An Integration by Parts Formula in a Markovian Regime Switching Model and Application to Sensitivity Analysis

An integration by parts formula in a Markovian regime switching model and application to sensitivity analysis

Yue Liu Nicolas Privault School of Finance and Economics School of Physical and Mathematical Sciences Jiangsu University Nanyang Technological University Zhenjiang 212013 21 Nanyang Link P.R. China Singapore 637371

August 5, 2017

Abstract We establish an integration by parts formula for the random functionals of a continuous-time Markov chain, based on partial diﬀerentiation with respect to jump times. In comparison with existing methods, our approach does not rely on the Girsanov theorem and it imposes less restrictions on the choice of directions of diﬀerentiation, while assuming additional continuity conditions on the considered functionals. As an application we compute sensitivities (Greeks) using stochastic weights in an asset price model with Markovian regime switching.

Key words: Integration by parts, Markov chains, regime switching, sensitivity analysis. Mathematics Subject Classiﬁcation (2010): 60G40; 35R35; 93E20; 60J28; 91G80.

1 Introduction

Integration by parts methods on the Wiener space have been successfully applied to the sensitivity analysis of diﬀusion models in ﬁnance, cf. [10]. This framework has also been implemented for jump processes, using the absolute continuity of jump sizes,

1 cf. [1], or the process jump times as in [13].

More generally, integration by parts formulas for discrete and jump processes can be obtained using multiple stochastic integral expansions and finite difference operators, or the absolute continuity of jump times. In the setting of continuous-time Markov chains, integration by parts formulas have recently been proposed in [19] and [6], based on finite difference and differential operators. The construction of [6] is based on the Girsanov approach as in [3], and it uses time shifts instead of space shifts of the under- lying process, while [19] uses the representation of Markov chains as semimartingales, cf. Appendix B of [8]. For discrete-time Markov chains and point processes, multiple stochastic integral expansions for random functionals have been built in e.g. [14], [2], [18], [4], [7].

The result of [19] is stated for functions of the numbers of chain transitions, cf. The- orem 4 therein, using the characterization of pure jump martingales under change of measure. However, this characterization cannot be applied to a shift of a Markov jump process (βt)t∈[0,T ] as claimed in Lemma 2 therein, otherwise it would actually entail the absolute continuity of the discrete random variable βt.

In this paper we derive an integration by parts formula similar to that of [6] for continous-time Markov processes. Our approach uses time changes based on the intensity of the point process as in [15], [16]. In comparison with the construction of [6], we impose less constraints on directions of diﬀerentiation as we do not use the Girsanov theorem and assume instead a smoothness condition on random variables.

Namely, we deﬁne the gradient DF of a smooth functional F of a continuous-time

Markov chain (βt)t∈IR+ with state space M in such a way that the integration by parts formula Z T Z T β β IE Dt F utdt = IE F ut(dNt − αβt dt) (1.1) 0 0 holds for (ut)t∈[0,T ] a square-integrable adapted process, see Proposition 4.2, where

2 β (Nt )t∈[0,T ] is the birth process counting the transitions of (βt)t∈IR+ , and (αl)l∈M is the set of parameters of the exponentially distributed interjump times of (βt)t∈IR+ . In this way, a “partial” Malliavin calculus is established with respect to the absolutely continuous transition times, while the spatial discrete jump component remains un- changed. This approach has some similarities to the partial Malliavin calculus of [5], in which only the Brownian component of a jump-diﬀusion proces is subjected to a random perturbation.

In comparison with the construction of [6], which is also based on functionals of transition times, we do not constrain (ut)t∈[0,T ] to satisfy the a.s. vanishing condition R T 0 utdt = 0. Instead, we assume a continuity condition on the random variables F in (1.1), see the condition (2.3) below, which nevertheless does not prevent us from including βT in the domain of D, with the relation DβT = 0.

As an application, we consider the sensitivity analysis of option prices in a diffusion model with regime-switching. Our method follows the Malliavin calculus approach to the fast computation of Greeks for option hedging [10], and in addition we take both sources of Markovian and Gaussian noises into account. In case the diffusion component vanishes, our result still allows us to estimate sensitivities by using Marko- vian noise, see Proposition 6.1. In addition, the absence of vanishing requirement on R T 0 utdt makes it easier to satisfy necessary integrability requirements on the associated stochastic weights, cf. Proposition 5.1. Indeed, such conditions are more difficult to satisfy in regime-switching models due to the changing signs of drifts, making it easier for denominators to vanish in the estimation of weights.

In Section2 we recall some background notation and results on integration by parts and gradient operators defined by infinitesimal perturbation of jump times. Section3 extends this construction to birth processes, using time changes based on the intensity of the considered birth process. Section4 treats the general case of finite continuous- time Markov chains by partial differentiation. In Section5 we apply the construction to sensitivity analysis of option prices in a jump-diffusion model driven by a geometric

3 Brownian motion with regime switching. Section6 deals with an extension to non- smooth payoﬀ functions.

2 Integration by parts for the Poisson process

In this section we start by reviewing integration by parts for a standard Poisson process (Nt)t∈IR+ with jump times (Tn)n≥1, and whose interjump times (τk)k≥1 are independent exponentially distributed random variables, cf. e.g. § 7.3 of [17].

Deﬁnition 2.1 Let ST denote the space of Poisson functionals of the form

∞ X F = f01{NT =0} + 1{NT =k}fk(T1,...,Tk), (2.1) k=1 where each function fk is weakly diﬀerentiable on the simplex

k k ST := (t1, . . . , tk) ∈ [0,T ] : 0 ≤ t1 ≤ · · · ≤ tk ≤ T ,

k k and is such that kfkk∞ ≤ A and k∂lfkk∞ ≤ A , 1 ≤ l ≤ k, k ≥ 1, for some constant A > 0.

The next statement comes from Deﬁnition 7.3.2 in [17]. We let ∂lf denote the partial derivative of a function f with respect to its l-th variable.

