Itô integrals for fractional Brownian motion and applications to option pricing
IIT ˆO INTEGRALS FOR FRACTIONAL BROWNIAN MOTIONAND APPLICATIONS TO OPTION PRICING
ZHONGMIN QIAN AND XINGCHENG XU
Abstract.
In this paper, we develop an Itˆo type integration theory for fractional Brownianmotions with Hurst parameter H ∈ ( , ) via rough path theory. The Itˆo type integrals arepath-wise defined and have zero expectations. We establish the fundamental tools associatedwith this Itˆo type integration such as Itˆo formula, chain rule and etc. As an application weapply this path-wise Itˆo integration theory to the study of a fractional Black-Scholes modelwith Hurst parameter less than half, and prove that the corresponding fractional Black-Scholes market has no arbitrage under a class of trading strategies. Introduction
Since Black and Scholes proposed a financial model for pricing options in their seminalpaper [8] in 1973 (see also Merton [27] for a continuous model), most of papers publishedin the area of quantitative finance are written in the setting of Black-Scholes’ model andits generalizations, where Brownian motion, L´evy processes and semi-martingales are usedto model sources of noise in financial markets. The Markovian property of Black-Scholes’model is essential for a complete financial market, and is required by the market efficiencyassumption. For financial markets, however, statistics of financial data show that value andprice processes exhibit correlations in time. According to the theory of behavioral finance,people in general recognize that past information affects reactions of market participants.In order to model the memory effects of markets, it is natural to develop financial dynamicmodels with memory over time. A simple and good candidate of simulating market noisewith certain memory is a special family of Gaussian processes called fractional Brownianmotions (fBM for abbreviation), introduced by Mandelbrot and van Ness [26] in 1968. FBMsare extensions of the standard Brownian motion, which are different from Brownian motionin that fBMs capture dependencies with decay rate of polynomial order in time.Many researchers have investigated integration theories with respect to fBM (see e.g. [12,13, 5, 15, 21, 28, 2, 3] and the literature therein). Among them, Duncan, Hu and Pasik-Duncan [14] who are the first to propose Wick product approach to define fractional stochasticintegrals against fBM, called Wick-Itˆo integrals. Hu and Oksendal [21], Elliott and van derHoek [15] extended the idea of the Wick product approach, developed a fractional whitenoise calculus and applied their theory to financial models. The approach in [14] is limitedto the persistent case, i.e. fBM with Hurst parameter
H > , while the approach in [15]works for the anti-persistent case, i.e. fBM with Hurst parameter H < . In these papers,ordinary multiplications are replaced by Wick products to define stochastic integrals. The Mathematics Subject Classification.
Primary 60H05, 60H10; Secondary 91B70, 91G80.
Key words and phrases.
Rough paths, Fractional Brownian motions, Itˆo integration, Itˆo formula, FractionalBlack-Scholes model, Non-arbitrage.Xingcheng Xu is supported by the China Scholarship Council. a r X i v : . [ m a t h . P R ] M a r ZHONGMIN QIAN AND XINGCHENG XU idea of Wick products is also applied to the study of portfolios and to the study of self-financing in fractional Black-Scholes’ markets introduced in these papers, see below for amore precise definition. By a conceptual innovation, they showed the non-arbitrage propertyand completeness of the market where Black-Scholes model is replaced by the geometricfBM with Hurst parameter
H > . These papers also initiated an intense debate becauseof conceptual difference in the definition of non-arbitrage from the standard Black-Scholes’model. Severe critiques arose concerning the economic meaning of Wick products applyingto the fundamental economic concepts, such as values of portfolio and self-financing (see, e.g.[6]). Wick products however are only defined as multiplications of two random variables,and cannot be computed in a path-wise sense, hence, if one only knows realizations X ( ω ), Y ( ω ) of two random variables X, Y , one is unable to work out the Wick product of X ( ω )and Y ( ω ).There are other approaches to define stochastic integrals with respect to fBM. The firstis a path-wise approach by using the H¨older continuity of sample paths of fBM, developedin Ciesielski, Kerkyacharian and Roynette [11] and Z¨ahle [32]. Ciesielski et al [11] proposedan integration theory based on the Besov-Orlicz spaces using wavelet expansions, and Z¨ahle[32] developed a theory using fractional calculus and a generalization of the integration byparts formula. These integration theories are suitable however only for fBM with Hurstparameter H > . The second approach, developed by Decreusefond and ¨Ust¨unel [13], relieson the Malliavin calculus for fBM. Alos and Nualart [1], Carmona, Coutin and Montseny[9], and Cheridito and Nualart [10] investigated the theory proposed in [13] further. Thethird approach is an integration theory via the rough path theory, which was worked out byCoutin and Qian [12]. The rough path theory works for fBM with Hurst parameter H > ,which can be considered as an integration theory in Stratonovich’s sense.When applying Stratonovich’s fractional integration to financial models such as optionpricing, it has an obvious shortcoming, namely the expectations of Stratonovich’s integralsare not zero in general, which makes it possible to construct explicitly an arbitrage tradingstrategy (see e.g. [31, 4]) in fractional Black-Scholes’ markets. Since fBM with Hurst parame-ter H (cid:54) = is not a semi-martingale, there is no hope to apply Itˆo’s theory of semi-martingalesto fBM. The best we can hope is to define stochastic integrals in such way so that integralsagainst fBM have zero mean. This feature is at least more promising than the situation inthe Stratonovich integration theory when considering arbitrage problems, and it implies thatthere occurs no systemic bias in price processes over all possible paths.What we are going to do in this paper is to construct an Itˆo type fractional path-wiseintegration theory with respect to fBM with Hurst parameter H ∈ ( , ], by using therough path approach. It is therefore path-wise defined and has zero expectation property,and thus meets our initial expectations. It seems that the natural language to deal withfBM with Hurst parameter H < is based on the rough path theory of T. Lyons (see e.g.