Fractional smoothness of functionals of diffusion processes under a change of measure

Let $v:[0,T]\times \R^d \to \R$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [\sigma \sigma^\perp]_{il} \partial_{x_i \partial_{x_l} v + \sum_{i} b_i \partial_{x_i}v + kv =0$ with terminal condition $g$, where the coefficients are time- and state-dependent, and satisfy certain regularity assumptions. Let $X=(X_t)_{t\in [0,T]}$ be the associated $\R^d$-valued diffusion process on some appropriate $(\Omega,\cF,\Q)$. For $p\in [2,\infty)$ and a measure $d\P=\lambda_T d\Q$, where $\lambda_T$ satisfies the Muckenhoupt condition $A_\alpha$ for $\alpha \in (1,p)$, we relate the behavior of $\|g(X_T)-\ept g(X_T) \|_{L_p(\P)}$, $\|\nabla v(t,X_t) \|_{L_p(\P)}$ and $\|D^2 v(t,X_t) \|_{L_p(\P)}$ to each other, where $D^2v:=(\partial_{x_i \partial_{x_l}v)_{i,l}$ is the Hessian matrix.


Introduction
For a fixed time-horizon T > 0 let (Ω, F , (F t ) t∈[0,T ] , É) be a filtered probability space where (Ω, F , É) is complete, F = F T , the filtration (F t ) t∈[0,T ] is right-continuous, F 0 is generated by the null sets of F and where all local martingales are continuous (see Section 2). Assume for some d ≥ 1 that the process B = (B t ) t∈[0,T ] is a d-dimensional (F t ) t∈[0,T ] -standard Brownian motion starting in zero. We consider an Ê d -valued diffusion process X = (X t ) t∈[0,T ] , solution to the stochastic differential equation for some smooth bounded coefficients b and σ, and we focus on the rate of convergence of R X p (t) := g(X T ) − (g(X T )|F t ) p for p ∈ [2, ∞) as t → T , where g satisfies a suitable growth condition ensuring g(X T ) ∈ L p . The behavior of R X p (t) as t → T is a measure of the fractional smoothness of g, see [4] for an overview. Actually it is now well-known [3,6,10,5] that there is a precise correspondence between the irregularity of the terminal function g and the time-singularity of the L p -norms of ∇v(t, X t ) as t ↑ T where v(t, x) = (g(X T )|X t = x).
The aim of this paper is to extend these quantitative equivalence results to situations where the L p -norms are computed under different measures. The theory of probabilistic Muckenhoupt weights, developed as a counterpart to the deterministic ones from [14] and other papers, gives a natural way to extend various martingale inequalities to equivalent measures, see exemplary [12,1,13] and the references therein. A typical situation is a change of measure initiated by a Girsanov transformation, i.e. a change of the drift of X. Applying the results of this paper in this particular case, gives -without going into full details-the following: if the process Y differs from X by another bounded drift and if θ ∈ (0, 1), then we have (1) which follows from Theorem 1 below for q = ∞ as explained in Remark 2 (7). The parameter θ is the degree of fractional smoothness.
Regarding the references in the literature related to (1), a 1-dimensional diffusion case with X = Y is considered in [3], the extension to multidimensional processes is performed in [6] in the case X = Y being a Brownian motion and in [10] for diffusion processes. In [5] path-dependent functionals are considered. For an overview the reader is referred to [4]. Actually our main result (Theorem 1) takes a more general form than (1): • we consider an additional potential factor k in our parabolic problem to define v; • the change of measure, described in (1) by the change from X to Y , is described by Muckenhoupt weights; • we also state results regarding the second derivatives.
Applications. The tight control of the behavior of the norms ∇v(t, X t ) L 2 as t → T is an issue that has been raised in [3], where the purpose was to analyze discrete approximations of stochastic integrals coming from the representation Discretizing the above stochastic integral and analyzing the resulting approximation error in L 2 , requires a better understanding how strongly the irregularity of the terminal function g transfers to the blow-up of the function t → ∇v(t, X t ) L 2 and higher derivatives of v as well. Major consequences of this analysis are the derivation of tight convergence rates for uniform time grids and the design of non-equidistant time grids to obtain optimal convergence rates.
Recently, similar results have been established in the context of Backward Stochastic Differential Equations [10,5] to pave the way for the development of more efficient numerical schemes. Finally, similar issues arise in the analysis of the Delta-Gamma hedging strategies in Finance, which typically result in a higher order approximation of the stochastic integral (2), see [11]. Within the applications in Stochastic Finance intrinsically two measures are involved: the historical measure for evaluating the risk, for example as L pmean, and the risk-neutral measure, under which the price and the hedging strategy are computed and which is related to the above function v. For this setting, the current results are particularly of interest. Moreover, the potential k may be interpreted as an interest rate.

