Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition

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Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition

Christel Geiss, Stefan Geiss, Emmanuel Gobet

To cite this version:

Christel Geiss, Stefan Geiss, Emmanuel Gobet. Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition. Stochastic Processes and their Applications, Elsevier, 2012, 122 (5), pp.2078–2116. �hal-00572496�

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Generalized fractional smoothness and L

_p

-variation of BSDEs with non-Lipschitz

terminal condition

Christel Geiss, Stefan Geiss^∗ and Emmanuel Gobet ^† Department of Mathematics

University of Innsbruck Technikerstraße 13/7

A-6020 Innsbruck Austria

[email protected] and [email protected] Centre de Math´ematiques Appliqu´ees^‡

Ecole Polytechnique F-91128 Palaiseau Cedex

France

[email protected] March 1, 2011

Abstract

We relate theLp-variation, 2 ≤p <∞, of a solution of a backward stochastic differential equation with a path-dependent terminal condition to a generalized notion of fractional smoothness. This concept of fractional smoothness takes into account the quantitative propagation of singularities in time.

MSC2010: 60H10, 46B70

∗The author is supported by the Project 133914 of theAcademy of Finland.

†This research is part of the Chair Financial Risks of the Risk Foundation, the Chair Derivatives of the Futureand the ChairFinance and Sustainable Development.

‡A significant part of this work was done when the third author was affiliated to Grenoble Institute of Technology.

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Introduction

During the last years the concept of fractional smoothness in the sense of function spaces has been used in the theory of stochastic processes to analyze approximation and variational properties. It turned out that phenomena known for special examples can be explained in terms of fractional smoothness. For example, approximation properties of certain stochastic integrals can be explained by the fractional smoothness of the integral itself, see [10, 11]. Similarly, variational properties of backward stochastic differential equations (BSDEs) can be upper bounded in case that the fractional smoothness of the terminal condition is known. To explain the latter aspect consider the BSDE

Yt=ξ+ Z T

t

f(s, Xs, Ys, Zs)ds− Z T

t

ZsdWs

with a Lipschitz generatorf, whereX = (Xt)t∈[0,T] is a forward diffusion, and define theLp-variation

varp(ξ, f, τ) := sup

i=1,...,n

sup

ti−1<s≤ti

kYs−Yti−1k^p+ Xn

i=1

Z ti

ti−1

kZt−Zti−1k²pdt

!¹₂

whereτ= (ti)ⁿ_i=0 is a deterministic time-net 0 =t0<· · ·< tn =T, Zti−1 := 1

ti−ti−1❊"Z ti

ti−1

Zsds|F^tⁱ⁻¹

# ,

and where 2 ≤ p < ∞, which we will assume throughout this paper. Note that by interchanging theLp- and L2-norms (where we use p≥ 2) and using the Burkholder-Davis-Gundy inequality, theLp-distance between the stochastic integralRT

0 ZsdWsand its discrete counterpartPn

i=1Zti−1(Wti−Wti−1) is upper bounded by a multiple of Pn

i=1

Rti

ti−1kZt−Zti−1k²pdt¹₂

. Hence the quantity varp(ξ, f, τ) is stronger compared to what is needed to quantify the discretization of the stochastic integral term of our BSDE. Besides the fact that this variation gives a strong insight into the quantitative behavior of the BSDE, in particular var2(ξ, f, τ) was used to describe the error in adapted backward Euler schemes forξ=g(XT) withgbeing a Lipschitz function; see [5, 23] for implicit schemes and [15, 16] for explicit schemes possibly with jump processes. In [14, Theorems 3.1 and 3.2] upper bounds for

Xn i=1

Z ti

ti−1

kZt−Zti−1k²2dt were obtained forξ=g(XT) satisfying

❊|g(XT)−❊(g(XT)|F^t)|²≤c²(T−t)^θ

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for some 0 < θ ≤ 1, where g is not assumed to be a Lipschitz function. On the other hand, path-dependent settings without taking into account fractional smoothness were considered, for example, in [19, 20, 25] In this paper, results are generalized and extended into the following directions:

• We consider a path-dependent setting by terminal conditions of the form ξ=g(Xr1, ..., XrL)

with 0 =r0 <· · ·< rL=T, whereg is not necessarily a Lipschitz function and introduce a corresponding path-dependent fractional smoothness in the Malliavin sense. This concept of smoothness extends the classical concepts, based on real interpolation, to a time-dependent one taking care about the propagation of smoothness in time. In the classical case one would assign to a random variableξsome 0< θ≤1 which describes the fractional smoothness of ξ while here we assign to the parameters (ξ, f) of our BSDE avectorΘ = (θ1, ..., θL), whereθl stands for the local smoothness of the BSDE at timerl. It turns out that this vector is com- pletely characterized by theLp-variation ofY andZ. In case our terminal condition depends onXT only our generalized smoothness coincides with earlier approaches from, for example, [10] and [14].

• Instead of theL2-variation we consider the strongerLp-variation with 2≤ p <∞. In addition, the integratedZ-variationPn

i=1

Rti

ti−1kZt−Zti−1k²2dt is replaced by the variationkZs−Ztk^p with sandtbeing fixed, and the Lp-variation of the process Y is included as well. To our knowledge the weaker criterion for 0 < p <2 in the context of this paper has not been considered yet and might require different arguments as some of our proofs rely on the condition thatp≥2.

• We provide equivalences showing that the results are sharp.

• In Corollary 2.4 we show, given the terminal conditionξ=g(Xr1, ..., XrL) has a certain fractional smoothness, how to obtain time-netsτⁿ of cardi- nalityLn+ 1 such that

sup

n

√nvarp(ξ, f, τⁿ)<∞.

These time-nets compensate the possible singularities of the Z-process when approaching a time-pointrlfrom the left.

Organization of the paper. After introducing the setting in Section 1, we for- mulate in Section 1.2 our concept of functional fractional smoothness of a BSDE and characterize this smoothness in various ways. Here we partly transfer the results from [10] and [14] from the caseξ=g(XT) to the path-dependent one.

In Section 2.2 we present two sufficient conditions for our fractional smoothness. The point of these two conditions (Corollary 2.6 and Theorem 2.10) is

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that they only involve the terminal conditionξ and do not use the solutionY nor the generatorf of our BSDE. The proofs of the main results are contained in Section 3.

Some notation. Given a vectorx∈❘^d we denote by |x| its Euclidean norm, for a linear operator D ∈ L(❘ⁿ,❘^m) the symbol |D| stands for the Hilbert- Schmidt norm, where ❘ⁿ and ❘^m are equipped with the standard Euclidean structure. GivenD(t, x)∈ L(❘ⁿ,❘^m) with (t, x)∈[0, T]×❘^d and 0< T <∞, we use

kDk∞:= sup

x∈❘^d,t∈[0,T]|D(t, x)|. Finally, B(η1, η2) :=R1

0 x^η¹⁻¹(1−x)^η²⁻¹dx where η1, η2 > 0, will denote the Beta-function.

1 Setting and basic concepts

1.1 Forward-backward stochastic differential equations

We fix a complete probability space (Ω,F,P),T >0,d≥1 and ad-dimensional standard Brownian motion W = (Wt)t∈[0,T] with W0 ≡ 0. Furthermore, we assume that (F^t)t∈[0,T] is the augmentation of the natural filtration ofW. The forward equation. Let

Xt=x0+ Z t

0

b(s, Xs)ds+ Z t

0

σ(s, Xs)dWs

with x0 ∈ ❘^d, where b : [0, T]×❘^d → ❘^d and σ : [0, T]×❘^d → L(❘^d,❘^d) satisfy the following conditions:

(Ab,σ) We have b, σ ∈C_b^0,2([0, T]×❘^d), where the derivatives up to order two are taken with respect to the space-variables and, for some γ ∈ (0,1], are assumed to be γ-H¨older continuous (w.r.t. the parabolic metric) on all compact subsets of [0, T]×❘^d. Moreover, there is aδ >0 such that hAx, xi ≥δ|x|² for x∈❘^d andb andσ are ¹₂-H¨older continuous in time, uniformly in space.

We work with the usual stochastic flow (X_s^t,x)s,t∈[0,T],x∈❘^dthat solves for (t, x)∈ [0, T]×❘^d the SDEXs=xon [0, t] and dX_s^t,x=σ(s, X_s^t,x)dW_s^t+b(s, X_s^t,x)ds on [t, T], whereW_s^t:=Ws−Wtand the augmented natural filtration (Fs^t)s∈[t,T]

of (W_s^t)s∈[t,T] is used (i.p. X =X^0,x⁰). With our assumptions we can assume that (X_s^t,x)_s,t_∈_[0,T],x_∈❘^d is a continuous process in (s, t, x).

Ifg:❘^d→❘is a polynomially bounded Borel function, 0< R≤T, and F(t, x) :=❊g(X_R^t,x) for 0≤t≤R, (1)

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thenF ∈C^1,2([0, R)×❘^d) and

∂

∂tF(t, x) +1 2

A(t, x), D²F(t, x)

+hb(t, x),∇^xF(t, x)i= 0 by Proposition B.1 below where

D²:=

∂²

∂xi∂xj

^d

i,j=1

.

The standard tail estimates for the transition density Γ are re-called in Propo- sition B.1. They ensure that _∂^∂

t∇^xF,∇^x∂^∂tF andD^m_xF with|m| ≤3 exist and are continuous on [0, R)×❘^d. For 0≤t≤r < R≤T one has that, a.s.,

∇^xF(r, X_r^t,x) = ❊

g(X_R^t,x)N_R^r,1,(t,x)|Fr^t

,

D²F(r, X_r^t,x) = ❊

g(X_R^t,x)N_R^r,2,(t,x)|Fr^t

for the Malliavin weightsN_R^r,i,(t,x) that satisfy, for any given 0< q <∞, that h❊

N_R^r,i,(t,x)

q|Fr^t

i¹_q

≤ κq

(R−r)ⁱ² a.s. and ❊

N_R^r,i,(t,x)|Fr^t

= 0 a.s.

for i = 1,2 and all 0 ≤t ≤r < R ≤ T with a constantκq >0 independent from (t, r, R, x) (see [17], [14, Proof of Lemma 1.1] and Remark B.2 below). A typical application of these estimates are the crucial inequalities

k∇^xF(r, X_r^t,x)k^p ≤ κp^′kg(X_R^t,x)−❊(g(X_R^t,x)|Fr^t)k^p

√R−r , (2)

kD²F(r, X_r^t,x)k^p ≤ κp^′kg(X_R^t,x)−❊(g(X_R^t,x)|Fr^t)k^p

R−r , (3)

for 1< p, p^′<∞with 1 = (1/p) + (1/p^′).

The backward equation. We are interested in the backward equation Yt=ξ+

Z T t

f(s, Xs, Ys, Zs)ds− Z T

t

ZsdWs for t∈[0, T] a.s.

and assume the following conditions:

(Af) The function f : [0, T]×❘^d×❘×❘^d → ❘ is continuous in (t, x, y, z) and continuously differentiable in x, y and z with uniformly bounded derivatives. In particular, there areKf >0 andLf >0 such that

|f(s, x1, y1, z1)−f(s, x2, y2, z2)| ≤ Lf[|x1−x2|+|y1−y2|+|z1−z2|],

|f(s, x, y, z)| ≤ Kf+Lf(|x|+|y|+|z|).

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(Ag) There are R = {r0, ..., rL} with 0 = r0 < r1 < · · · < rL = T and a measurable function of at most polynomial growth g : (❘^d)^L →❘such that

ξ:=g(Xr1, ..., XrL).

In this setting, the solution (Y, Z) to the above BSDE is uniquely defined in any Lp-space for 1< p <∞; see [6, Theorem 4.2]. Additionally, we assume in the paper that the solution (Y, Z) is realized such that, on [rl−1, rl),

Yt=ul(Xl−1;t, Xt) and Zt=vl(Xl−1;t, Xt)σ(t, Xt),

where we setXl−1:= (Xr1, . . . , Xrl−1). The above functionsul andvl are well defined due to the next proposition, which is an extension of [24, Theorem 3.2]

and follows from Lemma A.2, see also [19].

Proposition 1.1. Assume that (Ab,σ),(Af)and (Ag)are satisfied. Then, for l = 1, ..., L there exist measurable ul : (❘^d)^l⁻¹ ×[rl−1, rl)×❘^d → ❘ and vl: (❘^d)^l⁻¹×[rl−1, rl)×❘^d→❘¹^×^d and Borel sets Dl⊆❘^d(l⁻¹⁾,l= 2, ..., L, such thatD^c_l is of Lebesgue measure zero, and such that

(i) ul(xl−1;·,·) : [rl−1, rl)×❘^d→❘is continuous and continuously differentiable w.r.t. the space variable with∇^xul(xl−1;t, x) =vl(xl−1;t, x), where xl−1= (x1, . . . , xl−1),

(ii) there are αl, ql,1, ..., ql,l∈[1,∞)such that sup

t∈[rl−1,rl)|ul(xl−1;t, x)|+ sup

t∈[rl−1,rl)

√rl−t|vl(xl−1;t, x)|

≤αl(1 +|x1|^q^l,1+· · ·+|xl−1|^q^l,l−1+|x|^q^l,l), (iii) for alll= 1, ..., L,x1, ..., xl−1, x∈❘^d andrl−1≤s < rl the triplet

X_t^s,x, ul(xl−1;t, X_t^s,x), vl(xl−1;t, X_t^s,x)σ(t, X_t^s,x)

t∈[s,rl)

solves the BSDE with generatorf and terminal condition ul(xl−1;rl, X_r^s,x_l )

where

ul(xl−1;rl, x) :=

ul+1(xl−1, x;rl, x)χDl(xl−1) : 2≤l < L, g(xl−1, x)χDl(xl−1) : l=L.

andu1(r1, x) :=u2(x;r1, x).

In the above proposition we used the convention thath(x0;·) :=h(·). It should be noted that by Proposition 1.1 we modify at each levell= 2, ..., Lthe functional for theY-process on a nullset. However, because of

P(Xr1 ∈D2, ...,(Xr1, ..., XrL−1)∈DL) = 1, (4) this does not affect the Lp-solution of our BSDE so that Proposition 1.1 is sufficient for our purpose.

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Piece-wise linearization of the backward equation. We letFl(xl−1;·,·) : [rl−1, rl]×❘^d→❘be given by

Fl(x1, ..., xl−1;t, x) =Fl(xl−1;t, x) :=❊ul(x1, ..., xrl−1;rl, X_r^t,x_l ).

The functionFlsolves the backward PDE

∂Fl

∂t (xl−1;t, x) +1 2

A(t, x), D²Fl(xl−1;t, x)

+hb,∇^xFl(xl−1;t, x)i= 0 on the interval [rl−1, rl) for fixedx1, ..., xl−1∈❘^d.

Two facts that are frequently used in the paper. Firstly, for a filtered probability space (M,Σ,◗,(G^t)t∈[r,R]), 1≤q≤ ∞,r≤t≤Rand ξ∈Lq, one has that

kξ−❊(ξ|G^t)k^q ≤ sup

t≤s≤Rkξ−❊(ξ|G^s)k^q ≤2kξ−❊(ξ|G^t)k^q (5) as a consequence that ❊(·|F^s) is a contraction on Lq. Secondly, given the assumptions on our forward diffusion, a polynomially bounded Borel function g:❘^d→❘,r≤t≤R≤T and 1≤q <∞, we have that

kg(X_R^r,x)−❊(g(X_R^r,x)|Ft^r)kq

≤ Z

❘^d

Z

❘^d

Z

❘^d|g(ξ)−g(η)|^qΓ(r, x;t, y)Γ(t, y;R, ξ)Γ(t, y;R, η)dydξdη ¹_q

≤2kg(X_R^r,x)−❊(g(XR^r,x)|Ft^r)kq. (6)

1.2 Functional fractional smoothness

The usage of fractional smoothness in the investigation of variational properties of BSDEs is the central idea of this paper. Fractional smoothness can be defined in various ways. One way is the so-calledK-method, a method where functions are decomposed into differentiable parts and parts that are not differentiable.

A quantitative analysis of these decompositions leads to fractional smoothness.

To be more precise, assume two Banach spacesX0 and X1, where (say) X1 is continuously embedded intoX0, 0 < t < ∞ and x ∈ X0, and recall that the K-functional is given by

K(x, t;X0, X1) := inf{kx0k^X0+tkx1k^X1 :x=x0+x1}. For 0< θ <1 and 1≤q≤ ∞this leads to the real interpolation spaces

kxk(X0,X1)θ,q:=t⁻^θK(x, t;X0, X1)

Lq(^(0,∞),^dt_t) with

X1⊆(X0, X1)θ1,q₁^′ ⊆(X0, X1)θ1,q1⊆(X0, X1)θ0,q0 ⊆X0

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where 0< θ0< θ1<1 and 1≤q0, q1, q^′₁≤ ∞withq₁^′ ≤q1(see [2, 3]). Applying this concept to the Malliavin Sobolev spaceD1,p, we obtain the Malliavin Besov (or fractional Sobolev) spaces

B_p,q^θ := (Lp, D1,p)θ,q (7) where 0< θ <1 is the main parameter of the smoothness and 1< q ≤ ∞the fine-tuning parameter. In a context close to this paper these spaces and related ones have been exploited for example in [10, 11, 14, 21]. The classical setting of the Wiener space is changed in [10, 14] into a setting where the standard Gaussian measure is replaced by the distribution of the forward diffusion. Here we go one step ahead and replace 0< θ <1 by a vector Θ = (θ1, ...., θL), where θldescribes the smoothness at time rl:

Definition 1.2. Let Θ = (θ1, ..., θL)∈(0,1]^L, 2≤p <∞andξ∈Lp. IfY is the solution of the BSDE with generatorf and terminal conditionξ, then we let (ξ, f)∈B_p,^Θ_∞(X) provided that there is somec >0 such that

kYrl−❊(Yrl|F^s)k^p≤c(rl−s)^θl²

for alll = 1, ..., L and rl−1 ≤ s < rl. The infimum over all possible c > 0 is denoted by

cB^Θ_p,∞=cB^Θ_p,∞(ξ, f).

In the case thatf = 0 we will simply writeξ∈B_p,^Θ_∞(X).

Specializing top= 2 and to the linear one-step Gaussian case (X =W, T = L= 1 andf = 0) it holds (see [11, Corollary 2.3]) that

g(W1)∈B_2,^(θ)_∞(W) if and only if g∈B^θ_2,_∞(❘^d, γd),

where the Wiener space over the standard Gaussian measureγdon❘^dis considered. In particular, ford= 1 and for the orthonormal basis consisting of Hermite polynomials (hk)^∞_k=0⊆L2(❘, γ1) we obtain thatg=P_∞

k=0αkhk∈B_2,^θ_∞(❘, γ1) if and only if there is somec >0 such that for all 0≤t <1 one has that

X∞ k=1

kt^k⁻¹α²_k≤ c² (1−t)¹⁻^θ,

see [11, Theorem 2.2]. These connections explain the notation (p,∞) in Def- inition 1.2. For a more general connection between the speed of convergence of the conditional expectations used in Definition 1.2 and the real interpolation method the reader is referred to [11]. Our definition of fractional smoothness by an upper bound of

kYrl−❊(Yrl|F^s)k^p

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has the advantage that (2) and (3) give the upper bounds ∇^xFl(Xl−1;s, Xs)

p≤κp^′kYrl−❊(Yrl|F^s)k^p

√rl−s ≤κp^′cB_p,∞^Θ (rl−s)^θl⁻¹² and

D²Fl(Xl−1;s, Xs)_p≤κp^′kYrl−❊(Yrl|F^s)k^p

rl−s ≤κp^′cB^Θ_p,∞(rl−s)^θl²⁻² forrl−1 ≤s < rl and 1 = (1/p) + (1/p^′), so that we can control the gradient and the Hessian ofFl. For our paper the fine-tuning parameter q=∞in (the generalization of) (7) turns out to be the right one.

Finally, we want to mention the coincidence, that most of the relevant examples are naturally linked to this fine-tuning parameterq=∞in (7).

1.3 Time-nets, splines and entropy numbers

In our BSDE system theZ-process gets possibly singular at any of the particular time pointsrlwhenrlis approached from the left. The degree of this singularity is determined by the parameterθldescribing the fractional smoothness inrl. To keep the variation varp(g(Xr1, ..., XrL), f, τ) small, we have to choose time-nets which refine on the left ofrl with an order given by the fractional smoothness θlwhile each of the intervals [rl−1, rl] is divided intonsub-intervals.

Definition 1.3. For Θ∈(0,1]^L we letτ^n,Θ = (t^n,Θ_k )^nL_k=0 be given by t^n,Θ₀ := 0 and

t^n,Θ_k :=rl−1+ (rl−rl−1) 1−

1−k−(l−1)n n

_θl¹!

for (l−1)n < k≤ln.

Estimates on theLp-variation kYt−Ysk^p are close to estimates how good the processY can be approximated inLp by linear adapted splines, i.e. we simply compute adapted approximations ofY at the time-pointst0, ..., tn and interpo- late them linearly. So the notionadapted splinerefers to the fact that the knots are adapted, however the spline itself is not an adapted process. The adapted splines are typically used in complexity theory for stochastic processes to find efficient approximation schemes for stochastic processes where the whole path needs to be approximated but the adaptedness of the approximation is not fully needed, see [7]. Here we use the following notation:

Definition 1.4. Given a time-netτ = (tk)ⁿ_k=0 withr=t0<· · ·< tn=R≤T we say that the processS= (St)t∈[r,R]is an adapted spline based onτ provided thatStk isF^tk-measurable for allk= 0, ..., nand

St:= tk−t tk−tk−1

Stk−1+ t−ttk−1

tk−tk−1

Stk for tk−1≤t≤tk.

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Finally, we recall the notion of entropy numbers to measure and compare com- pactness properties ofY = (Ys)s∈[t,rl] ast↑rl where the process gets singular.

Definition 1.5. Given a normed spaceEandA⊆Ewe defineen(A|E) := infε, where the infimum is taken over allε >0 such that there arex1, ..., xn∈Ewith

A⊆ [n i=1

{xi+εBE} with BE:={x∈E:kxk ≤1}.

2 Functional fractional smoothness and BSDEs

2.1 A general equivalence

The basic result of this paper is

Theorem 2.1. Assume that(Ab,σ),(Af)and(Ag)are satisfied. For2≤p <∞ andfixedl∈ {1, ..., L} andθl∈(0,1]consider the following conditions:

(C1l) There is somec1>0 such that, forrl−1≤s < t < rl, kZt−Zsk^p ≤c1

Z t s

(rl−r)^θ^l⁻²dr ¹2

.

(C2l) There is somec2>0 withkZtk^p≤c2(rl−t)^θl⁻¹² forrl−1≤t < rl. (C3l) There is somec3>0 such that, forrl−1≤s < t≤rl,

kYt−Ysk^p≤c3

Z t s

(rl−r)^θ^l⁻¹dr ¹2

.

(C4l) There is somec4>0 such that, forrl−1≤s < rl, kYrl−❊(Yrl|F^s)k^p≤c4(rl−s)^θl². (C5l) There is somec5>0 such that, forrl−1≤t < rl,

Z t rl−1

|(D²Fl)(Xl−1;s, Xs)|²ds

!¹₂

p

≤c5(rl−t)^θl²⁻¹.

(C6l) There is some c6 > 0 such that for all n = 1,2, ... there is an adapted splineSⁿ = (S_tⁿ)_t_∈_[r_l−1_,r_l_] based on

rl−1+ (rl−rl−1) 1−

1−k n

_θl¹!!n

k=0

such that √

n sup

t∈[rl−1,rl]kYt−S_tⁿk^p≤c6.

The spline can be arranged such thatS_rⁿ_l−1 =Yrl−1 andS_rⁿ_l=Yrl.