[22, 24, 23, 17, 16]). Under the setting of the analysis of rough paths, we establish thefundamental tools associated with Itˆo type integration such as relations with Stratonovichfractional integrals, fractional Itˆo formula, chain rule and etc. In a general setting, roughpath integration theory for non-geometric rough paths have been also studied in [19, 20, 25]and etc. Here we however develop the theory only for fBM and for the propose of applicationsin Black-Scholes models. As an application we apply this path-wise Itˆo integration theory tofractional Black-Scholes model, and prove that the corresponding fractional Black-Scholes’market has no arbitrage under a class of allowed trading strategies which is more restrictive T ˆO ROUGH PATH INTEGRALS 3 than those allowed in a complete market. The study of arbitrage problems is just an example.There are many other problems which can be studied with this theory. We hope the presentwork can arise a new interest in applying geometric fBM to the study of quantitative finance.Now we give a summary of our work. In section 2, we will recall the most importantinsights concerning rough path theory, and in section 3 we will present the fundamental factsabout fBM and define the Itˆo factional Brownian rough path with Hurst parameter H ∈ ( , ]by the theory introduced in section 2. The material in section 2 and section 3 can be foundin standard literature such as [12, 24, 22, 16, 17]. Section 4 forms a core of this paper, wewill give a definition of integration with respect to our Itˆo fractional Brownian rough path,establish the relation between Stratonovich fractional integrals and Itˆo’s integrals, and workout the fractional Itˆo formula in our integration framework. Besides, we will also address thedifferential equation driven by rough paths and give several important examples of differentialequations driven by fractional Brownian rough path, especially the fractional Black-Scholesmodel. Then we build the chain rule for fractional Black-Scholes’ model, and finally show thezero expectation property of our Itˆo integration in this section. In section 5, we will applythe theory established above to financial market as an application, and show that there is noarbitrage in Itˆo fractional Black-Scholes’ market under a restriction on the class of tradingstrategies.As a remark, the path-wise Itˆo integration we construct is just for a one form F ( B Ht ) offBM B Ht . If the integrator is not a one form, the integral can still have a meaning underthis natural rough path lift for fBM but no longer have zero expectation. For example,the fractional Ornstein-Uhlenbeck process X t = (cid:82) t e − ( t − s ) dB Hs can not be a function of B Ht ,that is, not one form of fBM. If we want to define a path-wise Itˆo integral for fractionalOrnstein-Uhlenbeck process with respect to fBM, i.e. make the integral (cid:82) t X s dB Hs have zeroexpectation, we should find another rough path lift for fBM as Itˆo lift. As a matter of fact,for constructing general path-wise Itˆo integration with respect to fBM, the Itˆo rough path liftof fBM, in particular its L´evy area part, has to be made to depend on the integrator, whichis a striking difference from Itˆo’s theory for Brownian motion. Itˆo’s integrals with respect tofBM can be constructed, as we will demonstrate in the present paper, by using a single roughpath lift of fBM for all one forms. When the integrator is a solution of a rough differentialequation driven by fBM, however, the Itˆo rough path lift of fBM depends on the solutiontoo. 2. Preliminaries on rough paths
In this section, we recall some basic notions concerning rough paths to establish severalnotations which will be used throughout the paper. Our exposition follows closely thosepresented in rough path literature (see e.g. [12, 16, 17, 24]). In particular, we mention thefundamental framework needed to ensure that fBM with Hurst parameter H ∈ ( , ], whichwill be introduced in the next section, has a natural rough path lift.For N ∈ N , T ( N ) ( R d ) denotes the truncated tensor algebra defined by T ( N ) ( R d ) := ⊕ Nn =0 ( R d ) ⊗ n , ZHONGMIN QIAN AND XINGCHENG XU with the convention that ( R d ) ⊗ = R . The space T ( N ) ( R d ) is equipped with a vector spacestructure and a multiplication ⊗ defined by( X ⊗ Y ) k = k (cid:88) i =0 X k − i Y i , k = 0 , , · · · , N, where X = (1 , X , · · · , X N ), Y = (1 , Y , · · · , Y N ) ∈ T ( N ) ( R d ) . We will consider continuous R d -valued paths X on [0 , T ] with bounded variations, andtheir canonical lifts X s,t = (1 , X s,t , · · · , X Ns,t ) in the space T ( N ) ( R d ), where X s,t = X t − X s ,X s,t = (cid:90) s 1, where N = [ p ]), if it satisfies (2.1) andhas finite p -variations, that is, N (cid:88) i =1 sup D (cid:88) (cid:96) | X it (cid:96) − ,t (cid:96) | p/i < ∞ , where the sup runs over all finite partitions D = { t < t < · · · < t n = T } . The p -variation distance is defined to be d p ( X, Y ) = N (cid:88) i =1 (cid:32) sup D (cid:88) (cid:96) | X it (cid:96) − ,t (cid:96) − Y it (cid:96) − ,t (cid:96) | p/i (cid:33) i/p . Equivalently, X : ∆ → T ( N ) ( R d ) has finite p -variations if | X is,t | ≤ ω ( s, t ) i/p , ∀ i = 1 , · · · , N, ∀ ( s, t ) ∈ ∆for some function ω , where ω is a non-negative, continuous, super-additive function on ∆and satisfies ω ( t, t ) = 0. Such function ω is called a control of the rough path X . The space of all p -rough paths is denoted by Ω p ( R d ). A rough path X is called a geometricrough path if there is a sequence of X ( n ), where X ( n ) are the canonical lifts of their firstlevel X ( n ) which are continuous with