Setting
Notation. We denote by | · | the Euclidean norm of a vector. Given a matrix C considered as operator C : ℓ n 2 → ℓ N 2 , the expression |C| stands for the Hilbert-Schmidt norm and C ⊤ for the transposed of C. The L pnorm (p ∈ [1, ∞]) of a random vector Z : Ω → Ê n or a random matrix Z : Ω → Ê n×m is denoted by Z p = |Z| Lp . As usual, ∂ α x ϕ is the partial derivative of the order of an multi-index α (with length |α|) with respect to x ∈ Ê d . The Hessian matrix of a function ϕ : Ê d → Ê is abbreviated by D 2 ϕ and the gradient (as row vector) by ∇ϕ. In particular, this means that D 2 and ∇ always refer to the state variable x ∈ Ê d . If we mention that a constant depends on b, σ or k, then we implicitly indicate a possible dependence on T and d as well. Finally, letting h : [0, T ] × Ê d → Ê n×m we use the notation h ∞ := sup t,x |h(t, x)|.
The parabolic PDE. We fix T > 0 and consider the Cauchy problem where A := (a ij ) ij = σσ ⊤ . The assumptions on the coefficients and g are as follows: (C1) The functions σ i,j , b i , k are bounded and belong to C 0,2 and there is some γ ∈ (0, 1] such that the functions and their statederivatives are γ-Hölder continuous with respect to the parabolic metric on each compactum of [0, T ]×Ê d . Moreover, σ is 1/2-Hölder continuous in t uniformly in x. (C3) the terminal function g : Ê d → Ê is measurable and exponentially bounded: for some K g ≥ 0 and κ g ∈ [0, 2) we have t Γ exist in any order, are continuous, and satisfy For , for x ∈ Ê d and t ∈ [0, T ), where c > 0 depends at most on (κ g , K g , c (3) , T ). As we work on a closed time-interval we have to explain our understanding of a local martingale: we require that the localizing sequence of stopping for this is that we think about the extension of the filtration constantly by F T to (T, ∞) and that all local martingales (N t ) t∈[0,T ] (in our setting) are extended by N T to (T, ∞). This yields the standard notion of a local martingale. However this is not needed explicitly in our paper, we only need this implicitly whenever we refer to results about the Muckenhoupt weights A α (É) from [13].
To shorten the notation, we denote sometimes the conditional expectation (.|F t ) by Ft (.). The process X = (X t ) t∈[0,T ] is given as strong unique solution of Introducing the standing notation Moreover, lim almost surely and in any L r (É) with r ∈ [1, ∞). Using Proposition 1 for where c > 0 depends at most on (σ, b) and is, in particular, independent from the starting value so that Remark 1 applies as well. We will also use the following Let t ∈ (0, T ], h : Ê d → Ê be a Borel function satisfying (C3) and Γ X be the transition density of X, i.e. the function Γ from Proposition 1 in the case is right-continuous and G r is generated by the null sets of G. Then, for q ∈ (1, ∞) and s ∈ [r, t), one has a.s. that where κ q > 0 depends at most on (σ, b, q).
Conditions on the equivalent measure. In addition to the given measure É we will use an equivalent measure È ∼ É and agree about the following standing assumption: is a martingale and dÈ = λ T dÉ.
Definition 1. Assume that condition (P) is satisfied.
(i) For α ∈ (1, ∞) we say that λ T ∈ A α (É) provided that there is a constant c > 0 such that for all stopping times τ : Ω → [0, T ] one has that (ii) For β ∈ (1, ∞) we let λ T ∈ RH β (É) provided that there is a constant c > 0 such that for all stopping times τ : Ω → [0, T ] one has that The class A α (É) is the probabilistic variant of the Muckenhoupt condition and RH stands for reverse Hölder inequality. Next we need where N * t := sup s∈[0,t] |N s |.

Remark 2.
(1) Using [13,Corollary 3.3] it is sufficient to require that λ T ∈ A p (É) as in this case there is an ε ∈ (0, p − 1) such that λ T ∈ A p−ε (É). One the other hand, it would be of interest to investigate the case when λ T ∈ A α (É) with α > p. This is not done here.
(3) In the case X = B, È = É, T = 1 and k = 0 the conditions of Theorem 1 (neglecting the boundedness condition (C3)) are equivalent to that g belongs to the Malliavin Besov space B θ p,q on Ê d weighted by the standard Gaussian measure (see [8]).
under É it is natural to consider condition (i θ ) for the corresponding martingale under È as well: One can easily check that (i θ ) ⇐⇒ (i ′ θ ) for θ ∈ (0, 1] and q ∈ [1, ∞]: Indeed, for any random variables U and V , respectively bounded and in L p (È), observe that For U := e T 0 k(r,Xr)dr and V := g(X T ) we have and obtain This proves (i θ )=⇒ (i ′ θ ). The converse is proved similarly by letting U := e − T 0 k(r,Xr)dr and V := e T 0 k(r,Xr)dr g(X T ).
(8) In the case the drift term in item (7) is Markovian, i.e. β t = β(t, X t ) for an appropriate β : [0, T ] × Ê d → Ê d , and if we let Y t := v(t, X t ) and Z t := ∇v(t, X t )σ(t, X t ), then we get the BSDE Then it is proved in [5] under certain conditions the equivalence between the following assertions for p ∈ [2, ∞), θ ∈ (0, 1] and polynomially bounded g: These are the analogues of (i θ ) and (ii θ ) for q = ∞.

Proof of Theorem 1
Through the whole section we assume that the condition (P) is satisfied.

Preliminaries
To estimate L p norms under different measures, the following lemma is useful. Proof.
As simple consequences of this lemma for V ≡ 1 , observe that In the next step we will estimate ∇v(t, X t ) and D 2 v(t, X t ) in Lemmas 3 and 6 from above by conditional moments of M T = K X T g(X T ) and g(X T ), and extend therefore Lemma 1 to the case k = 0 and allow a change of measure by Muckenhoupt weights.
Now we follow a martingale approach (see, for example, [9]) and prove the statement for the measure É.
is a É-martingale. One way consists in using Itô's formula to verify that N is a martingale. In fact, the bounded variation term in the Itô-process decomposition of N is where t 0 C s ds is the bounded variation term of ∇v(t, X t )∇X t . Hence it is sufficient to show that The PDE for w = ∇v on [0, T ) × Ê d reads as By a simple computation this gives that the bounded variation term of (c) Exploiting the martingale property of N between t and some deterministic S ∈ (t, T ), we have At the last equality, we have used the É-martingale property of (M t ) t∈[0,T ] and the conditional Itô isometry (available for any square integrable and progressively measurable matrixvalued processes (A 1,r ) r and (A 2,r ) r , having d columns and an arbitrary number of rows). After simplifications, (12) writes For the following we let m(t, x) := v(t, x)k(t, x).