiliary results
We recall those results from the approach to stochastic integration developed by Métivier and Pellaumail that we need, refering to [18, 19] for detail. We also prove two additional lemmas pertaining to this theory that are indispensable for the proofs in the following sections.
Moreover, we provide a sufficient condition for a process to be weakly càdlàg and a generalized version of the uniform integrability criterion by de la Vallée Poussin.
4.1 Stochastic integration with respect to Hilbert-space-valued semimartingales
Let be a separable Hilbert space. An -valued process is elementary if there exist , sequences , , , and , , with and , such that
Then the stochastic integral of with respect to is defined as
A positive increasing adapted process is called a control process for if, for every separable Hilbert space , for every elementary -valued process , and for every stopping time , one has
It turns out that an adapted càdlàg -valued process is a semimartingale if and only if it admits a control process. In particular, the set of control processes for a semimartingale , that we shall denote by , is not empty. One can also show that, writing , with locally square integrable local martingale and a finite-variation process, a control process is given by
where is the predictable quadratic variation of , is the variation of , and is the quadratic variation of the pure-jump martingale part of , in the sense of [18, Definition 19.3].
We need to introduce some notation: for any control process and any strongly measurable adapted process with values in , let us define the process as
For any stopping time , let us define the measure on the predictable -algebra as
and note that is finite if . The space of strongly predictable processes with values in such that is finite coincides with the Bochner space with respect to the measure and values in , with norm
where the norm of is taken in . Denoting the Banach space of adapted càdlàg processes with values in such that by , with norm , the inequality in the definition of control process can thus be written as
The first step in the construction of the stochastic integral for more general integrands is as follows: suppose that there exists a stopping time such that , so that is a finite measure and the vector space of elementary processes is dense in . Then the mapping , initially defined on elementary processes, admits a unique extension to a linear continuous map from to . As a second step, assume that is a control process for and is a process with values in such that the process is finite, and introduce the sequence of stopping times defined as
so that as well as , i.e. . Then, by the previous step, one has for all . Since increases to as and it is not difficult to show that on for all , , one has a well-defined process . One then shows that such a process does not depend on the sequence . However, it may still depend on the control process . A final step shows that if admits two control processes and such that the processes and are finite, then the stochastic integrals constructed in the two possible ways coincide. The following definition is therefore meaningful.
A strongly predictable -valued process is integrable with respect to if there exists a control process for such that the process is finite.
We shall occasionally denote the set of strongly predictable -valued processes such that the process is finite by .
Note that the construction of implies that the inequality in the definition of control processes can be extended as follows: for every , , and stopping time , one has
We shall need a further maximal inequality for stochastic integrals with respect to a semimartingale, whose proof relies on the following deep inequality (see [10, Lemma 1.3]).
Let be a positive real-valued measurable process and an increasing predictable process such that, for every finite stopping time ,
for a constant . Then for every concave function and every finite stopping time one has
Let be a control process for and , so that
Since the process is left-continuous, hence predictable, the previous lemma yields, taking , ,
The following elementary lemma is essential in the last section.
Let be a control process for the semimartingale and a stopping time. Then is a control process for the semimartingale .
For every elementary -valued process and every stopping time one has , hence also
which in turns implies
where and . ∎
We also recall the following version of the dominated convergence theorem for stochastic integrals with respect to semimartingales (cf. [18, Theorem 26.3]).
Let , be predictable -valued processes such that in a.e. in . If there exists a control process for and such that
then for every , , and
in probability for every .
Finally, we recall, for the reader’s convenience, the following stochastic version of Gronwall’s lemma (cf. [18, Lemma 29.1]).
Let be an adapted, right-continuous, increasing, positive process defined on a stochastic interval , with with probability one. Let also be a real, increasing, adapted process such that, for every stopping time ,
for certain constants . Then,
4.2 Weak right-continuity of vector-valued functions
Throughout this section and denote two Banach spaces, with reflexive, densely and continuously embedded in . A classical result by Strauss (see ) states that
We are going to show that the result continues to hold replacing the spaces of weakly continuous functions by spaces of weakly càdlàg functions.
The inclusion of the space on the right-hand side in the space on the left-hand side is evident. Let . Since is negligible with respect to the Lebesgue measure on , it is not restrictive to suppose that (otherwise, we shall modify the value of in , obtaining a version of which is still in ). We first show that, in order for to belong to , it suffices to prove that there exists a constant such that for every .
Step 1. Assuming that is bounded in , let and be a sequence converging to . Then weakly in by assumption, and, since is reflexive, there exists a subsequence and such that weakly in . Therefore and weakly in , i.e. is weakly càd with values in . A completely analogous (in fact easier) argument shows that is also làg with values in .
Step 2. Let be a sequence of mollifiers in whose support is contained in . Denoting the extension of to zero outside by the same symbol, it follows from that . In particular, Minkowski’s inequality yields
for all and . Let be arbitrary but fixed. By reflexivity of , there exist and a subsequence of , denoted by the same symbol for simplicity, such that weakly in . Moreover, for any ,
where by assumption. In particular, is right-continuous at , i.e. for any there exists such that for all . Since the support of is contained in , for we have
i.e. as . Since this holds for any , we infer that weakly in . Moreover, as is bounded in and is reflexive, we easily deduce that weakly in , thus also, by weak lower semicontinuity of the norm, that
Since was arbitrary, this implies that for all . Moreover, since , we have that is bounded in , as required. ∎
4.3 A criterion for uniform integrability
We shall need a slightly generalized version of the de la Vallée-Poussin criterion for uniform integrability. For the purposes of this paragraph only, will denote a finite measure space, and stands for the product measure of , the Lebesgue measure, and on . For compactness of notation, we set
for any .
Let be proper, convex and lower semicontinuous in the third variable, measurable in the first two, and such that
If is such that there exists a constant for which
then is uniformly integrable in .
We need to show that is bounded in and that for every there exist such that, for any measurable set with , one has
Let be a constant. By assumption there exists such that with implies for every . Then one has, for any ,
Choosing it immediately follows that is bounded in . Moreover, for every there exists such that , hence satisfies the condition we are looking for. ∎
The same argument shows, keeping fixed, that if there exists a finite positive random variable such that, for -a.e. ,
then is uniformly integrable in for -a.e. .
5 The Itô formula
In this section we prove an Itô-type formula for the square of the -norm: this can be seen as an integration-by-parts formula in a generalized setting. We point out that the framework that we consider here is is “unusual”, as we work with processes with components in and simultaneously, for which Itô’s formula is not available using existing techniques. Let us recall also that the quadratic variation of is defined as the process