finite variations, such that X is the limit of X ( n )under p -variation distance d p . The space of geometric rough paths is denoted by G Ω p ( R d ).As our interest lies in fBMs with Hurst parameter H ∈ ( , ], which will be introducedlater, we consider only rough paths valued in T (2) ( R d ). Thus, in what follows, we will T ˆO ROUGH PATH INTEGRALS 5 assume that 3 > p ≥ p ] = 2. A rough path of roughness p can be written as X s,t = (1 , X s,t , X s,t ) for s < t , and the algebraic relation (Chen’s identity) now becomes(2.2) X s,t = X t − X s , and(2.3) X s,t − X s,u − X u,t = X s,u ⊗ X u,t , for all ( s, u ) , ( u, t ) ∈ ∆, where X should be (if it makes sense) considered as an iteratedintegral(2.4) (cid:90) ts X s,u dX u := X s,t which is of course not defined a priori.The most convenient tool to construct rough paths is through almost rough path s. Afunction Y = (1 , Y , Y ) from ∆ to T (2) ( R d ) is called an almost rough path if it has finite p -variation, and for some control ω and constant θ > | ( Y s,t ⊗ Y t,u ) i − Y is,u | ≤ ω ( s, u ) θ , i = 1 , , for all ( s, t ) , ( t, u ) ∈ ∆. According to Theorem 3.2.1, [24], given an almost rough path Y = (1 , Y , Y ), there exists a unique rough path X = (1 , X , X ) such that(2.6) | X is,t − Y is,t | ≤ ω ( s, t ) θ , i = 1 , , θ > ω , and all ( s, t ) ∈ ∆.3. Fractional Brownian motion Fractional Brownian motion (fBM) is a continuous-time Gaussian process B H ( t ) (where t ≥ t ≥ 0, and the co-variance function given by(3.1) E [ B H ( t ) B H ( s )] = 12 ( | t | H + | s | H − | t − s | H ) , where H is a real number in (0 , H > / 2, then the increments of the fBM are positively correlated, and in the case that H < / 2, then the increments of the fBM are negatively correlated. Therefore, for H < ,fBM has the property of counter persistence: that is, the persistence is increasing with respectto the past, it is more likely to decrease in the future, and vice versa. In contrast, for H > ,the fBM is persistent, it is more likely to keep trend than to break it. Therefore, fBM isa popular model for both short-range dependent and long-range dependent phenomena invarious fields, including physics, biology, hydrology, network research, financial mathematicsetc.The fBM with Hurst parameter H has an integral representation in terms of Brownianmotion(3.2) B H ( t ) = (cid:90) t K H ( t, s ) dW ( s ) , where W ( t ) is a standard Brownian motion and K H ( t, s ) = C H (cid:34) H − (cid:18) t ( t − s ) s (cid:19) H − − (cid:90) ts (cid:18) u ( u − s ) s (cid:19) H − duu (cid:35) (0 ,t ) ( s ) ZHONGMIN QIAN AND XINGCHENG XU which is a singular kernel, and C H is a normalised constant.A theory of integration with respect to fBM with Hurst parameter H > may be estab-lished by using Young’s integration theory or functional integration theories (see e.g.[11, 32]),while stochastic calculus with respect to fBM with Hurst parameter H < is better to bestudied in the framework of rough path analysis. In [12], a construction of a canonical level-3rough path B H with Hurst parameter H > / p -variation distance. Here as our main concern is the fBM with Hurstparameter H ∈ ( , ], so we only need the level-2 results about fBM. The method of [12] toconstruct the fractional Brownian rough B H = (1 , ( B H ) , ( B H ) ) implies that(3.3) ( B H ) s,t := lim m → (cid:90) ts B H, ( m ) s,u ⊗ dB H, ( m ) u , a.s., exists in p -variation distance as long as pH > 1, where the dyadic piece-wise linear approxi-mations B H, ( m ) t on interval [ s, t ] is defined by B H, ( m ) r := B Ht m(cid:96) − + 2 m r − t m(cid:96) − t m(cid:96) − t m(cid:96) − ∆ m(cid:96) B H , with ∆ m(cid:96) B H = B Ht m(cid:96) − B Ht m(cid:96) − , t m(cid:96) := s + (cid:96) m ( t − s ) for (cid:96) = 1 , , · · · , m . Proposition 3.1. ( [12] , Theorem 2; [16] , Theorem 10.4)Let B H = ( B H, , · · · , B H,d ) be a d -dimensional fBM with the Hurst parameter H ∈ ( , ] .Then B H , restricted to an finite interval [0 , T ] , lifts via (3.3) to a geometric rough path B H = (1 , ( B H ) , ( B H ) ) ∈ G Ω p ([0 , T ] , R d ) , for all p ∈ ( H , . The details of the proof of this proposition may be found from [12, 16]. We mentionthat the random rough path B H = (1 , ( B H ) , ( B H ) ) is called the canonical lift. It canbe viewed as the Stratonovich lift of fBM. Since we will introduce a new natural lift offBM in Itˆo’s sense in this paper, we rewrite the Stratonovich fractional Brownian roughpath B H = (1 , ( B H ) , ( B H ) ) as B S = (1 , ( B S ) , ( B S ) ). The fBM B H is denoted by B forsimplicity.The Itˆo rough path associated with fBM B H is defined by(3.4) B Is,t := (1 , ( B I ) s,t , ( B I ) s,t ) = (cid:18) , B t − B s , ( B S ) s,t − 12 ( t H − s H ) (cid:19) , where the Hurst parameter H ∈ ( , ]. We may verify that B Is,t = (1 , ( B I ) s,t , ( B I ) s,t ) is stilla random rough path but not a geometric one. We call this non-geometric rough path asItˆo fractional Brownian rough path as we can see from below that the rough path and itsintegration theory for B I is an extension of standard Brownian motion and Itˆo stochasticintegration. For simplicity, we denote it by B s,t = (1 , B s,t , B s,t ) if no confusion may arise.4. Itˆo Integration against fractional Brownian motion Itˆo integrals of one forms against fBM. The goal of this section is to give ameaning for Itˆo integrals of one forms with respect to fBM such as (cid:82) ts F ( B ) dB , where B = ( B (1) , · · · , B ( d ) ) is a d -dimensional fBM with a Hurst parameter H ∈ ( , ]. T ˆO ROUGH PATH INTEGRALS 7 In order to define (cid:82) ts F ( B ) dB , where F : R d → L ( R d , R e ) satisfies some smooth conditions,according to the rough path theory of T. Lyons, we should take B as a rough path. Actuallythe symbol (cid:82) ts F ( B ) dB , to some extent, is a misleading statement. Recall that the roughintegral (cid:82) F ( X ) dX against a rough path X = (1 , X , X ) ∈ Ω p ( R d ) with 2 ≤ p < (cid:98) Y s,t = F ( X s ) X s,t + DF ( X s ) X s,t , (4.1) (cid:98) Y s,t = F ( X s ) ⊗ F ( X s ) X s,t , (4.2)(one can find a proof in [24], Theorem 5.2.1), and the integral (cid:82) F ( X ) dX is defined to bethe rough path Y uniquely associated with the almost rough path (cid:98) Y . The integral can bewritten in compensated Riemann sum form, that is, Y s,t = (cid:90) ts F ( X ) dX = lim | D |→ (cid:88) (cid:96) (cid:16) F ( X t (cid:96) − ) X t (cid:96) − ,t (cid:96) + DF ( X t (cid:96) − ) X t (cid:96) − ,t (cid:96) (cid:17) (4.3)and the second level Y s,t = (cid:90) ts F ( X ) dX = lim | D |→ (cid:88) (cid:96) (cid:16) Y s,t (cid:96) − ⊗ Y t (cid:96) − ,t (cid:96) + F ( X t (cid:96) − ) ⊗ F ( X t (cid:96) − ) X t (cid:96) − ,t (cid:96) (cid:17) . (4.4)These limits exist in the p -variation distance.Apply the general definition above, we get the integral against the Stratonovich fractionalBrownian rough path B S , X S := (cid:82) F ( B S ) dB S . For simplicity we will use X S = (cid:82) F ( B ) ◦ dB to denote this Stratonovich integral, which is by definition the integral against rough path B S .Respectively, we also define the Itˆo integral (cid:82) F ( B I ) dB I against rough path B I (denotedby B in what follows), and denote it as X I := (cid:82) F ( B ) dB .4.1.1. Relation between Stratonovich and Itˆo rough integrals (I). Now we want to establish arelation between Stratonovich and Itˆo integrals. Theorem 4.1. The relation between Stratonovich and Itˆo integral is given as the following.(i) For the first level, (4.5) ( X S ) s,t − ( X I ) s,t = 12 (cid:90) ts DF ( B u ) du H , ZHONGMIN QIAN AND XINGCHENG XU (ii) For the second level, ( X S ) s,t − ( X I ) s,t = 12 (cid:90) ts F ( B u ) ⊗ F ( B u ) du H + 12 (cid:90) ts (cid:18)(cid:90) us DF ( B r ) dr H (cid:19) ⊗ d ( X S ) ,u + 12 (cid:90) ts ( X S ) s,u ⊗ DF ( B u ) du H − (cid:90) ts (cid:18)(cid:90) us DF ( B r ) dr H (cid:19) DF ( B u ) du H , (4.6) where the last four integrals are Young integrals. This theorem is a corollary of Theorem 4.2 below.4.1.2. Relation between Stratonovich and Itˆo rough integrals (II). Let us introduce the space-time Stratonovich/Itˆo path (cid:101) B = ( B, t ), where the first level is given by (cid:101) B s,t = ( B s,t , t − s ) , and the second level is given by (cid:101) B s,t = (cid:18) B s,t , (cid:90) ts B s,u du, (cid:90) ts ( u − s ) dB u , 12 ( t − s ) (cid:19) , where the cross integrals are Young integrals, and B is the Stratonovich or Itˆo lift of fBM.Naturally(4.7) (cid:90) F ( B, t ) dB := (cid:90) f ( (cid:101) B ) d (cid:101) B, with f ( x, t )( ξ, τ ) = F ( x, t ) ξ and the right hand side is well defined as an integral for roughpaths. We use the symbol (cid:82) F ( B, t ) dB =: X I as Itˆo integral and the symbol (cid:82) F ( B, t ) ◦ dB =: X S as Stratonovich integral. We can establish the relationship between the Stratonovich andItˆo integrals, too. Theorem 4.2. The relation between Stratonovich and It integral is given as the following.(i) For the first level, (4.8) ( X S ) s,t − ( X I ) s,t = 12 (cid:90) ts D x F ( B u , u ) du H , (ii) For the second level, ( X S ) s,t − ( X I ) s,t = 12 (cid:90) ts F ( B u , u ) ⊗ F ( B u , u ) du H + 12 (cid:90) ts (cid:18)(cid:90) us D x F ( B r ) dr H (cid:19) ⊗ d ( X S ) ,u + 12 (cid:90) ts ( X S ) s,u ⊗ D x F ( B u ) du H − (cid:90) ts (cid:18)(cid:90) us D x F ( B r , r ) dr H (cid:19) D x F ( B u , u ) du H , (4.9) T ˆO ROUGH PATH INTEGRALS 9 where the last four integrals are Young integrals.Proof. (i) By definition of our integral, we have (cid:90) ts F ( B, u ) dB = (cid:90) ts f ( (cid:101) B ) d (cid:101) B = lim |P|→ (cid:88) [ u,v ] ∈P f ( (cid:101) B u ) (cid:101) B u,v + Df ( (cid:101) B u ) (cid:101) B u,v = lim |P|→ (cid:88) [ u,v ] ∈P F ( B u , u ) B u,v + D x F ( B u , u ) B u,v + D u F ( B u , u ) (cid:90) vu B u,r dr. Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) vu B r dr − B u ( v − u ) (cid:12)(cid:12)(cid:12)(cid:12) = o ( | v − u | ) = o ( |P| ) , so that (cid:90) ts F ( B, u ) dB = lim |P|→ (cid:88) [ u,v ] ∈P F ( B u , u ) B u,v + D x F ( B u , u ) B u,v . Therefore we may conclude that (cid:90) ts F ( B, u ) dB = lim |P|→ (cid:88) [ u,v ] ∈P F ( B u , u ) B u,v + D x F ( B u , u )(( B S ) u,v − I ( v H − u H ))= lim |P|→ (cid:88) [ u,v ] ∈P (cid:0) F ( B u , u ) B u,v + D x F ( B u , u )( B S ) u,v (cid:1) − 12 lim |P|→ (cid:88) [ u,v ] ∈P D x F ( B u , u )( v H − u H )= (cid:90) ts F ( B, u ) ◦ dB − (cid:90) ts D x F ( B u , u ) du H . (ii) Now we show the second relation (4.9). It follows in the similar way as (i). (cid:90) ts F ( B, u ) dB = (cid:90) ts f ( (cid:101) B ) d (cid:101) B = lim |P|→ (cid:88) [ u,v ] ∈P ( X I ) s,u ⊗ ( X I ) u,v + f ( (cid:101) B u ) ⊗ f ( (cid:101) B u ) (cid:101) B u,v . By eqn (4.8), and f ( (cid:101) B u ) ⊗ f ( (cid:101) B u ) (cid:101) B u,v = F ( B u , u ) ⊗ F ( B u , u ) B u,v , we obtain (cid:90) ts F ( B, u ) dB = lim |P|→ (cid:88) [ u,v ] ∈P (cid:18) ( X S ) s,u − (cid:90) us D x F ( B r , r ) dr H (cid:19) ⊗ (cid:18) ( X S ) u,v − (cid:90) vu D x F ( B r , r ) dr H (cid:19) + F ( B u , u ) ⊗ F ( B u , u ) (cid:18) ( B S ) u,v − 12 ( v H − u H ) (cid:19) = lim |P|→ (cid:88) [ u,v ] ∈P ( X S ) s,u ⊗ ( X S ) u,v + F ( B u , u ) ⊗ F ( B u , u )( B S ) u,v − F ( B u , u ) ⊗ F ( B u , u ) (cid:0) v H − u H (cid:1) − (cid:90) us D x F ( B r , r ) dr H ⊗ ( X S ) u,v − ( X S ) s,u ⊗ (cid:90) vu D x F ( B r , r ) dr H + 14 (cid:90) us D x F ( B r , r ) dr H (cid:90) vu D x F ( B r , r ) dr H . Since B t has finite p -variation with p > /H , therefore B is of finite p/ F ( B t , t ), D x F ( B t , t ) have finite p -variations, (cid:82) · D x F ( B r , r ) dr H has finite 1 / H -variation and ( X S ) has finite p -variation. Since H < p < < H < , so that p + 2 H > 1, and the lastfour sums on the right hand side converge to the Young integral. Eqn (4.9) therefore followsimmediately. (cid:3) The Itˆo formula. Homogeneous Itˆo type formula. Let X = (1 , X , X ) ∈ Ω p ( R d ), 2 ≤ p < p -roughpath, denote X t := X ,t , and F : R d → L ( R d , R e ) be a Lip( γ ) function for some γ > p . Sincewe often do composition with rough paths, we want to lift the function F ( X ) to a roughpath F R ( X ) = (1 , F R ( X ) , F R ( X ) ). In terms of rough path integrals, we use the formula(4.10) F R ( X ) = (cid:90) DF ( X ) dX as a definition of image F R of function F . Actually F R sends a rough path to another roughpath, so that F R : Ω p ( R d ) → Ω p ( R e ), 2 ≤ p < 3. By the definition of rough path integrals, F R ( X ) s,t = (cid:90) ts DF ( X ) dX = lim | D |→ (cid:88) (cid:96) (cid:16) DF ( X t (cid:96) − ) X t (cid:96) − ,t (cid:96) + D F ( X t (cid:96) − ) X t (cid:96) − ,t (cid:96) (cid:17) , (4.11) T ˆO ROUGH PATH INTEGRALS 11 F R ( X ) s,t = (cid:90) ts DF ( X ) dX = lim | D |→ (cid:88) (cid:96) (cid:16) F R ( X ) s,t (cid:96) − ⊗ F R ( X ) t (cid:96) − ,t (cid:96) + DF ( X t (cid:96) − ) ⊗ DF ( X t (cid:96) − ) X t (cid:96) − ,t (cid:96) (cid:17) . (4.12)Let X ( φ ) s,t = (1 , X ( φ ) s,t , X ( φ ) s,t ) := (1 , X s,t , X s,t − φ s,t ) be a perturbation of the roughpath X = (1 , X , X ), where φ s,t = φ t − φ s (additive) is finite q -variation with q ≤ p . Assumethat X is a geometric rough path, then X ( φ ) is no longer a geometric rough path in general.Define the composition F R ( X ( φ )) = (1 , F R ( X ( φ )) , F R ( X ( φ )) ) as(4.13) F R ( X ( φ )) = (cid:90) DF ( X ( φ )) dX ( φ ) . We have the following Itˆo type formula. Theorem 4.3. (Itˆo type formula) Assume that X ∈ G Ω p ( R d ) with ≤ p < , X ( φ ) is aperturbation of the rough path X as above, and F : R d → L ( R d , R e ) is a Lip( γ ) function forsome γ > p , then(i) F R ( X ) s,t = F ( X t ) − F ( X s ) .(ii) For the first level, F ( X t ) − F ( X s ) = (cid:90) ts DF ( X ( φ )) dX ( φ ) + (cid:90) ts D F ( X u ) dφ u . (iii) For the second level, F R ( X ) s,t = (cid:90) ts DF ( X ( φ )) dX ( φ ) + (cid:90) ts DF ( X u ) ⊗ DF ( X u ) dφ u + (cid:90) ts (cid:18)(cid:90) us D F ( X r ) dφ r (cid:19) ⊗ dF ( X u )+ (cid:90) ts ( F ( X u ) − F ( X s )) ⊗ D F ( X u ) dφ u + (cid:90) ts (cid:18)(cid:90) us D F ( X r ) dφ r (cid:19) D F ( X u ) dφ u , where the last four integrals are Young integrals, and F R ( X ) s,t can be viewed as a kind ofgeometric increments of the second level process.Proof. (i) The equality holds for any continuous path X with finite variation and its canonicallift as rough paths. Then by definition of geometric rough paths, it can be approximated bya sequence of path with finite variation in p -variation. By continuity of both sides, we knowthe equality still holds. (ii) and (iii) can be proved by the same arguments as in Theorem4.2. (cid:3) Remark 4.4. In integration theory for rough paths, if X ∈ Ω p , ≤ p < , X t = X ,t , onecannot just write a symbol dF ( X t ) as the ordinary case. There is no meaning for this symbolunless under the sense of Young integrals, if it is well defined. We can see an example belowwhich says that the dF ( X t ) is undefined in general. Actually if we want to make sense the differential symbol, we should lift F ( X t ) to a rough path F R ( X ) as above and then understandthe differential symbol as dF R ( X ) = d ( F R ( X ) , F R ( X ) ) . In order to clarify the remark above, first we give a lemma below. Lemma 4.5. Let Y = (1 , Y , Y ) be a rough path. Then the integral (4.14) (cid:90) ts dY = (1 , Y s,t , Y s,t ) as expected. Example 1. Take Y as F R ( X ), F R ( X ( φ )). Then by lemma 4.5, we have (cid:90) ts dF R ( X ) = (1 , F R ( X ) s,t , F R ( X ) s,t ) , (cid:90) ts dF R ( X ( φ )) = (1 , F R ( X ( φ )) s,t , F R ( X ( φ )) s,t ) . Actually F ( X ( φ ) t ) = F ( X t ), but dF R ( X ) (cid:54) = dF R ( X ( φ )), even for the first level as we cansee that F R ( X ) s,t (cid:54) = F R ( X ( φ )) s,t by It formula above. Therefore the symbol dF ( X t ) or dF ( X ( φ ) t ) for rough paths can lead to confusion. Remark 4.6. If X is a geometric rough path, by Theorem 4.3, we have F R ( X ) s,t = F ( X t ) − F ( X s ) , i.e. F R ( X ) s,t = F ( X ) s,t . However, for the non-geometric rough path we do not have theequality alike. In fact, in general, F R ( X ( φ )) s,t (cid:54) = F ( X ( φ ) t ) − F ( X ( φ ) s )(= F ( X t ) − F ( X s )) . We next want to establish an Itˆo type formula for integrals against Itˆo fractional Brownianrough path. As a corollary, we have Corollary 4.7. (Itˆo formula for fBM) Let B S be fractional Brownian rough path with Hurstparameter < H ≤ enhanced under Stratonovich sense, B be the Itˆo fractional Brownianrough path, and F : R d → L ( R d , R e ) be a Lip( γ ) function for some Hγ > . Then(i) F ( B t ) − F ( B s ) = F R ( B S ) s,t = (cid:82) ts DF ( B ) ◦ dB .(ii) For the first level, F ( B t ) − F ( B s ) = (cid:90) ts DF ( B ) dB + 12 (cid:90) ts D F ( B u ) du H . (iii) For the second level, F R ( B S ) s,t = (cid:90) ts DF ( B ) dB + 12 (cid:90) ts DF ( B u ) ⊗ DF ( B u ) du H + 12 (cid:90) ts (cid:18)(cid:90) us D F ( B r ) dr H (cid:19) ⊗ dF ( B u )+ 12 (cid:90) ts ( F ( B u ) − F ( B s )) ⊗ D F ( B u ) du H − (cid:90) ts (cid:18)(cid:90) us D F ( B r ) dr H (cid:19) ⊗ D F ( B u ) du H , T ˆO ROUGH PATH INTEGRALS 13 where the last four integrals are Young integrals. Inhomogeneous Itˆo formula. In the following, we want to make sense for F R (( X, t ))when the inhomogeneous function F ( x, t ) applied to a rough path X = (1 , X , X ) ∈ Ω p andestablish Itˆo formula for it. First, recall the space-time rough path (cid:101) X = ( X, t ), where thefirst level is given by (cid:101) X s,t = ( X s,t , t − s ) , and the second level is given by (cid:101) X s,t = (cid:18) X s,t , (cid:90) ts X s,u du, (cid:90) ts ( u − s ) dX u , 12 ( t − s ) (cid:19) , where the cross integrals are Young integrals. Define the rough path F R (( X, t )) by(4.15) F R (( X, t )) := F R ( (cid:101) X ) = (cid:90) DF ( (cid:101) X ) d (cid:101) X, where DF ( x, t )( ξ, τ ) = D x F ( x, t ) ξ + D t F ( x, t ) τ .Note that if X ( φ ) is a perturbation of the rough path X , and (cid:101) X ( φ ) is its associatedspace-time rough path, then (cid:101) X ( φ ) s,t = (cid:101) X s,t , (cid:101) X ( φ ) s,t = (cid:18) X s,t − φ s,t , (cid:90) ts X s,u du, (cid:90) ts ( u − s ) dX u , 12 ( t − s ) (cid:19) . Note that only the first d × d dimensional components of the second level of (cid:101) X are changed. Theorem 4.8. (Itˆo formula) Assume X ∈ G Ω p ( R d ) with ≤ p < , X ( φ ) is a perturbationof the rough path X , and (cid:101) X , (cid:101) X ( φ ) are their associated space-time rough path respectively, F : R d +1 → L ( R d +1 , R e ) be a Lip( γ ) function for some γ > p .(i) We have the basic calculus formula: (4.16) F ( X t , t ) − F ( X s , s ) = (cid:90) ts DF ( (cid:101) X ) d (cid:101) X . (ii) For the first level, (4.17) F ( X t , t ) − F ( X s , s ) = (cid:90) ts DF ( (cid:101) X ( φ )) d (cid:101) X ( φ ) + (cid:90) ts D x F ( X u , u ) dφ u . (iii) For the second level, F R (( X, t )) s,t = (cid:90) ts DF ( (cid:101) X ( φ )) d (cid:101) X ( φ ) + (cid:90) ts D x F ( X u , u ) ⊗ D x F ( X u , u ) dφ u + (cid:90) ts (cid:18)(cid:90) us D x F ( X r , r ) dφ r (cid:19) ⊗ dF ( X u , u )+ (cid:90) ts ( F ( X u , u ) − F ( X s , s )) ⊗ D x F ( X u , u ) dφ u + (cid:90) ts (cid:18)(cid:90) us D x F ( X r , r ) dφ r (cid:19) ⊗ D x F ( X u , u ) dφ u , where the last four integrals are Young integrals. Proof. (i) The proof is same as (i) of Theorem 4.3, first for p = 1 it holds, then the result forany geometric rough path follows from the continuity. For (ii) and (iii), note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ts X s,u du (cid:12)(cid:12)(cid:12)(cid:12) = o ( | t − s | )and (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ts ( u − s ) dX u (cid:12)(cid:12)(cid:12)(cid:12) = o ( | t − s | ) , the rest of proof is same as Theorem 4.2. (cid:3) Note that if t → X t := X ,t is a continuous path with finite variation, then eqn (4.16)reads as(4.18) F ( X t , t ) − F ( X s , s ) = (cid:90) ts D x F ( X u , u ) dX u + (cid:90) ts D u F ( X u , u ) du, and eqn (4.17) becomes F ( X t , t ) − F ( X s , s ) = (cid:90) ts D x F ( X u , u ) dX u + (cid:90) ts D u F ( X u , u ) du + (cid:90) ts D x F ( X u , u ) dφ u . (4.19)These equations are just like Itˆo formulas in terms of the Stratonovich and Itˆo integrals instochastic calculus.Now set (cid:101) B S , (cid:101) B are the associated space-time rough paths of Stratonovich fractional Brow-nian rough path B S and Itˆo rough path B , respectively. Corollary 4.9. (Itˆo formula for fractional Brownian rough path) Let B S be fractional Brow-nian rough path with Hurst parameter < H ≤ enhanced under Stratonovich sense, B I bethe Itˆo fractional Brownian rough path (say B for short), and F : R d +1 → L ( R d +1 , R e ) be a Lip( γ ) function for some Hγ > , then(i) F ( B t , t ) − F ( B s , s ) = F R (( B S , t )) s,t = (cid:82) ts DF ( (cid:101) B ) ◦ d (cid:101) B .(ii) For the first level, (4.20) F ( B t , t ) − F ( B s , s ) = (cid:90) ts DF ( (cid:101) B ) d (cid:101) B + 12 (cid:90) ts D x F ( B u , u ) du H . (iii) For the second level, F R (( B S , t )) s,t = (cid:90) ts DF ( (cid:101) B ) d (cid:101) B + 12 (cid:90) ts D x F ( B u , u ) ⊗ D x F ( B u , u ) du H + 12 (cid:90) ts (cid:18)(cid:90) us D x F ( B r , r ) dr H (cid:19) ⊗ dF ( B u , u )+ 12 (cid:90) ts ( F ( B u , u ) − F ( B s , s )) ⊗ D x F ( B u , u ) du H − (cid:90) ts (cid:18)(cid:90) us D x F ( B r , r ) dr H (cid:19) ⊗ D x F ( B u , u ) du H , (4.21) where the last four integrals are Young integrals. T ˆO ROUGH PATH INTEGRALS 15 Differential equations driven by rough paths. Basics of differential equations. In this subsection, we readdress some basic facts aboutdifferential equations driven by rough paths, following the description of [24]. We recall adefinition of differential equations driven by rough paths below. Definition 4.10. Let f : W → L ( V, W ) be a vector field on W . Let y ∈ W and let X ∈ Ω p ( V ) , for ≤ p < . Then we say that a rough path Y ∈ Ω p ( W ) is a solution to thefollowing initial value problem: (4.22) dY = f ( Y ) dX, Y = y if there is a rough path Z ∈ Ω p ( V ⊕ W ) such that π V ( Z ) = X , π W ( Z ) = Y and (4.23) Z = (cid:90) (cid:98) f ( Z ) dZ, where (cid:98) f : V ⊕ W → L ( V ⊕ W, V ⊕ W ) is defined by (cid:98) f ( x, y )( ξ, η ) = ( ξ, f ( y + y ) ξ ) , ∀ ( x, y ) , ( ξ, η ) ∈ V ⊕ W. If the vector field f ∈ C ( W, L ( V, W )) satisfies the linear growth and Lipschitz continuousconditions, then the existence and uniqueness of a solution are ensured (see [24], Theorem6.2.1, Corollary 6.2.2). We will use Φ( y , X ) to denote the unique solution Y , call the map X → Φ( y , X ) Itˆo map defined by differential equation (4.22), whose Lipschitz continuityin p -variation topology can be proved under the same conditions above (see [24], Theorem6.2.2). This is an important result of Itˆo maps in the framework of differential equationsdriven by rough paths.4.3.2. Relation between differential equations driven by different rough paths. In this subsec-tion, our main goal is to show the relationship of differential equations driven by Stratonovichfractional Brownian rough path and Itˆo fractional Brownian rough path, respectively. DefineΦ( x, B S ) as the Itˆo map of the differential equation(4.24) dX = f ( X ) dB S , X = x, where B S is the Stratonovich fractional Brownian rough path, and we use dB S to denote theequation driven by Stratonovich rough path, which sometime we use ◦ dB instead. Respec-tively, let I ( x, B ) denote the Itˆo map of the differential equations driven by the Itˆo fractionalrough path(4.25) dX = f ( X ) dB, X = x. Now we want to ask what is the relationship between I ( x, B ) and Φ( x, B S ) or if the Itˆodifferential equation has a representation in terms of a Stratonovich differential equation.First, we introduce a geometric rough path B S,ϕ defined by( B S,ϕ ) s,t := ( B s,t , t H − s H ) , and ( B S,ϕ ) s,t := (cid:18) ( B S ) s,t , (cid:90) ts B s,u du H , (cid:90) ts ( u H − s H ) dB u , 12 ( t H − s H ) (cid:19) , where the cross integrals are Young integrals. Let Φ (cid:101) f ( x, B S,ϕ ) be the Itˆo map defined by thedifferential equation(4.26) dX = (cid:101) f ( X ) dB S,ϕ , X = x, where (cid:101) f : R e → L ( R d ⊕ R , R e ), (cid:101) f ( x )( ξ, η ) := f ( x ) ξ − Df ( x ) f ( x )( η ) , for all x ∈ R e , ( ξ, η ) ∈ R d ⊕ R . Namely, Φ (cid:101) f ( · , B S,ϕ ) is the Itˆo map of the rough differentialequation(4.27) dX = f ( X ) dB S − Df ( X t ) f ( X t ) dt H . Theorem 4.11. Let f : R e → L ( R d , R e ) be a C vector field, and I ( x, B ) , Φ (cid:101) f ( x, B S,ϕ ) be theItˆo map defined by (4.25), (4.26), respectively. Then (4.28) I ( x, B ) s,t = Φ (cid:101) f ( x, B S,ϕ ) s,t and (4.29) I ( x, B ) s,t = Φ (cid:101) f ( x, B S,ϕ ) s,t − (cid:90) ts f ( X u ) ⊗ f ( X u ) du H , where X t := Φ (cid:101) f ( x, B S,ϕ ) ,t . As a remark, we can see that the differential equations (4.25) and (4.26) (or (4.27)) arenot equivalent. As far as the first level concerned, they define the same solution. This agreeswith classical stochastic differential equations (SDE) driven by standard Brownian motion,i.e. we translate SDE (4.25) into SDE (4.27) when H = 1 / 2. However, in terms of the secondlevel, two differential equations are different. The reason is that I ( x, B ) can be viewed asthe iterated integral of the first level in Itˆo’s sense, while Φ (cid:101) f ( x, B S,ϕ ) is the iterated integralof the first level in Stratonovich sense.Besides, the relationship between Φ( x, B S ) and Φ (cid:101) f ( x, B S,ϕ ) is obvious. They are all un-derstood in the Stratonovich sense, the difference is their drift terms. By the continuity ofItˆo’s maps, they can all be approximated in variation topology by the solution of differentialequations driven by piece-wise linear approximations of fBM and its iterated path integrals.In summary, we can establish the relationship between I ( x, B ), Φ( x, B S ) and Φ (cid:101) f ( x, B S,ϕ ). Proof. Suppose { B t , t ≥ } is a piece-wise linear/smooth approximation with finite variationof fractional Brownian motion { B t , t ≥ } . Set B s,t = B t − B s and let B s,t be the differenceof the iterated integral over [ s, t ] of B and ( t H − s H ). Consider the rough differentialequation(4.30) dX = f ( X ) dB, that is, the integral equation(4.31) Z = (cid:90) (cid:98) f ( Z ) dZ, π d ( Z ) = B, T ˆO ROUGH PATH INTEGRALS 17 where (cid:98) f ( x, y )( ξ, η ) := ( ξ, f ( y ) ξ ), and π d is projection operator to R d , which is solved by thePicard iteration, that is Z ( n + 1) = (cid:90) (cid:98) f ( Z ( n )) dZ ( n ) , Z (0) = ( B, . More precisely, define almost rough paths(4.32) (cid:98) Z ( n + 1) s,t := (cid:98) f ( Z ( n ) s ) Z ( n ) s,t + D (cid:98) f ( Z ( n ) s ) Z ( n ) s,t , (4.33) (cid:98) Z ( n + 1) s,t := (cid:98) f ( Z ( n ) s ) ⊗ (cid:98) f ( Z ( n ) s ) Z ( n ) s,t , and define the corresponding rough paths(4.34) Z ( n + 1) s,t = lim |P|→ (cid:88) [ u,v ] ∈P (cid:98) Z ( n + 1) u,v , (4.35) Z ( n + 1) s,t = lim |P|→ (cid:88) [ u,v ] ∈P Z ( n ) s,u ⊗ Z ( n ) u,v + (cid:98) Z ( n + 1) u,v , where P is a partition of the interval [ s, t ]. Then(4.36) | Z ( n ) is,t − (cid:98) Z ( n ) is,t | ≤ ω ( s, t ) θ , i = 1 , , θ > , for some control ω .(i) Now we prove (4.28). Set (cid:98) Z ( n ) s,t = ( B s,t , X ( n ) s,t ), by definition of (cid:98) f and (4.32), wehave (cid:98) Z ( n + 1) s,t = ( B s,t , f ( X ( n ) s ) B s,t ) + D (cid:98) f ( Z ( n ) s ) Z ( n ) s,t (cid:39) ( B s,t , f ( X ( n ) s ) B s,t ) + D (cid:98) f ( Z ( n ) s ) (cid:98) Z ( n ) s,t (cid:39) ( B s,t , f ( X ( n ) s ) B s,t ) + (0 , Df ( X ( n ) s ) f ( X ( n ) s ) B s,t (cid:39) ( B s,t , f ( X ( n ) s ) B s,t ) − 12 (0 , Df ( X ( n ) s ) f ( X ( n ) s )( t H − s H ) , where (cid:39) means the error can be controlled by ω ( s, t ) θ with θ > 1. Hence, X ( n + 1) s,t (cid:39) f ( X ( n ) s ) B s,t − Df ( X ( n ) s ) f ( X ( n ) s )( t H − s H ) . Since B has finite variation and (cid:80) [ u,v ] ∈P X ( n ) u,v = X ( n ) s,t , the formula above implies that X ( n + 1) s,t = lim |P|→ (cid:88) [ u,v ] ∈P f ( X ( n ) u ) B u,v − Df ( X ( n ) u ) f ( X ( n ) u )( v H − u H ) , as n goes to infinity, the limit above is identified as X t = X s + (cid:90) ts f ( X u ) dB u − (cid:90) ts Df ( X u ) f ( X u ) du H . On the other hand, lim n →∞ X ( n ) s,t = Φ( · , B ) s,t , so we get (4.28) when the system is driven by B . By the continuity of Itˆo’s maps, we concludethat (4.28) holds in real fractional Brownian rough path case. (ii) Similarly, by (4.35) and the continuity of Itˆo’s maps again, we conclude that (4.29)holds. (cid:3) Examples and applications. In this subsection, we will give some interesting exam-ples as applications of fractional Brownian rough paths. Example 2. Consider the differential equation in dimension d = 1, dX = σX ◦ dB. Then X = (1 , X , X ) is the solution of this differential equation, where X t = exp( σB t ), X s,t = X t − X s = exp( σB t ) − exp( σB s ) ,X s,t = 12 ( X s,t ) = 12 (exp( σB t ) − exp( σB s )) . Example 3. Now we consider the Itˆo rough differential equation dX = σXdB. (i) Set X t = exp( σB t − σ t H ) =: F ( B t , t ) and X s,t := X t − X s . Then X s,t = (cid:90) ts σXdB , and X s,t := (cid:90) ts σF ( B, u ) ◦ dB − σ (cid:90) ts ( F ( B u , u )) du H − σ (cid:90) ts (cid:18)(cid:90) us F ( B r , r ) dr H (cid:19) dY u − σ (cid:90) ts Y s,u F ( B u , u ) du H + σ (cid:90) ts (cid:18)(cid:90) us F ( B r , r ) dr H (cid:19) F ( B u , u ) du H , where Y s,t = (cid:82) ts σF ( B, u ) ◦ dB , Y t := Y ,t and the last four integrals are Young integrals. Example 4. Geometric fBM (or fractional Black-Scholes model ). First consider the frac-tional Black-Scholes model in Stratonovich sense(4.37) dX = µX t dt + σX ◦ dB, where µ, σ are constants. The solution X can be constructed as above. Set(4.38) X t = exp ( σB t + µt ) =: F ( B t , t ) , and X s,t = X t − X s ,X s,t = lim n →∞ (cid:90) ts F ( B nu , u ) dF ( B nu , u ) , where B nu is the linear approximation of B , i.e. B nu = B t (cid:96) − + B t (cid:96) − B t (cid:96) − t (cid:96) − t (cid:96) − ( u − t (cid:96) − )on interval [ t (cid:96) − , t (cid:96) ] and { t n(cid:96) , (cid:96) = 0 , , · · · , n } is any partition of [ s, t ]. We can verify X =(1 , X , X ) is the solution. T ˆO ROUGH PATH INTEGRALS 19 In this situation, the corresponding fractional Black-Scholes market has arbitrage. Wechange the Stratonovich integral into an Itˆo integral, i.e. we consider the fractional differentialequation in Itˆo’s sense(4.39) dX = µX t dt + σXdB. We demonstrate that the corresponding Itˆo fractional Black-Scholes market is arbitragefree in a restricted sense.(i) Let(4.40) X t = exp (cid:18) σB t + µt − σ t H (cid:19) =: F ( B t , t ) . Then X s,t := X t − X s and X s,t = (cid:90) ts σXdB + (cid:90) ts µX u du. By the relation between Stratonovich integrals and Itˆo integrals in time dependent case, wehave RHS = (cid:90) ts σF ( B, u ) dB + (cid:90) ts µF ( B u , u ) du = (cid:90) ts σF ( B, u ) ◦ dB − σ (cid:90) ts F ( B u , u ) du H + (cid:90) ts µF ( B u , u ) du = (cid:90) ts D x F ( B, u ) ◦ dB + (cid:90) ts D u F ( B u , u ) du = (cid:90) ts DF ( (cid:101) B ) ◦ d (cid:101) B = F ( (cid:101) B t ) − F ( (cid:101) B s )= F ( B t , t ) − F ( B s , s ) = X t − X s = LHS. (ii) Now set X s,t := Z s,t + (cid:90) ts µZ s,u F ( B u , u ) du + (cid:90) ts (cid:18)(cid:90) us µF ( B r , r ) dr (cid:19) dZ u + (cid:90) ts (cid:18)(cid:90) us µF ( B r , r ) dr (cid:19) µF ( B u , u ) du, where Z s,t = X s,t − (cid:90) ts µF ( B u , u ) du, Z t = Z ,t Z s,t = (cid:90) ts σF ( B, u ) ◦ dB − σ (cid:90) ts ( F ( B u , u )) du H − σ (cid:90) ts (cid:18)(cid:90) us F ( B r , r ) dr H (cid:19) dY u − σ (cid:90) ts Y s,u F ( B u , u ) du H + σ (cid:90) ts (cid:18)(cid:90) us F ( B r , r ) dr H (cid:19) F ( B u , u ) du H , and Y s,t = (cid:82) ts σF ( B, u ) ◦ dB , Y t = Y ,t , and all the integrals except ones involving ◦ dB areYoung integrals, and the integral against ◦ dB is Stratonovich rough integral which can be computed by linear approximations. Actually, by the relation between Stratonovich roughintegrals and It rough integrals, we have Z s,t = (cid:90) ts σF ( B, u ) dB , and Z s,t = (cid:90) ts σF ( B, u ) dB . If we define f ( x, t )( ξ, τ ) := σF ( x, t ) ξ + µF ( x, t ) τ , (cid:101) B = ( B, t ) the space-time rough path of B ,then X s,t = (cid:82) ts f ( (cid:101) B ) d (cid:101) B . Combining (i) and (ii), we have verified that X s,t = (cid:82) ts f ( (cid:101) B ) d (cid:101) B. Theright hand side indeed coincides with the right hand side of the differential equation (4.39).So we have constructed the solution of the Itˆo fractional Black-Scholes equation (4.39). Remark 4.12. As a remark, we have two ways to understand the integrals on the righthand side of above differential equation with drift (4.39). On the one hand, we can define f ( x, t )( ξ, τ ) := σF ( x, t ) ξ + µF ( x, t ) τ , then X s,t = (cid:82) ts f ( (cid:101) B ) d (cid:101) B is well defined. On the otherhand, we can define g ( x, t )( ξ, τ ) := σF ( x, t ) ξ , h t := (cid:82) t µF ( B u , u ) du and see (cid:82) ts σF ( B, u ) dB as (cid:82) ts g ( (cid:101) B ) d (cid:101) B (This integral is well defined). Then view the right hand side of differentialequation (4.39) as a perturbation of (cid:82) g ( (cid:101) B ) d (cid:101) B by h . We want to say that the two ways areconsistent, they give the same results, i.e. (4.41) (cid:90) ts f ( (cid:101) B ) d (cid:101) B = (cid:90) ts g ( (cid:101) B ) d (cid:101) B + (cid:90) ts µF ( B u , u ) du, (cid:90) ts f ( (cid:101) B ) d (cid:101) B = (cid:90) ts g ( (cid:101) B ) d (cid:101) B + (cid:90) ts µZ s,u F ( B u , u ) du + (cid:90) ts (cid:18)(cid:90) us µF ( B r , r ) dr (cid:19) dZ u + (cid:90) ts (cid:18)(cid:90) us µF ( B r , r ) dr (cid:19) µF ( B u , u ) du, (4.42) where Z s,t := (cid:82) ts g ( (cid:101) B ) d (cid:101) B , Z t := Z ,t , and the last three integrals are Young Integral. Now wegive a proof of eqn (4.41), (4.42) below.Proof. (i) For the first level, we have (cid:90) ts f ( (cid:101) B ) d (cid:101) B = lim |P|→ (cid:88) [ u,v ] ∈P f ( (cid:101) B u ) (cid:101) B u,v + Df ( (cid:101) B u ) (cid:101) B u,v = lim |P|→ (cid:88) [ u,v ] ∈P σF ( B u , u ) B u,v + µF ( B u , u )( v − u )+ σD x F ( B u , u ) B u,v + σD u F ( B u , u ) (cid:90) vu ( B r − B u ) dr + µD x F ( B u , u ) (cid:90) vu ( r − u ) dB r + µ D u F ( B u , u )( v − u ) , Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) vu B r dr − B u ( v − u ) (cid:12)(cid:12)(cid:12)(cid:12) = o ( | v − u | ) = o ( |P| )and (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) vu rdB r − u ( B v − B u ) (cid:12)(cid:12)(cid:12)(cid:12) = o ( | v − u | ) = o ( |P| ) , T ˆO ROUGH PATH INTEGRALS 21 we therefore have (cid:90) ts f ( (cid:101) B ) d (cid:101) B = lim |P|→ (cid:88) [ u,v ] ∈P σF ( B u , u ) B u,v + µF ( B u , u )( v − u )+ σD x F ( B u , u ) B u,v = lim |P|→ (cid:88) [ u,v ] ∈P g ( (cid:101) B u ) (cid:101) B u,v + Dg ( (cid:101) B u ) (cid:101) B u,v + µF ( B u , u )( v − u )= (cid:90) ts g ( (cid:101) B ) d (cid:101) B + (cid:90) ts µF ( B u , u ) du, which completes the proof of eqn (4.41).(ii) For the second level, (cid:90) ts f ( (cid:101) B ) d (cid:101) B = lim |P|→ (cid:88) [ u,v ] ∈P X s,u X u,v + f ( (cid:101) B u ) ⊗ f ( (cid:101) B u ) (cid:101) B u,v = lim |P|→ (cid:88) [ u,v ] ∈P (cid:18) Z s,u + (cid:90) us µF ( B r , r ) dr (cid:19) (cid:18) Z u,v + (cid:90) vu µF ( B r , r ) dr (cid:19) + σ ( F ( B u , u )) B u,v (Some terms go to zero here as above.)= (cid:90) ts g ( (cid:101) B ) d (cid:101) B + (cid:90) ts µZ s,u F ( B u , u ) du + (cid:90) ts (cid:18)(cid:90) us µF ( B r , r ) dr (cid:19) dZ u + (cid:90) ts (cid:18)(cid:90) us µF ( B r , r ) dr (cid:19) µF ( B u , u ) du, which yields eqn (4.42). (cid:3) Chain rule of fBM. This subsection is the continuity of Example 4 which plays animportant role in the remainder of the paper. Let X be the solution of fractional Black-Scholes equation (4.39) and G is a good function, then the integral (cid:82) G ( X, t ) dX is welldefined. Since X t has the explicit representation (4.40), (cid:90) σG ( X, t ) XdB + (cid:90) µG ( X t , t ) X t dt is defined in terms of the rough path B , which can be defined as (cid:82) f ( (cid:101) B ) d (cid:101) B , where f ( x, t )( ξ, η ) := f ( x, t ) ξ + f ( x, t ) η,f ( x, t ) = σG ( F ( x, t ) , t ) F ( x, t ) ,f ( x, t ) = µG ( F ( x, t ) , t ) F ( x, t ) , and F ( x, t ) = exp( σx − σ t H / µt ) . Heuristically, the two integrals should equal. Our aim in this subsection is to show that theyare indeed the same in this case. This kind of formula is usually called Chain Rule . