On $L_p$-Solvability of Stochastic Integro-Differential Equations

A class of (possibly) degenerate stochastic integro-differential equations of parabolic type is considered, which includes the Zakai equation in nonlinear filtering for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces.


Introduction
We consider the equation for (t, x) ∈ [0, T ] × R d := H T with initial condition on a filtered probability space (Ω, F, P, (F t ) t≥0 ), carrying a sequence w = (w i t ) ∞ i=1 of independent F t -Wiener processes and an F t -Poisson martingale measureπ(dz, dt) = π(dz, dt) − µ(dz)⊗dt, where π(dz, dt) is an F t -Poisson random measure with a σ-finite intensity measure µ(dz) on a measurable space (Z, Z) with countably generated σ-algebra Z. We note that here, and later on, the summation convention is used with respect to repeated (integer-valued) indices and multi-numbers.
In the above equation A t is an integro-differential operator of the form A t = L t + N ξ t + N η t with a second order differential operator and integral operators N ξ t and N η t defined by for a suitable class of real-valued functions ϕ on R d for each t ∈ [0, T ], where ν(dz) is a fixed σ-finite measure on (Z, Z). For each integer r ≥ 1 the operator M r t is a first order differential operator of the form M r t = σ ir t (x)D i + β r t (x). The coefficients a ij , b i , c, σ ir and β r are real functions on Ω × H T for i, j = 1, 2, ..., d and integers r ≥ 1, and η = (η i t,z (x)) and ξ = (ξ i t,z (x)) are R d -valued functions of (ω, t, x, z) ∈ Ω × H T × Z. The free terms f and g r are real functions defined on Ω × H T for every r ≥ 1, and h is a real function defined on Ω × H T × Z. The stochastic differentials in equation (1.1) are understood in Itô's sense, see the definition of a solution in the next section.
We are interested in the solvability of the above problem in L p -spaces. We note that equation (1.1) may degenerate, i.e., the pair of linear operators (L, M) satisfies only the stochastic parabolicity condition, Assumption 2.1 below, and the operator N ξ may also degenerate. Our main result, Theorem 2.1 states that under the stochastic parabolicity condition on the operators (L, M), N ξ , N η , and appropriate regularity conditions on their coefficients and on the initial and free data, the Cauchy problem (1.1)-(1.2) has a unique generalised solution u = (u t ) t∈[0,T ] for a given T . Moreover, this theorem describes the temporal and spatial regularity of u in terms of Bessel potential spaces H n p , and presents also a supremum estimate in time. The uniqueness of the solution is proved by an application of a theorem on Itô's formula from [19], which generalises a theorem of Krylov in [25] to the case of jump processes. The existence of a generalised solution is proved in several steps. First we obtain a priori estimates in Sobolev spaces W n p for integers n ∈ [0, m] if p = 2 k for an integer k ≥ 1, where m is a parameter measuring the spatial smoothness of the coefficients and the data in (1.1)-(1.2). These estimates allow us to construct a generalised solution by standard methods of approximating (1.1)-(1.2) with non-degenerate equations with smooth coefficients and compactly supported smooth data in x ∈ R d , and passing to the weak limit in appropriate spaces. Thus we see that a solution operator, mapping the initial and free data into the solution of (1.1)-(1.2), exists and it is a bounded linear operator in appropriate L p -spaces if p = 2 k for an integer k ≥ 1. Hence by interpolation we get our a priori estimates in Bessel potential spaces H n p for any given p ≥ 2 and real number n ∈ [0, m]. We obtain essential supremum estimates in time for the solution from integral estimates, by using the simple fact that the essential supremum of Lebesgue functions over an interval [0, T ] is the limit of their L r ([0, T ])-norm as r → ∞. Hence we get the temporal regularity of the solution formulated in our main theorem by using Theorem 2.2 on Itô's formula in [19], an extension of Lemma 5.3 in [10] and a well-known interpolation inequality, Theorem 4.1(v) below.
Concerning the above construction of a generalised solution in L p -spaces we would like to emphasise that first we can get the necessary a priori estimates only if p = 2 k for an integer k ≥ 1 and we need to use interpolation via the solution operator to get these estimates for arbitrary p ≥ 2. We note that a similar situation arose in L p -estimates in finite difference approximations for stochastic PDEs in [13].
In the literature there are many results on stochastic integral equations with unbounded operators, driven by jump processes and martingale measures. A general existence and uniqueness theorem for stochastic evolution equations with nonlinear operators satisfying stochastic coercivity and monotonicity conditions is proved in [17], which generalises some results in [32] and [26] to stochastic evolution equations driven by semimartingales and random measures. This theorem implies the existence of a unique generalised solution to (1.1)- (1.2) in L 2 -spaces when instead of the stochastic parabolicity condition (2.1) in Assumption 2.1 below, the strong stochastic parabolicity condition, |z i | 2 for all z = (z 1 , z 2 , ..., z d ) ∈ R d with a constant λ > 0 is assumed on (L, M). Under the weaker condition of stochastic parabolicity the solvability of (1.1)-(1.2) in L 2 -spaces is investigated and existence and uniqueness theorems are presented in [9] and [28]. The first result on solvability in L p -spaces for the stochastic PDE problem (1.1)-(1.2) with ξ = η = 0 and h = 0 was obtained in [27], and was improved in [18]. However, there is a gap in the proof of the crucial a priori estimate in [27]. This gap is filled in and more general results on solvability in L p -spaces for systems of stochastic PDEs driven by Wiener processes are presented in [14]. As far as we know Theorem 2.1 below is the first result on solvability in L p -spaces of stochastic integro-differential equations (SIDEs) without any non-degeneracy conditions. It generalises the main result of [10] on deterministic integro-differential equations to SIDEs. Our motivation to study equation (1.1) comes from nonlinear filtering of jump-diffusion processes, and we want to apply Theorem 2.1 to filtering problems in a continuation of the present paper.
We note that under non-degeneracy conditions SIDEs have been investigated with various generalities in the literature, and very nice results on their solvability in L p -spaces have recently been obtained. In particular, L p -theories for such equations have been developed in [22], [23], [29], [30] and [31], which extend some results of the L p theory of Krylov [24] to certain classes of equations with non local operators. See also [7], [11] and [35] in the case of deterministic equations. Nonlinear filtering problems and the related equations describing the conditional distributions have been extensively studied in the literature. For results in the case of jump-diffusion models see, for example, [2], [4], [12] and [16].
In conclusion, we introduce some notions and notations used throughout this paper. All random elements are given on the filtered probability space (Ω, F, P, (F t ) t≥0 ). We assume that F is P -complete, the filtration (F t ) t≥0 is right-continuous, and F 0 contains all P -zero sets of F. The σ-algebra of the predictable subsets of Ω × [0, ∞) is denoted by P. For notations, notions and results concerning Lévy processes, Poisson random measures and stochastic integrals we refer to [1], [3] and [21].
For vectors v = (v i ) and w = (w i ) in R d we use the notation vw = m i=1 v i w i and |v| 2 = i |v i | 2 . For real-valued Lebesgue measurable functions f and g defined on R d the notation (f, g) means the integral of the product f g over R d with respect to the Lebesgue measure on R d . A finite list α = α 1 α 2 , ..., α n of numbers α i ∈ {1, 2, ..., d} is called a multinumber of length |α| := n, and the notation .., d}. We use also the multi-number of length 0 such that D means the identity operator. For an integer n ≥ 0 and functions v on R d , whose partial derivatives up to order n are functions, we use the notation D n v for the collection {D α v : |α| = n}, and define For differentiable functions v = (v 1 , ..., v d ) : R d → R d the notation Dv means the Jacobian matrix whose j-th entry in the i-th row is D j v i .
The space of smooth functions ϕ = ϕ(x) with compact support on the d-dimensional The notation L p,q means the space L p ∩ L q with the norm The space of sequences ν = (ν 1 , ν 2 , ...) of real numbers ν k with finite norm The Borel σ-algebra of a separable Banach space V is denoted by B(V ), and for p ≥ 0 the notations L p ([0, T ], V ) and L p (R d , V ) are used for the space of V -valued Borel-measurable functions f on [0, T ] and g on R d , respectively, such that |f | V and |g| V have finite Lebesgue integral over [0, T ] and R d , respectively. For p ≥ 1 and g ∈ L p (R d , V ) we use the notation |f | Lp , defined by In the sequel, V will be R, l 2 or L p,q . For integer n ≥ 0 the space of functions from L p (R d , V ), whose generalised derivatives up to order n are also in L p (R d , V ), is denoted by and we use L p to denote W 0 p . For m ∈ R and p ∈ (1, ∞) we use the notation H m p = H m p (R d ; V ) for the Bessel potential space with exponent p and order m, defined as the space of V -valued generalised functions ϕ on R d such that We will often omit the target space V in the notations W n p (V ), H m p (V ), W n p (V ) and H m p (V ) for convenience if V = R. When V = L p,q we use W n p,q , H m p,q W n p,q and H m p,q to denote W n p (L p,q ), H m p (L p,q ) W n p (L p,q ) and H m p (L p,q ) respectively, and we use L p,q to denote W 0 p,q . Remark 1.1. If V is a UMD space, see for example [20] for the definition of UMD spaces, then by Theorem 5.6.11 in [20] for p > 1 and integers n ≥ 1 we have W n p (V ) = H n p (V ) with equivalent norms. Clearly, L p,q is a UMD space for p, q ∈ (1, ∞), which implies W n p,q = H n p,q for non-negative integers n and p, q ∈ (1, ∞).

Formulation of the results
To formulate our assumptions we fix a constant K, a nonnegative number m, an exponent p ∈ [2, ∞), and non-negative Z-measurable functionsη andξ on Z such that they are bounded by K and We denote by m the smallest integer which is greater than or equal to m, and m the largest integer which is less than or equal to m.
for every v ∈ C ∞ 0 , where to ease notation we do not write the arguments t and z and write v k instead of D k v for functions v. Due to Assumption 2.3 these equations extend to v ∈ W 1 p for p ≥ 2 as well. Hence after changing the order of integrals, by integration by parts we obtain where for the sake of short notation the arguments t, z of η and η k have been omitted, and Operators J k ξ and J 0 ξ are defined as J k η and J 0 η in (2.2) and (2.3) but with ξ everywhere in place of η.

Preliminaries
acting on functions ϕ and φ defined on R d such that the generalised derivative is a function of x ∈ R d then the notation T v , I v and J v mean the operators defined by (3.1) with v(x) in place v. For example, J ξ and J η mean for each ω ∈ Ω, t ∈ [0, T ] and z ∈ Z the operators defined on differentiable functions ϕ on R d by for each fixed variable (ω, t, z) suppressed in this notation. We will often use the Taylor formulas and with ϕ i := D i ϕ and φ ij := D i D j φ, which hold for every x ∈ R d when ϕ ∈ C 1 (R d ) and φ ∈ C 2 (R d ), and they hold for dx-almost every x ∈ R d when ϕ ∈ W 1 p and φ ∈ W 2 p . We fix a non-negative smooth function k = k(x) with compact support on R d such that For ε > 0 and locally integrable functions v of x ∈ R d we use the notation v (ε) for the mollification of v, defined by is a locally Bochner-integrable function on R d taking values in a Banach space, the mollification of v is defined as (3.4) in the sense of Bochner integral.
The following lemmas are taken from [10] and for their proof we refer to [10]. to W k p respectively, for k = 0, 1, ..., m, such that T η ϕ, I η f and J η g are P ⊗ Z-measurable W k p -valued functions of (ω, t, z), and The following lemma is a slight generalisation of Lemma 3.4 in [10].
Proof. We show first that | det Dρ ε,ϑ | is separated away from zero for sufficiently small ε > 0.
To this end observe that for bounded Lipschitz where L is the Lipschitz constant of v and K is a bound for |v|. Using this observation and taking into account that D i ρ l is bounded by N and it is Lipschitz continuous with a Lipschitz constant not larger than N , we get Thus setting ε = M/(2d! N d ), for ε ∈ (0, ε ) and ϑ ∈ [0, 1] we have for all x, y ∈ R d and ε > 0 and ϑ ∈ [0, 1]. Observe that which implies lim |x|→∞ |ρ ε,ϑ (x)| = ∞, i.e., under ρ ε,ϑ the pre-image of any compact set is a compact set for each ε ∈ (0, ε ) and ϑ ∈ [0, 1]. A continuous function with this property is called a proper function, and by Theorem 1 in [15] Now we can complete the proof of the lemma by noting that since For fixed ε > 0 and ϑ ∈ [0, 1] let ρ ε,ϑ denote any of the functions T ] × Z, and assume that Assumptions 2.2 and 2.3 hold. Then by the inverse function theorem ρ is a local C 1 (R d )-diffeomorphism for each t, θ and z. Since , for each t ∈ [0, T ], z ∈ Z and θ ∈ [0, 1], by Theorem 1 in [15]. Note that by the formula on the derivative of inverse functions a C 1 (R d )-diffeomorphism and its inverse have continuous derivatives up to the same order. Thus Lemmas 3.2 and 3.3 imply the following lemma, which is a slight generalisation of Corollary 3.6 in [10].
Proof. This lemma is well-known. Its proof can be found, e.g., in [19], see Lemma 4.4 therein.
Moreover, we will often use the following characterisation of W n p (L q 1 ∩ L q 2 ). Lemma 3.6. Let v ∈ L p (L p ∩ L q ) for some p, q ∈ (1, ∞), and let α be a multi-index. Then the following statements hold. ( where the integrals are understood as Bochner integrals of L p ∩ L q -valued functions. Hence for all ϕ ∈ C ∞ 0 and bounded Z-measurable functions ψ supported on sets of finite µ-measure. We can use Fubini's theorem to get Notice that for µ-almost every z ∈ Z the functionsv α (·, z) and v(·, z) belong to L p (R d , R). Hence there is a set S ⊂ Z of full µ-measure such that for z ∈ S equation (3.8) holds for all ϕ ∈ C ∞ 0 , which proves that for z ∈ S the functionv α (·, z) is generalised D α -derivative of v(·, z). To prove (ii) notice that if for µ-almost every z ∈ Z the function v α (·, z) belongs to L p (R d , R) and it is the D α generalised derivative of the function v(·, z), then for µ-almost every z ∈ Z we have for every ϕ ∈ C ∞ 0 . Using condition (3.7) and that v ∈ L p (L p ∩ L q ), it is easy to check that, as functions of z, both sides of the a bove equation are functions in L p ∩ L q , and hence that these integrals define the same functions as the corresponding L p ∩ L q -valued Bochner integrals. This proves that v α is the L p ∩ L q -valued generalised D α -derivative of v. ( Proof. The proof of (3.11) and (3.12) is given in [9] and [10]. For the convenience of the reader we prove each of the above estimates here. We may assume that
Next we present two important Itô's formulas from [19] for the p-th power of the L p -norm of a stochastic process.
holds almost surely for all t ∈ [0, T ].
The following slight generalisation of Lemma from [18] will play an essential role in obtaining supremum estimates.
Lemma 3.10. Let T ∈ [0, ∞] and let f = (f t ) t≥0 and g = (g t ) t≥0 be nonnegative F t -adapted processes such that f is a cadlag and g is a continuous process. Assume for any constant c > 0 and bounded stopping time τ ≤ T . Then, for any bounded stopping Proof . This lemma is proved in [18] when both processes f and g are continuous. A word by word repetition of the proof in [18] extends it to the case when f to be cadlag. For the convenience of the reader we present the proof below. By replacing f t and g t with f t∧T and g t∧T , respectively, we see that we may assume that T = ∞. Then we replace g t with max s≤t g s and see that without losing generality we may assume that g t is nondecreasing. In that case fix a constant c > 0 and let θ f = inf{t ≥ 0 : f t ≥ c}, θ g = inf{t ≥ 0 : g t ≥ c}. Then In the light of (3.17) we replace the expectation with Hence Now it only remains to substitute c 1/γ in place of c and integrate with respect to c over (0, ∞). The lemma is proved.  Proof. Since S is dense in [0, T ], for a given t ∈ [0, T ) there is a sequence {t n } ∞ n=1 with elements in S such that t n ↓ t. Due to sup n∈N |f (t n )| V < ∞ and the reflexivity of V there is a subsequence {t n k } such that f (t n k ) converges weakly in V to some element v ∈ V . Since f is weakly cadlag in U , for every continuous linear functional ϕ over U we have lim k→∞ ϕ(f (t n k )) = ϕ(f (t)). Since the restriction of ϕ in V is a continuous functional over V we have lim k→∞ ϕ(f (t n k )) = ϕ(v). Hence f (t) = v, which proves that f is a V -valued function over [0, T ). Moreover, by taking into account that Let φ be a continuous linear functional over V . Due to the reflexivity of V , the dual U * of the space U is densely embedded into V * , the dual of V . Thus for φ ∈ V * and ε > 0 there is φ ε ∈ U * such that |φ − φ ε | V * ≤ ε. Hence for arbitrary sequence t n ↓ t, t n ∈ [0, T ] we have Letting here n → ∞ and then ε → 0, we get which proves that f is right-continuous in the weak topology in V . We can prove in the same way that at each t ∈ [0, T ] the function f has weak limit in V from the left at each t ∈ (0, T ], which finishes the proof of the lemma.

Some results on interpolation spaces
A pair of complex Banach spaces A 0 and A 1 , which are continuously embedded into a Hausdorff topological vector space H, is called an interpolation couple, and A θ = [A 0 , A 1 ] θ denotes the complex interpolation space between A 0 and A 1 with parameter θ ∈ (0, 1). For an interpolation couple A 0 and A 1 the notations A 0 ∩A 1 and A 0 +A 1 is used for the subspaces respectively. Then the following theorem lists some well-known facts about complex interpolation, see e.g., 1.
We will also use the following theorem on the interpolation spaces between the interpolation couple L q ∩ L p 0 and L q ∩ L p 1 , for 1 ≤ p 0 ≤ p 1 and a fixed q / ∈ (p 0 , p 1 ), where the notation L p means the L p -space of real functions on a measure space (Z, Z, µ) with a σ-finite measure µ on a σ-algebra Z.
This theorem is proved in [33] only in the special case when Z is a domain in R d , Z is the σ-algebra of the Borel subsets of R d , µ is the Lebesgue measure on R d and q = 2 ≤ p 0 ≤ p 1 , but the same proof works also in our situation. For the convenience of the reader we present here the very nice argument from [33] in our more general setting. The key role is played by the following lemma, which is an adaptation of Theorem 4 from [33]. The notation L p (a, b) means the L p space of Borel-measurable real-functions on an interval (a, b) with respect to the Lebesgue measure on (a, b) for −∞ ≤ a < b ≤ ∞. Lemma 4.3. Let f ∈ L 1 + L ∞ be a fixed function. Then there are bounded linear operators S 1 and S 2 mapping L 1 + L ∞ to L 1 (0, 1) and l ∞ , respectively, and there are also bounded linear operators T 1 and T 2 mapping L 1 (0, 1) and l ∞ , respectively into L 1 + L ∞ , such that for all u ∈ L p , v ∈ L p (0, 1) and w ∈ l p .
Thus f * ≤ g * , and one can apply Calderon's theorem again to get a bounded linear operator H : L 1 (0, ∞) + L ∞ (0, ∞) → L 1 + L ∞ such that f = Hg, and H is a bounded operator from L 1 (0, ∞) to L 1 and from L ∞ (0, ∞) to L ∞ , with operator norms not larger than 1. Hence it is easy to check that S i := V i L and T i := HW i , i = 1, 2 satisfy (4.18) and (4.19) for p = 1, ∞, and hence for all p ∈ [1, ∞] by the Riesz-Thorin theorem.
Use also the notationL r := L q ∩ L r . For an f ∈L p = L q ∩ L p ⊂ L 1 + L ∞ let S i and T i denote the operators from the previous lemma. Then clearly, S 1 :L p →L p (0, 1) = L p (0, 1) = [L p 0 (0, 1),L p 1 (0, 1)] θ , S 2 :L p →l p = l q = [l p 0 ,l p 1 ] θ and by interpolation, are bounded operators with operator norms not greater than 1. Hence taking V := [L p 0 ,L p 1 ] θ norm in both sides of equation (4.18) we get Let now f ∈ [L p 0 ,L p 1 ] θ ⊂ L 1 + L ∞ , and denote again by S i and T i for i = 1 the linear operators corresponding to f by the above lemma. Then clearly, T 1 :L p (0, 1) →L p , and T 2 :l p →L p , and by interpolation and =l p are bounded operators with operator norm not greater than 1. Hence which finishes the proof of the theorem when 1 ≤ q ≤ p 0 ≤ p 1 . The theorem in the case 1 ≤ p 0 ≤ p 1 ≤ q can be proved in the same way with obvious changes.  ∈ (1, ∞). Then for p 0 , p 1 ∈ (1, ∞) and θ ∈ (0, 1) Proof. We have equation   [H m p 0 (L p 0 ,q ), H m p 1 (L p 1 ,q )] θ = H m p θ (L p θ ,q ) with equivalent norms for any m ∈ (−∞, ∞), where p θ = ((1 − θ)/p 0 + θ/p 1 ) −1 and L r,q = L r ∩ L q for any r, q ∈ [1, ∞).

L p estimates
Let Assumptions 2.1 and 2.3 hold, and let u = (u t ) t∈[0,T ] be a W n+2 p -valued solution to (1.1)-(1.2) for some integer n ≥ 0. Then by an application of Lemma 3.8 we have , ω ∈ Ω and t ∈ [0, T ]. Recall that the notation v α = D α v is often used. In order to estimate the right-hand side of (5.1), we also define for integers n ∈ [0, m] and p ≥ 2 the "p-form" , for each ω ∈ Ω and t ∈ [0, T ].
Proof. This estimate is proved in [14] in a more general setting.
Noticing that for i = 1, 2, ..., k by induction on the length of multi-numbers α we get We will prove Proposition 5.3 by the help of the following lemmas.
Lemma 5.4. Let Assumption 2.3 hold. Then for p ≥ 2 and integers n ∈ [0, m] we have Proof. Clearly, For the next lemmas consider for integers n ≥ 0 the expressions for v ∈ W n+1 p and h ∈ W n+1 p (L p,2 ), where G (α) is defined in (5.11).
Lemma 5.5. Let Assumption 2.3 hold with m ≥ 0. Then for Proof. Let x = (x α ) and y = (y α ) denote the vectors with coordinates for multi-numbers α of length n. Then taking into account (5.12) we have and by Taylor's formula there is a constant N = N (d, p, n) such that Hence writing G for the vector (G (α) ) |α|=n and noticing with a constant N = N (d, n, p).

Hence by Hölder's inequality
with N = N (K, d, m), which proves the first inequality in (5.14). Let p ≥ 4. Then with and By Fubini's theorem and Hölder's inequality . Since p ≥ 4, by Taylor's formula, Fubini's theorem, change of variables, integration by parts and using Assumption 2.3 we get . Hence by Hölder's inequality, change of variables, Fubini's theorem and using |η| ≤ K we obtain Similarly, with a constant N = N (K, d, m, p, K η ) we have By integration by parts, using Assumption 2.3, Cauchy-Schwarz and Hölder inequalities we get . Combining this with (5.16) through (5.18) and using Young's inequality we get (5.15).

Proof of Proposition 5.3. Set
Then by Lemma 5.4 By Lemma 5.5 for we have B ≤ I η |D n v| p + pT η |D n v| p−2 T η v α G (α) + pI η (|D n v| p−2 v α )h α +N T η |D n v| p−2 |G| 2 + N T η |D n v| p−2 |D n h| 2 + N |G| p + N |D n h| p with a constant N = N (d, n, p). Thus introducing the notations N (d, n, p). Combining this with equation (5.19) and noticing that By Lemma 3.7 (i) and (ii) we have a constant N = N (K, d) such that and Due to Assumption 2.3, taking into account that p ≥ 2 and using Young's inequality we get with a constant N = N (d, n, p). Combining this with estimates (5.20) through (5.22) and using Lemma 5.6 we finish the proof of the proposition.
Introduce also the expressions , ω ∈ Ω and t ∈ [0, T ], where repeated indices α mean summation over all multi-numbers of length n.
(i) If Assumption 2.1 is satisfied then for all v ∈ W n+1 p and g ∈ W n+1 , with a constant N = N (d, m, p, K).

(ii) If Assumption 2.3 is satisfied then
for all v ∈ W n+1 p and h ∈ W n p (L p,2 ) with N = N (d, m, p, K, K η ).
Proof. Noticing that p|D n v| p−2 v α σ ir D i v α = σ ir D i |D n v| p , by integration by parts and by Minkowski's and Hölder's inequalities we obtain that can be estimated by the right-hand side of (5.24). By Minkowski and Hölder's inequalities it is easy to see that P 2 n,p (t, v, g) −P 2 n,p (t, v, g) can also be estimated by the right-hand side of (5.24). To prove (ii) let denote the integrand in (5.23). Using Taylor's formula for |x + y| p − |x| p − p|x| p−2 x α y α with vectors x α := D α v, y α := D α (I η v + h), α ∈ {1, 2, ..., d} n , we have the estimate 0 ≤ A s (x, z) ≤ N |x| p−2 |y| 2 + N |y| p ≤ N |D n v| p−2 |D n I η v| 2 +N |D n v| p−2 |D n h| 2 + N |D n h| p + N |D n I η v| p with constants N and N depending only on d,p and n. By Fubini's theorem and Hölder's inequality . By Hölder's inequality and Lemma 3.1 we obtain Moreover, by Assumption 2.3 and Lemma 3.1 we have Combining these inequalities and using Young's inequality we get (5.25).
for all t ∈ [0, T ], where ζ 1 and ζ 2 are local martingales defined by Q(s, ·), Q η (s, ·) and P η (s, z, ·) are functionals on W 1 p , for each (ω, s) and z, defined by andQ ξ (s, ·) is defined asQ η (s, ·) in (6.2), but with ξ in place of η. Recall thatb i = b − D j a ij and J i η , J 0 η are defined by (2.2)-(2.3). Note that due to the convexity of the function |r| p , r ∈ R, we have for real-valued functions v = v(x), x ∈ R d . Together with the above functionals we need also to estimate the functionals Q(s, ·) andQ(s, ·) defined for each (ω, s) ∈ Ω × [0, T ] by for v ∈ W 1 p . Proposition 6.1. Let Assumptions 2.1, 2.2 and 2.3 hold with m = 0. Then for p ≥ 2 there are constants for all v ∈ W 1 p and (ω, s) ∈ Ω × [0, T ]. Proof. Notice that the estimate (6.5) is the special case of Proposition 5.7 (iii), and for v ∈ W 2 p the second and third estimates in (6.4) follow from the estimate (5.8) in Proposition 5.2 (ii). Notice also that for v ∈ W 2 p the first estimate in (6.4) is a special case of (5.6) in Proposition 5.1. If v ∈ W 2 p then by Assumption 2.3 and estimate (3.11) in Lemma 3.7 we haveQ with a constant C only depending on K and d. It is an easy exercise to show that the functionals in the left-hand side of the inequalities in (6.4) are continuous in v ∈ W 1 p , that completes the proof of the proposition.
Define now the stochastic process for every integer n ≥ 1, where (ρ n ) ∞ n=1 is an increasing sequence of stopping times, converging to infinity such that (ζ i (t∧ρ n )) t∈[0,T ] is a martingale for each n ≥ 1 and i = 1, 2. Then clearly, Eζ i (t ∧ τ n ) = 0 for t ∈ [0, T ] and i = 1, 2. Due to (6.3) and the estimate in (6.5) we have which implies Thus, substituting t ∧ τ n in place of t in (6.1) and then taking expectation and using Proposition 6.1 we obtain Hence by Gronwall's lemma Ey(t ∧ τ n ) = 0 for each t ∈ [0, T ] and integer n ≥ 1, which implies almost surely y t = 0 for all t ∈ [0, T ] and completes the proof of the uniqueness. with a constant N = N (m, d, p, T, K, K ξ , K η ).
Proof. We may assume that the right-hand side of the inequality (6.6) is finite. For multinumbers |α| ≤ m and ϕ ∈ C ∞ Recall, see (5.1), that by Lemma 3.8 on Itô's formula for each integer n ∈ [0, m] d|D n u t | p Lp = (Q n,p (v t , t, f t , g t ) + Q ξ n,p (v t ) +Q n,p (v t , h t )) dt + where the Q n,p , Q ξ n,p andQ n,p are defined in (5.2), (5.3) and (5.5), and ζ i = (ζ i (t)) t∈[0,T ] is a cadlag local martingale starting from zero for each i = 1, 2, 3, such that and for v ∈ W m+2 p and h ∈ W m+2 p,2 . By Propositions 5.1 and 5.2 we obtain Hence using the estimate (5.25) in Proposition 5.7 we have E|D n u t∧τ k | p ≤ E|D n ψ| p + N with a constant N = N (ε, d, m, p, T, K, K ξ , K η ). The proof of (6.10) is well-known and it goes as follows. By the Davis inequality, using the estimate in (5.24) we obtain E sup t≤T |ζ 1 (t)| ≤ 3E   which gives (6.10) by virtue of (6.9). To prove (6.11) we first assume that µ is a finite measure. Then taking into account Lemma 5.5 we have ζ(t) := ζ 2 (t) + ζ 3 (t) = for i = 4 in the same way as estimate in (6.10) is proved. Using Lemma 3.7 (iii) we get which allows us to get the estimate (6.13) for i = 5. Using Hölder's inequality we get W n p ds, which gives the estimate (6.13) for i = 6. By Lemma 5.6 and the estimate in (6.9) we have for each m ≥ 1 with a constant N = N (d, m, p, T, K m , K ξ,m , K η,m ). Since u is a cadlag process with values in H n p , with almost surely no jump at T , we can change the essential supremum to supremum here, which finishes the proof of the proposition. 6.3. Existence of a generalised solution. In the whole section we assume that the conditions of Theorem 2.1 are in force. By standard stopping time argument we may assume that EK p p,m (T ) < ∞. First we additionally assume that Assumption 6.1 holds and that m is an integer. Under these conditions we approximate the Cauchy problem (1.1)-(1.2) by mollifying all data and coefficients involved in it. For ε ∈ (0, ε 0 ) we consider the equation t (x, z) π(dz, dt), (6.17) with initial condition v 0 (x) = ψ (ε) , (6.18) where ε 0 is given in Lemma 3.4, with operators L ε = a εij D ij + b (ε)i D i + c (ε) , a ε = a (ε) + εI, and N ξ (ε) and N η (ε) defined as N ξ and N η in (1.3) with ξ (ε) and η (ε) in place of ξ and η, respectively. Recall that v (ε) denotes the mollification v (ε) = S ε v of v in x ∈ R d defined in (3.4). Note that by virtue of standard properties of mollifications and by Lemmas 3.4 and 3.5 the conditions of Proposition 6.3 are satisfied. Hence for u ε , the solution of (6.17)-(6.18) we have |u ε | V n r,p ≤ N (|ψ| Ψ n p + |f | H n p + |g| H n+1 p (l 2 ) + |h| H n+i p (L p,2 ) ) for n = 0, 1, 2, ..., m (6.19) for every integer r > 1 with a constant N = N (d, p, m, T, K, K ξ , K η ), where i = 1 when p = 2 and i = 2 when p > 2. Since V n r,p is reflexive, there exists a sequence {ε k } ∞ k=1 and a process u ∈ V n,r p such that lim k→∞ ε k = 0 and u ε k converges weakly to some u in V n,r p . To show that a modification of u is a solution to (1.1)-(1.2) we pass to the limit in the equation s,z (x)) − u ε s (x) + h (ε) s (z) ϕ(x) dxπ(dz, ds) (6.20) where ϕ ∈ C ∞ 0 . To this end we take a bounded predictable real-valued process ζ = (ζ t ) t∈[0,T ] , multiply both sides of equation (6.20) with ζ t and then integrate the expression we get against P ⊗ dt over Ω × [0, T ]. Thus we obtain