Deﬁnition 2.2 The gradient operator D is deﬁned on F ∈ ST of the form (2.1) by

∞ k X X DtF := − 1{NT =k} 1[0,Tl](t)∂lfk(T1,...,Tk), t ∈ IR+. (2.2) k=1 l=1

c Let now ST denote the subspace of ST made of Poisson functionals of the form (2.1) that satisfy the continuity condition

fk(t1, . . . , tk) = fk+1(t1, . . . , tk, tk+1), 0 ≤ t1 ≤ · · · ≤ tk ≤ T ≤ tk+1, k ∈ IN. (2.3)

c We note that given F ∈ ST of the form (2.1) and such that

fn(t1, . . . , tn) = fk(t1, . . . , tk), 0 ≤ t1 ≤ · · · ≤ tk, k ≥ n,

4 for some n ≥ 1, the function gn deﬁned by

gn(t1, . . . , tn) (2.4) n−1 X := f01{T

F = gn(T1,...,Tn), (2.5) and D can be written on F ∈ ST of the form (2.5) as

n X DtF = − 1[0,Tk](t)∂kgn(T1,...,Tn), t ∈ IR+. (2.6) k=1 We also deﬁne the space of simple processes

( n ) c X c UT := Gihi : G1,...,Gn ∈ ST , h1, . . . , hn ∈ C([0,T ]), n ≥ 1 , i=1 where C([0,T ]) denotes the space of continuous functions on [0,T ], and let the diver- c gence or Skorohod integral operator δ be deﬁned on UT by Z T Z T δ(Gh) := G h(t)(dNt − dt) − h(t)DtGdt, (2.7) 0 0

G ∈ ST , h ∈ C([0,T ]). The duality relation (2.8) below between D and δ relies on standard integration by parts on [0,T ]n and on the expression

∞ Z T Z tk Z t2 −T −T X IE[F ] = e f0 + e ··· fk(t1, . . . , tk)dt1 ··· dtk, k=1 0 0 0

c for F ∈ ST of the form (2.1), cf. Proposition 7.3.3 of [17] and the appendix Section7 for a proof.

Proposition 2.3 The operators D and δ deﬁned by (2.2) and (2.7) satisfy the duality relation Z T c c IE utDtF dt = IE[F δ(u)],F ∈ ST , u ∈ UT . (2.8) 0

5 As a consequence of the Proposition 2.3, the operators D and δ are closable, and (2.8) c c c extends to their closed domains DomT (D) and DomT (δ) deﬁned as the closure of ST c and UT respectively.

Letting (Ft)t∈IR+ denote the ﬁltration generated by (Nt)t∈IR+ , we note that for any 2 Ft-measurable random variable F ∈ L (Ω) we have

DsF = 0, s ∈ [t, ∞), and that δ coincides with the stochastic integral with respect to the compensated Poisson process, i.e. Z T δ(u) = ut(dNt − dt), (2.9) 0 2 for all Ft-adapted square-integrable process u ∈ L (Ω × [0,T ]), as in e.g. Proposi- tion 7.2.9 of [17].

c The condition F ∈ ST in the integration by parts (2.8) of Proposition 2.3 can be R T relaxed to F ∈ ST as in [6] under the additional condition 0 utdt = 0, a.s., cf. also (7.3.6) in [17], on the space of simple processes

( n ) X Z T UT := Gihi : Gi ∈ ST , hi ∈ C([0,T ]), hi(t)dt = 0, 1 ≤ i ≤ n . i=1 0

Proposition 2.4 The operators D and δ deﬁned by (2.2) and (2.7) satisfy the duality relation Z T

IE utDtF dt NT ≥ 1 = IE[F δ(u) NT ≥ 1],F ∈ ST , u ∈ UT . 0 Proof. The same argument as in the proof of Proposition 2.3 in the appendix Section7 shows that Z T Z T

IE 1{NT ≥1} utDtF dt = IE utDtF dt = IE[F δ(u)],F ∈ ST , u ∈ UT , 0 0

R T and we note that δ(u) = 0 on {NT = 0} due to the condition 0 utdt = 0.

6 From Proposition 2.4 the operators D and δ can be extended to larger domains

DomT (D) and DomT (δ), by completion of the spaces ST and UT . In this case,

DomT (D) contains non-smooth functionals such as NT itself and we retain the equality (2.9) between δ and the (non compensated) Poisson stochastic integral over square- integrable adapted processes, however this requires the additional (a.s.) vanishing R T condition 0 utdt = 0.

3 Integration by parts for birth processes

α In this section we consider a pure birth process (Nt )t∈IR+ whose interjump times α (τk )k∈IN are independent exponentially distributed random variables with respective parameters (αk)k∈IN satisfying the bound

0 < αk ≤ C, k ∈ IN, (3.1)

α for some C > 0. In other words, the jump times of (Nt )t∈IR+ can be written as

α α α τ0 τn−1 Tn = τ0 + ··· + τn−1 = + ··· + , n ≥ 1, (3.2) α0 αn−1 with the relation

α α Nt = min n ≥ 0 : t < Tn+1 , t ∈ IR+.

Deﬁning Z t α λ(t) := αNs ds (3.3) 0 α Nt α X α α = αN α (t − T α ) + αk−1(T − T ) t Nt k k−1 k=1 α Nt α X = αN α (t − T α ) + τk−1 t Nt k=1 α = αN α (t − T α ) + TN α , t ∈ IR+, t Nt t

α it follows from (3.2)-(3.3) that (Nt )t∈IR+ can be written as the time-changed Poisson process α Nt = Nλ(t), t ∈ IR+,

7 with n α X α α λ(Tn ) = αk−1(Tk − Tk−1) = Tn, n ≥ 1. k=1 1 k k For all k ≥ 1 we denote by Cb (ST ) the space of functions on IR which are continuously k diﬀerentiable (with bounded derivatives) on the simplex ST .

α Deﬁnition 3.1 Let ST denote the class of functionals of the form ∞ α X α α α F = f 1 α + 1 α f (T ,...,T ), (3.4) 0 {NT =0} {NT =k} k 1 k k=1

α α 1 k α k α k where f0 ∈ IR and fk ∈ Cb (ST ), are such that kfk k∞ ≤ A and k∂lfk k∞ ≤ A , 1 ≤ l ≤ k, k ≥ 1, for some constant A > 0.

c,α α We also let ST denote the subspace of ST made of functionals of the form (3.4) that satisfy the continuity condition

α α fk (t1, . . . , tk) = fk+1(t1, . . . , tk, tk+1), 0 ≤ t1 ≤ · · · ≤ tk ≤ T < tk+1, k ∈ IN. (3.5)

The next lemma will be used for Deﬁnition 3.3 and in the proof of Proposition 3.4 below.

α c,α c Lemma 3.2 We have the inclusions ST ⊂ SCT and ST ⊂ SCT . Proof. We proceed in three steps. α (i) Given F ∈ ST of the form (3.4), we let i X 1 tα := (t − t ), i ≥ 1, (3.6) i α m m−1 m=1 m−1 and k X α α α f (t , . . . , t ) := 1 α α α f (t , . . . , t ), k ≥ 1, (3.7) k 1 k {0≤t1 ≤···≤ti ≤T

0 ≤ t1 ≤ · · · ≤ tk ≤ CT < tk+1

α implies tk+1 > T , hence we have

k−1 α α α X α α α f (t , . . . , t ) = 1 α α α f (t , . . . , t ) + 1 α α α f (t , . . . , t ) k 1 k {0≤t1 ≤···≤tk ≤T

8 α α α α α α α = 1 α α f (t , . . . , t ) + 1 α α α f (t , . . . , t ) + ··· + 1 α f , {0≤t1 ≤···≤tk ≤T } k 1 k {0≤t1 ≤···≤tk−1≤T

(3.5), we find that fk is weakly differentiable on {0 ≤ t1 ≤ · · · ≤ tk ≤ CT < tk+1} for every k ≥ 1. α (ii) Next, by (3.4) and the relation NT ≤ NCT we find ∞ X f01{NCT =0} + 1{NCT =k}fk(T1,...,Tk) k=1 ∞ k X X α α α = f 1 + 1 T α f (T ,...,T ) 0 {NCT =0} {NCT =k} {NT =i} i 1 i k=1 i=0 ∞ k X X α α α = f 1 + f 1 T α + 1 T α f (T ,...,T ) 0 {NCT =0} 0 {NCT >0} {NT =0} {NCT =k} {NT =i} i 1 i k=1 i=1 ∞ k X X α α α = f 1 α + 1 T α f (T ,...,T ) 0 {NT =0} {NCT =k} {NT =i} i 1 i k=1 i=1 ∞ ∞ X X α α α = f 1 α + 1 T α f (T ,...,T ) 0 {NT =0} {NCT =k} {NT =i} i 1 i i=1 k=i ∞ α X α α α = f 1 α + 1 α f (T ,...,T ) 0 {NT =0} {NT =i} i 1 i i=1 = F, hence (2.1) is satisfied and we conclude that F ∈ SCT . c,α (iii) Finally, taking F ∈ ST , it remains to check that (fk)k∈IN satisfies the continuity condition (2.3) on the domain [0,CT ]. For any k ∈ IN and 0 ≤ t1 ≤ · · · ≤ tk ≤ CT < α tk+1 we have tk+1 > T , therefore, k+1 X α α α f (t , . . . , t ) = 1 α α α f (t , . . . , t ) k+1 1 k+1 {0≤t1 ≤···≤ti ≤T

9 α Deﬁnition 3.3 For F in the space ST of functionals of the form (3.4) we deﬁne the α time-changed gradient Dt as

α D F := α α D F, t ∈ IR , (3.8) t Nt λ(t) + where D is applied to F ∈ SCT on the time interval [0,CT ].

α α α Denoting by (Ft )t∈IR+ the ﬁltration generated by (Nt )t∈IR+ , the process (Nt −λ(t))t∈IR+ α is an Ft -martingale, and the stochastic integral Z T α u (dN − α α dt) t t Nt 0

α is deﬁned for square-integrable (Ft )t∈[0,T ]-adapted processes (ut)t∈[0,T ] via the isometry formula "Z T 2# Z T α 2 IE u (dN − α α dt) = IE |u | α α dt . t t Nt t Nt 0 0 Next, we show that under Deﬁnition 3.3, Dα is dual to the stochastic integral with α R t respect to N − α α ds , as a consequence of e.g. Proposition 6 of [16] or t 0 Ns t∈[0,T ] Proposition 1 of [15].

c α α Proposition 3.4 Given F ∈ DomT (D ) and (ut)t∈[0,T ] a square-integrable (Ft )t∈[0,T ]- adapted process, we have the integration by parts formula

Z T Z T α α IE D F u dt = IE F u (dN − α α dt) . t t t t Nt 0 0 α Proof. Let u be an (Ft )t∈[0,T ]-adapted simple process in

( n ) c,α X α UT := Fiui : u1, . . . , un ∈ C([0,T ]),F1,...,Fn ∈ ST , n ≥ 1 . i=1

10 c By Lemma 3.2 we have u ∈ UCT , hence we can apply Proposition 2.3 on [0,CT ] to c F ∈ SCT since λ(T ) ≤ CT . Hence by (3.3), (3.8), (3.1), we have Z T Z ∞ α −1 IE D F utdt = IE 1[0,T ](λ )DtF u −1 dt t t λt 0 0 Z CT = IE 1[0,λ(T )](t)DtF u −1 dt λt 0 Z CT = IE F 1[0,λ(T )](t)u −1 (dNt − dt) λt 0 Z ∞ −1 = IE F 1[0,T ](λ )u −1 (dNt − dt) t λt 0 Z T = IE F u (dN − α α dt) . t λ(t) Nt 0

c,α α We conclude by the denseness of UT in the space of square-integrable (Ft )t∈[0,T ]- adapted process.

c α α c,α c We denote by DomT (D ) the domain of D obtained by completion of ST in DomCT (D). α Deﬁnition 3.3 can be restated on ST as in the next proposition whose proof is given in the appendix Section7.

α Proposition 3.5 For any F ∈ ST of the form (3.4) we have

∞ i α X X α α α D F = − 1 α 1 α (t)∂ f (T ,...,T ), t ∈ IR . t {NT =i} [0,Tj ] j i 1 i + i=1 j=1

c,α Given n ≥ 1 and F ∈ ST a functional of the form (3.4) such that

α α fn (t1, . . . , tn) = fk (t1, . . . , tk), 0 ≤ t1 ≤ · · · ≤ tk, k ≥ n,

α we note that similarly to (2.4), the function gn deﬁned by

α gn (t1, . . . , tn) (3.10) n−1 α X α α := f0 1{T

n is weakly diﬀerentiable on ST = {0 ≤ t1 ≤ · · · ≤ tn ≤ T }, and satisﬁes

α α α F = gn (T1 ,...,Tn ). (3.11)

11 Consequently, similarly to (2.6), Proposition 3.5 admits the following corollary which shows that the gradient operator Dα coincides with that of [6], cf. Proposition 3.1 therein, under certain conditions.

c,α Corollary 3.6 For F ∈ ST of the form (3.11) we have

n α X α α α D F = − 1 α (t)∂ g (T ,...,T ), t ∈ IR . (3.12) t [0,Tk ] k n 1 n + k=1 4 Integration by parts for Markov chains

Consider a (right-continuous) Markov chain (βt)t∈IR+ with state space M := {1, . . . , m} β and transition rate matrix Q = (qi,j)1≤i,j≤m. Let Tn denote the n-th transition time β of (βt)t∈IR+ with T0 := 0, and let

β β Nt := min n ∈ IN : t < Tn+1

denote the number of transitions of (βt)t∈IR+ up to time t ∈ IR+. The Markov chain

(βt)t∈IR+ can be represented as

βt = Z β , t ∈ IR+, Nt where (Zn)n∈IN deﬁned by

Zn := β β , n ∈ IN, Tn

denotes the embedded chain of (βt)t∈IR+ , with Z0 := β0. It is known in addition β β that, given (Zn)n∈IN, the interjump times (Tn+1 − Tn )n∈IN form a sequence of independent exponentially distributed random variables with respective parameters (αZn )n≥1, where r X αk := qk,l = −qk,k, k = 1, . . . , m, (4.1) l=1 l6=k cf. e.g. § 4.2 and § 6.4 of [9] for related representations of (βt)t∈IR+ . Continuous-time Markov chains can also be represented using stochastic integrals with respect to a Poisson random measure, cf. Chapter II of [20] or Section 3 of [11].

12 In the sequel we also assume that the sequence (αk)k≥0 satisfies the condition (3.1) for some C > 0. We will also make use of the following Definition 4.1 in order to extend Definition 3.3 to the setting of Markov processes.

β Deﬁnition 4.1 Let ST denote the space of functionals of the form

∞ β X β β β F = f0 1 β + 1 β fk (T1 ,...,Tk ,Z0,...,Zk), (4.2) {NT =0} {NT =k} k=1

β β 1 k β k where f0 ∈ IR and fk (·, z0, z1, . . . , zk) ∈ Cb (ST ) are such that kfk k∞ ≤ A and β k k∂lfk k∞ ≤ A for all z0, z1, . . . , zk ∈ M, 1 ≤ l ≤ k ≤ n, for some A > 0.

We denote by β α Dt F := Dt F, t ∈ IR+,

α α β the partial gradient Dt applied to F ∈ DomT (D ). In other words, for F ∈ ST of the form (4.2), as in Proposition 3.5 we have

∞ k β X X β β β Dt F = − 1 β 1 β (t)∂lfk (T1 ,...,Tk ,Z0,...,Zk), t ∈ IR+. (4.3) {NT =k} [0,Tl ] k=1 l=1 In addition, as in (3.9) we have

Z T Z λ(T ) Z CT β 2 2 2 2 2 2 kD F kL ([0,T ]) = |Dλ(t)F | αβt dt = |DtF | αβt dt ≤ C |DtF | dt, 0 0 0

β c hence the deﬁnition of D extends to all F ∈ DomCT (D), with the bound Z CT β 2 2 kD F kL2(Ω×[0,T ]) ≤ C IE |DtF | dt . 0

c,β β Given n ≥ 1, let ST denote the subspace of ST made of functionals of the form (4.2) that satisfy the continuity condition

β β fk (t1, . . . , tk, z0, z1, . . . , zk) = fk+1(t1, . . . , tk+1, z0, z1, . . . , zk+1), (4.4)

0 ≤ t1 ≤ · · · ≤ tk ≤ T < tk+1, z0, z1, . . . , zk+1 ∈ M, k ∈ IN.

c β β c,β We will denote by DomT (D ) the domain of D obtained by completion of ST in c DomCT (D). From Proposition 3.4 we obtain the integration by parts formula of the

13 β β next Proposition 4.2, in which DomT (D ) and the ﬁltration (Ft )t∈IR+ are deﬁned α α analogously to DomT (D ) and (Ft )t∈IR+ . As in (2.9) above, the stochastic integral

Z T β ut(dNt − αβt dt) 0

β is deﬁned for square-integrable (Ft )t∈IR+ -adapted processes (ut)t∈IR+ via the isometry

"Z T 2# Z T β 2 IE ut(dNt − αβt dt) = IE |ut| αβt dt . 0 0

c β β Proposition 4.2 Given F ∈ DomT (D ) and (ut)t∈[0,T ] a square-integrable (Ft )t∈IR+ - adapted process we have the integration by parts formula

Z T Z T β β IE Dt F utdt = IE F ut(dNt − αβt dt) . (4.5) 0 0

β As in (3.10), the (random) function gn deﬁned by

β gn(t1, . . . , tn,Z0,...,Zn) n−1 β X β := f0 1{T

n is a.s. weakly diﬀerentiable on ST = {0 ≤ t1 ≤ · · · ≤ tn ≤ T } and satisﬁes

β β β F = gn(T1 ,...,Tn ,Z0,...,Zn). (4.6)

Hence Corollary 3.6 yields the following result, see also Proposition 3.1 of [6].

c,β Proposition 4.3 For F ∈ ST of the form (4.6) for which there exists n ≥ 1 such that β β fn (t1, . . . , tn, z0, z1, . . . , zn) = fk (t1, . . . , tk, z0, z1, . . . , zk),

0 ≤ t1 ≤ · · · ≤ tk, z0, z1, . . . , zk ∈ M, k ≥ n, we have

n β X β β β Dt F = − 1 β (t)∂kgn(T1 ,...,Tn ,Z0,...,Zn), t ∈ IR+. [0,Tk ] k=1

14 5 Sensitivity analysis in a regime switching model

In this section we apply the integration by parts formula (4.5) to the computation of Greeks. Consider a geometric Brownian motion (Yt)t∈[0,T ] with regime switching, given by Z t Z t Yt = Y0 exp µ β (s, βs)ds + σ(βs)dBs , t ∈ [0,T ], (5.1) Ns 0 0 where (Bt)t∈[0,T ] is a standard Brownian motion independent of the ﬁnite-state continuous- time Markov chain (βt)t∈[0,T ], and µ, σ are deterministic functions on the state space

M := {1, 2, . . . , m} of (βt)t∈[0,T ].

Consider the expected value function

V (y, z) := IE[φ(YT ) | Y0 = y, β0 = z], y > 0, z ∈ M, where r > 0 denotes the risk-free rate and φ is an integrable payoff function. In Proposition 5.1 we compute the sensitivity ∂V ∆(y, z) := (y, z), y > 0, z ∈ M, ∂y 1 with respect to the initial price y when the payoff function φ ∈ Cb (IR) has bounded derivative (an extension to non-differentiable payoff functions is given in Section6).

XT +WT In the sequel we will rewrite (5.1) as YT = Y0e , where Z T WT := σ(βs)dBs, 0 and Z T Z T 2 2 XT := µ β (s, βs)ds, Σ := σ (βs)ds (5.2) Ns T 0 0 c,β belong to ST . We also let Z T β β β 2 Du F := utDs F ds, F ∈ DomT (D ), u ∈ L (Ω × [0,T ]). 0 R T In the next proposition, the absence of hypothesis on the vanishing of 0 utdt allows us to keep DuXT strictly positive, therefore ensuring integrability of the weight ΓT together with a better stability of the associated numerical implementation.

15 1 Proposition 5.1 Assume that µk(·, l) ∈ C ([0,T ]) satisﬁes

µk(T, l) = 0, k ≥ 0, l = 1, . . . m,

β β β β and consider u ∈ C([0,T ]) such that 1 β /Du XT and 1 β Du Du XT /Du XT {NT ≥1} {NT ≥1} belong to L2(Ω). Then we have 1 W T ∆(y, z) = IE φ(YT ) 1{N β ≥1}ΓT + 1{N β =0} Y0 = y, β0 = z , y > 0, y T T T σ2(z) (5.3) 1 for all φ ∈ Cb (IR), where the weight ΓT is given by Z T β β 2 2 1 β Du Du XT ΣT − (WT ) β 2 ΓT := β ut(dNt − αβt dt) + β + 4 Du ΣT , (5.4) Du XT 0 Du XT 2ΣT and (αl)l=1,...,m is given in (4.1).

Proof. We write ∆(y, z) as ∆(y, z) = ∆1(y, z) + ∆2(y, z), where ∂ h i ∆1(y, z) := IE φ(YT )1{N β ≥1} Y0 = y, β0 = z (5.5) ∂y T and ∂ h i ∆2(y, z) := IE φ(YT )1{N β =0} Y0 = y, β0 = z . (5.6) ∂y T

Taking Y0 = y and β0 = z, we have ∂ h i ∆1(y, z) = IE 1{N β ≥1}φ(YT ) Y0 = y, β0 = z ∂y T ∂ XT +WT = IE 1{N β ≥1} φ(ye ) (5.7) T ∂y Z ∞ ∂ x+XT = IE 1{N β ≥1} φ(ye )ϕ(x, ΣT )dx , (5.8) T −∞ ∂y where XT is deﬁned in (5.2) and 1 ϕ(x, Σ ) := exp −x2/(2Σ2 ) , x ∈ IR, T p 2 T 2πΣT

2 is the probability density function of WT given ΣT . We note that XT can be written as

β β NT T ∧T X Z i+1 XT = µi(s, Zi)ds, β i=0 Ti ∧T

16 where (Zn)n∈IN is the discrete-time embedded chain of (βt)t∈IR+ . Consequently, XT is c,β an element of ST which is expressed in the form of (4.2) as ∞ β X β β β XT = f0 1 β + 1 β fk (T1 ,...,Tk ,Z0,...,Zk), {NT =0} {NT =k} k=1

β R T where f0 = 0 µ0(s, Z0)ds and

k Z ti+1∧T β X fk (t1, . . . , tk,Z0,...,Zk) = µi(s, Zi)ds, i=0 ti∧T k ≥ 1, satisfy the continuity condition (4.4), and by (4.3) we have

β β NT −1 NT β X β X β Dt XT = − 1 β (t)µi(Ti+1,Zi) + 1 β (t)µi(Ti ,Zi), t ∈ IR+, [0,Ti+1] [0,Ti ] i=0 i=1

β c,β XT c,β and Dt XT ∈ ST . Similarly, ﬁnd that the functional e belongs to ST , with

β XT XT β Du e = e Du XT . (5.9)

Hence by (5.9), taking Y0 = y and β0 = z we have ∂ h i XT +x XT +x 0 XT +x IE 1{N β ≥1}ϕ(x, ΣT ) φ(ye ) = IE 1{N β ≥1}ϕ(x, ΣT )e φ (ye ) T ∂y T 1 Dβφ(yeXT +x) XT +x u = IE 1 β ϕ(x, ΣT )e {NT ≥1} β X +x y Du e T 1 ϕ(x, Σ ) T β XT +x = IE 1 β Du φ(ye ) . (5.10) {NT ≥1} β y Du XT Now, for all ε > 0 we have DβX u T β XT +x IE 1 β ϕ(x, ΣT )Du φ(ye ) {NT ≥1} β 2 (Du XT ) + ε DβX u T β XT +x = IE ϕ(x, ΣT )D φ(ye ) β 2 u (Du XT ) + ε DβX DβX β u T XT +x XT +x β u T = IE D φ(ye )ϕ(x, ΣT ) − IE φ(ye )D ϕ(x, ΣT ) , u β 2 u β 2 (Du XT ) + ε (Du XT ) + ε DβX u T XT +x c,β where the functional φ(ye )ϕ(x, ΣT ) belongs to S for all x ∈ IR. β 2 T (Du XT ) + ε B Hence, denoting by (Ft )t∈[0,T ] the ﬁltration generated by (Bt)t∈[0,T ], by Proposition 4.2

17 we obtain DβX β u T XT +x IE D φ(ye )ϕ(x, ΣT ) u β 2 (Du XT ) + ε DβX β u T XT +x B = IE IE D φ(ye )ϕ(x, ΣT ) F u β 2 T (Du XT ) + ε DβX Z T u T XT +x β B = IE IE φ(ye )ϕ(x, ΣT ) ut(dN − αβ dt) F β 2 t t T (Du XT ) + ε 0 DβX Z T u T XT +x β = IE φ(ye )ϕ(x, ΣT ) ut(dN − αβ dt) β 2 t t (Du XT ) + ε 0 DβX Z T u T XT +x β = IE 1 β φ(ye )ϕ(x, ΣT ) ut(dNt − αβt dt) , {NT ≥1} β 2 (Du XT ) + ε 0

β β since Du XT = 0 on {NT = 0}. Next, we have

β β β β NT −1 NT Z Ti+1 Z Ti β X β X β Du XT = − utdtµi(Ti+1,Zi) + utdtµi(Ti ,Zi) i=0 0 i=1 0 β β NT Z Ti X β β = utdt µi(Ti ,Zi) − µi−1(Ti ,Zi−1) (5.11) i=1 0 ∞ β X β β β = g0 1 β + 1 β gk (T1 ,...,Tk ,Z0,...,Zk), {NT =0} {NT =k} k=1

β where g0 = 0 and

k Z ti∧T β X gk (t1, . . . , tk,Z0,...,Zk) = utdt (µi(ti,Zi) − µi−1(ti,Zi−1)) , i=1 0

β c,β k ≥ 1, satisfy the continuity condition (4.4), hence Du XT ∈ ST and by (4.3) we have

β β β NT −1 NT −1 Z Ti+1 β β X β X 0 β Dt Du XT = 1 β (t)u β µi(Ti+1,Zi) + utdt1 β (t)µi(Ti+1,Zi) [0,Ti+1] Ti+1 [0,Ti+1] i=0 i=0 0 β β β NT NT Z Ti X β X 0 β − 1 β (t)u β u β µi(Ti ,Zi) − 1 β (t) utdtµi(Ti ,Zi), [0,Ti ] Ti Ti [0,Ti ] i=1 i=1 0 which yields

β β β β β NT −1 NT −1 Z Ti+1 Z Ti+1 Z Ti+1 β β X β X 0 β Du Du XT = usdsu β µi(Ti+1,Zi) + utdt usdsµi(Ti+1,Zi) Ti+1 i=0 0 i=0 0 0

18 β β β β β NT NT Z Ti Z Ti Z Ti X β X 0 β − u β usdsu β µi(Ti ,Zi) − usds utdtµi(Ti ,Zi). Ti Ti i=1 0 i=1 0 0 Hence we see that DβX XT +x β u T IE φ(ye )D ϕ(x, ΣT ) u β 2 (Du XT ) + ε XT +x β 2 φ(ye ) β β (Du XT ) − ε β β = IE (D XT )D ϕ(x, ΣT ) − ϕ(x, ΣT )D D XT β 2 u u β 2 u u (Du XT ) + ε (Du XT ) + ε XT +x 2 2 β 2 φ(ye ) x − ΣT β 2 β (Du XT ) − ε β β = IE ϕ(x, ΣT )D Σ D XT − ϕ(x, ΣT )D D XT . β 2 4 u T u β 2 u u (Du XT ) + ε 2ΣT (Du XT ) + ε (5.12)

Combining (5.5), (5.8) and (5.10)-(5.12) and letting ε tend to zero we ﬁnd, by dominated convergence,

∆1(y, z) (5.13) Z ∞ h = IE 1{N β ≥1} −∞ T XT +x Z T β β 2 2 φ(ye ) β Du Du XT ΣT − x β 2 dx × β ϕ(x, ΣT ) ut(dNt − αβt dt) + β + 4 Du ΣT Du XT 0 Du XT 2ΣT y Z T β β 2 2 1 φ(YT ) β Du Du XT ΣT − (WT ) β 2 = IE 1 β ut(dNt − αβt dt) + + Du ΣT . {NT ≥1} β β 4 y Du XT 0 Du XT 2ΣT

Regarding the computation of ∆2(y, z), we note that by the classical Malliavin calculus for Brownian motion, or by a standard integration by parts with respect to the Gaussian density, we have ∂ h i 1 WT ∆2(y, z) = IE 1{N β =0}φ(YT ) = IE 1{N β =0}φ(YT ) , (5.14) ∂y T y T T σ2(z) as in e.g. [10]. The proof is completed by (5.13)-(5.14).

The integrability assumptions in Proposition 5.1 can be satisﬁed by choosing µk(t, l) β β so that Du XT in (5.11) remains strictly positive. However, positivity of Du XT is not necessary, as shown in the following numerical illustration in which we consider a two-state Markov chain (βt)t∈IR+ with values in {1, 2}, with

γ µk(t, l)L = (k + 1)(T − t) ηl, k ≥ 0, l = 1, 2,

19 and the parameters

γ = 1,T = 1,Y0 = 1, α1 = 40, α2 = 20, η1 = 1.5, η2 = 1, σ1 = 0.2, σ2 = 0.5.

Figure1 shows the faster convergence of (5.3) for a digital option with payoﬀ φ(x) =

1[K,∞)(x), strike K = 150, and ut := 1, t ∈ [0,T ], compared to a standard ﬁnite diﬀerence scheme and to an application of the partial Wiener-Malliavin calculus as ∂ h i 1 W ∆(y, z) = IE φ(Y ) N β ≥ 1,Y = y, β = z = IE φ(Y ) T Y = y, β = z , T T 0 0 T 2 0 0 ∂y y ΣT cf. e.g. [5].

0.31 Markov-Malliavin 0.3 Finite diﬀerences Wiener-Malliavin 0.29

0.28

0.27

0.26 Delta 0.25

0.24

0.23

0.22

0.21 0 5 10 15 20 25 30 35 40 45 50 samples x 106 Figure 1: Monte Carlo convergence graph. In case the diﬀusion term σ vanishes, Proposition 6.1 still allows us to estimate the β conditional sensitivity given that NT ≥ 1, as

∂ h β i IE φ(YT ) N ≥ 1,Y0 = y, β0 = z ∂y T " T T β β # 1 1{N β ≥1} Z Z D D X = IE φ(Y ) T u dN β − u α dt + u u T N β ≥ 1,Y = y, β = z . T β t t t βt β T 0 0 y Du XT 0 0 Du XT

6 Extension to non-diﬀerentiable payoﬀ functions

In this section, following the approach of in [12] we show that Proposition 5.1 can be extended to non-diﬀerentiable payoﬀ functions in the class ( n ) X Λ(IR):= f : IR → IR : f = fi1Ai , fi ∈ CL(IR),Ai intervals of IR, n ≥ 1 , i=1 (6.1)

20 where

CL(IR):= f ∈ C(IR) : |f(x) − f(y)| ≤ k|x − y| for some k ≥ 0 . (6.2)

Proposition 6.1 Under the hypotheses of Proposition 5.1 we have 1 WT ∆(y, z) = IE φ(YT ) 1 β ΓT + 1 β Y0 = y, β0 = z , (6.3) {NT ≥1} {NT =0} 2 y ΣT for all φ ∈ Λ(IR), where ΓT is given in (5.4).

N Proof. Since φ ∈ Λ(IR), there exists N ≥ 1 and a sequence (k1, . . . , kN ) ⊂ IR+ and a family (A1,...,AN ) of disjoint intervals such that

N X φ(x) = fi(x)1Ai (x), x ∈ IR, i=1 where fi(x) ∈ CL(IR) with

|f(x) − f(y)| ≤ ki|x − y|, x, y ∈ Ai, i = 1,...,N.

We denote Ai = (ai−1, ai], i = 1,...,N with a0 = −∞ and aN = ∞. Let y + ε Y ε := Y t ∈ [0,T ], ε ∈ IR, t y t where (Yt)t∈[0,T ] is deﬁned in (5.1) with Y0 = y > 0. 1 (i) Assuming that φ ∈ Λ(IR) ∩ C (IR) we show that φ(YT ) is integrable, with

ε ε φ(YT ) − φ(YT ) φ(YT ) − φ(YT ) lim IE Y0 = y, β0 = z = IE lim Y0 = y, β0 = z . ε→0 ε ε→0 ε (6.4)

Since φ is continuous, we see that

21 which is integrable, hence the integrability of φ(YT ) is proved. On the other hand, ε ε φ(YT ) − φ(YT ) YT − YT 1 lim ≤ max ki lim = max ki, ε→0 ε 1≤i≤N ε→0 ε y 1≤i≤N which is uniformly bounded. Therefore, (6.4) follows by dominated convergence.

(ii) Next, we note that (6.3) holds for φ ∈ Λ(IR) ∩ C1(IR), as by (6.4) we have ε ∂V φ(YT ) − φ(YT ) (y, z) = lim IE Y0 = y, β0 = z ∂y ε→0 ε ε φ(YT ) − φ(YT ) = IE lim Y0 = y, β0 = z ε→0 ε ∂ ∂

= IE 1{N β ≥1} φ(YT ) Y0 = y, β0 = z + IE 1{N β =0} φ(YT ) Y0 = y, β0 = z , T ∂y T ∂y (6.5) which shows that (6.3) holds for φ ∈ Λ(IR) ∩ C1(IR)by repeating the arguments from (5.7) to the end of proof of Proposition 5.1.

1 (iii) Finally, we extend the result from an increasing sequence (φn)n∈IN ⊂ Λ(IR)∩C (IR) to its pointwise limit φ ∈ Λ(IR). By e.g. (3.6)-(3.7) in [12], it suﬃces to show that for all compact subsets K ⊂ (0, ∞) we have

lim sup IE[φn(YT ) | Y0 = y, β0 = z] − IE[φ(YT ) | Y0 = y, β0 = z] = 0, (6.6) n→∞ y∈K and ∂ lim sup IE[φn(YT ) | Y0 = y, β0 = z] − IE[φ(YT )J | Y0 = y, β0 = z] = 0. (6.7) n→∞ y∈K ∂y where the weight J given by Z T Z T 1 β BT J := 1 β utdNt − vtdt + 1 β {NT ≥1} β {NT =0} Du XT 0 0 σ(z)

h β −2i is square-integrable under the condition IE 1 β |Du XT | < ∞. Since φ ∈ Λ(IR) {NT ≥1} is continuous on every interval (ai−1, ai), i = 1,...,N, there exists a pointwise in- 1 creasing sequence (φn)n∈N ∈ Λ(IR) ∩ C (IR) such that

lim φn(x) = φ(x), x ∈ IR \{a1, . . . , aN−1}. n→∞

22 The increasing sequence (fn)n∈N of continuous functions deﬁned by fn(y) := IE φn(YT ) Y0 = y, β0 = z , n ∈ N, y ∈ K, satisﬁes lim fn(y) = IE φ(YT ) Y0 = y, β0 = z n→∞ uniformly in y ∈ K by dominated convergence and Dini’s theorem, which proves (6.6). 1 Regarding (6.7), since (φn)n∈N ∈ Λ(IR) ∩ C (IR), by point (ii) above we have ∂ IE[φ (Y ) | Y = y, β = z] − IE[φ(Y )J | Y = y, β = z] ∂y n T 0 0 T 0 0 = | IE[φn(YT )J | Y0 = y, β0 = z] − IE[φ(YT )J | Y0 = y, β0 = z] |

≤ IE[(φ(YT ) − φn(YT ))|J| | Y0 = y, β0 = z]

p 2 p 2 ≤ IE[(φ(YT ) − φn(YT )) | Y0 = y, β0 = z] IE[|J| | Y0 = y, β0 = z].

Similarly to the above we conclude by noting that the sequence (gn)n∈N of continuous functions deﬁned by

2 gn(y) := IE (φ(YT ) − φn(YT )) Y0 = y, β0 = z , n ∈ N, y ∈ K, is decreasing to 0 uniformly on K.

7 Appendix

In this appendix we provide a proof of Proposition 2.3 that does not rely on the symmetry condition on the functions fk(t1, . . . , tk) assumed in Proposition 7.3.3 of [17].

Proof of Proposition 2.3. First, taking h ∈ C([0,T ]) we have

Z T Z T β β IE h(t)Dt F dt = IE 1{NT ≥1} h(t)Dt F dt (7.1) 0 0 " ∞ k # X X Z Tl = − IE 1{NT =k} ∂lfk(T1,...,Tk) h(t)dt k=1 l=1 0 ∞ " k # X T k X Z Tl = −e−T IE ∂ f (T ,...,T ) h(t)dt N = k k! l k 1 k T k=1 l=1 0

23 ∞ k Z T Z tk Z t2 Z tl −T X X = −e ··· ∂lfk(t1, . . . , tk) h(t)dtdt1 ··· dtk. (7.2) k=1 l=1 0 0 0 0 We start with the ﬁrst term when l = 1 and apply the chain rule of derivation in the following integration by parts on [0, t2]:

Z T Z tk Z t2 Z t1 ··· ∂1fk(t1, . . . , tk) h(t)dtdt1 ··· dtk 0 0 0 0 Z T Z tk Z t2 ∂ Z t1 = ··· h(t)dtfk(t1, . . . , tk) dt1 ··· dtk 0 0 0 ∂t1 0 Z T Z tk Z t2 − ··· h(t1)fk(t1, . . . , tk)dt1 ··· dtk 0 0 0 Z T Z tk Z t3 Z t2 = ··· h(t)dtfk(t2, t2, t3, . . . , tk)dt2 ··· dtk 0 0 0 0 Z T Z tk Z t2 − ··· h(t1)fk(t1, . . . , tk)dt1 ··· dtk. (7.3) 0 0 0

Similarly we have, by integration by parts on [tl−1, t2+1], l ∈ {2, . . . , k − 1},

Z T Z tk Z t2 Z tl ··· ∂lfk(t1, . . . , tk) h(t)dtdt1 ··· dtk 0 0 0 0 ˆ Z T Z tk Z tl Z t2 Z tl+1 = ··· ··· h(t)dtfk(t1, . . . , tl−1, tl+1, tl+1, . . . , tk)dt1 ··· dtˆl ··· dtk 0 0 0 0 0 Z T Z tk Z t2 − ··· h(tl)fk(t1, . . . , tk)dt1 ··· dtk 0 0 0 ˆ Z T Z tk Z tl Z t2 Z tl − ··· ··· h(t)dtfk(t1, . . . , tl, tl, tl+1, . . . , tk)dt1 ··· dtˆl−1 ··· dtk, 0 0 0 0 0

ˆ R tl R tl ˆ where 0 denotes the absence of 0 , dtl denotes the absence of dtl. Finally, by integration by parts on [tk,T ] we ﬁnd

Z T Z tk Z t2 Z tk ··· ∂kfk(t1, . . . , tk) h(t)dtdt1 ··· dtk (7.4) 0 0 0 0 Z T Z tk−1 Z t2 Z T = ··· fk(t1, . . . , tk−1,T )dt1 ··· dtk−1 h(t)dt 0 0 0 0 Z T Z tk Z t2 − ··· h(tk)fk(t1, . . . , tk)dt1 ··· dtk 0 0 0 Z T Z tk−1 Z t2 Z tk − ··· fk(t1, . . . , tk−2, tk, tk)dt1 ··· dtk−2dtk h(t)dt. 0 0 0 0

24 Plugging (7.3)-(7.4) into (7.1) yields

Z T β IE h(t)Dt F dt 0 ∞ k Z T Z tk Z t2 −T X X = e ··· h(tl)fl(t1, . . . , tk)dt1 ··· dtk k=1 0 0 0 l=1 Z T Z tk−1 Z t2 Z T − ··· fk(t1, . . . , tk−1,T )dt1 ··· dtk−1 h(t)dt 0 0 0 0 ∞ Z T Z T X = − IE 1{NT =k−1}fk−1(T1,...,Tk−1) h(t)dt + IE F h(t)dNt k=1 0 0 Z T = IE F h(t)(dNt − dt) , 0

c where we applied the continuity condition (2.3). Next, if u = Gh ∈ UT , by (2.7) we have Z T Z T β β IE 1{NT ≥1} utDt F dt = IE utDt F dt 0 0 Z T β = IE G h(t)Dt F dt 0 Z T Z T β β = IE h(t)Dt (FG)dt − F h(t)Dt Gdt 0 0 Z T Z T β = IE FG h(t)(dNt − dt) − F h(t)Dt Gdt 0 0 = IE[F δ(u)].

α Proof of Proposition 3.5. By Deﬁnition 2.2, Lemma 3.2 and the relations NT ≤ NCT and k X α α α f (t , . . . , t ) = 1 α α α f (t , . . . , t ), k ∈ IN, k 1 k {0≤t1 ≤···≤ti ≤T

α β D F = α α D F t Nt λ(t) ∞ k X X = −α α 1 1 (λ(t))∂ f (T ,...,T ) Nt {NCT =k} [0,Tl] l k 1 k k=1 l=1

25 ∞ k k X X X ∂ α α α = −α α 1 1 α (t) 1 α f (T ,...,T ) Nt {NCT =k} [0,Tl ] {NT =i} i 1 i ∂tl k=1 l=1 i=0 ∞ ∞ i X X X ∂ α α α = −α α 1 1 α (t)1 α f (T ,...,T ) Nt {NCT =k} [0,Tl ] {NT =i} i 1 i ∂tl i=1 k=i l=1 ∞ i X X ∂ α α α = −α α 1 1 α (t)1 α f (T ,...,T ) Nt {NCT ≥i} [0,Tl ] {NT =i} i 1 i ∂tl i=1 l=1 ∞ i X X ∂ α α α = −α α 1 α 1 α (t) f (T ,...,T ) Nt {NT =i} [0,Tl ] i 1 i ∂tl i=1 l=1 ∞ i X X = −α α 1 α 1 α (t) Nt {NT =i} [0,Tl ] i=1 l=1 i ! 1 α α α 1 1 X α α α × ∂lfi (T1 ,...,Ti ) + − ∂jfi (T1 ,...,Ti ) αl−1 αl−1 αl j=l+1 ∞ i X X 1 α α α = −α α 1 α 1 α (t) ∂ f (T ,...,T ) Nt {NT =i} [0,Tl ] l i 1 i αl−1 i=1 l=1 ∞ i i X X 1 1 X α α α −α α 1 α 1 α (t) − ∂ f (T ,...,T ) Nt {NT =i} [0,Tl ] j i 1 i αl−1 αl i=1 l=1 j=l+1 ∞ i X X 1 α α α = −α α 1 α 1 α (t) ∂ f (T ,...,T ) Nt {NT =i} [0,Tl ] l i 1 i αl−1 i=1 l=1 ∞ i j−1 X X α α α X 1 1 −α α 1 α ∂ f (T ,...,T ) 1 α (t) − Nt {NT =i} j i 1 i [0,Tl ] αl−1 αl i=1 j=2 l=1 ∞ i X X 1 α α α = −αN α 1{N α=i} 1[0,T α](t) ∂jf (T ,...,T ) t T j α i 1 i i=1 j=1 j−1 ∞ i j−1 X X α α α X 1 −α α 1 α ∂ f (T ,...,T ) 1 α (t) Nt {NT =i} j i 1 i [0,Tl ] αl−1 i=1 j=2 l=1 ∞ i j X X α α α X 1 +α α 1 α ∂ f (T ,...,T ) 1 α (t) Nt {NT =i} j i 1 i [0,Tl−1] αl−1 i=1 j=2 l=2 ∞ X 1 α α α = −αN α 1{N α=i}1[0,T α](t) ∂1f (T ,...,T ) t T 1 α i 1 i i=1 0

26 ∞ i j X X α α α X 1 −α α 1 α ∂ f (T ,...,T ) 1 α (t) Nt {NT =i} j i 1 i [0,Tl ] αl−1 i=1 j=2 l=1 ∞ i j X X α α α X 1 +α α 1 α ∂ f (T ,...,T ) 1 α (t) Nt {NT =i} j i 1 i [0,Tl−1] αl−1 i=1 j=2 l=1 ∞ X 1 α α α = −αN α 1{N α=i}1[0,T α](t) ∂1f (T ,...,T ) t T 1 α i 1 i i=1 0 ∞ i j X X α α α X 1 −α α 1 α ∂ f (T ,...,T ) 1 α Nt {NT =i} j i 1 i {Nt =l−1} αl−1 i=1 j=2 l=1 ∞ X α α α = − 1 α 1 α (t)∂ f (T ,...,T ) {NT =i} [0,T1 ] 1 i 1 i i=1 ∞ i j X X α α α X − 1 α ∂ f (T ,...,T ) 1 α {NT =i} j i 1 i {Nt =l−1} i=1 j=2 l=1 ∞ X α α α = − 1 α 1 α (t)∂ f (T ,...,T ) {NT =i} [0,T1 ] 1 i 1 i i=1 ∞ i X X α α α − 1 α ∂ f (T ,...,T )1 α (t) {NT =i} j i 1 i [0,Tj ] i=1 j=2 ∞ i X X α α α = − 1 α 1 α (t)∂ f (T ,...,T ), t ∈ IR . {NT =i} [0,Tj ] j i 1 i + i=1 j=1

Acknowledgement This research was supported by the Singapore MOE Tier 2 Grant MOE2016-T2-1-036 and by the grant 71673117 from the Natural Science Foundation of China.

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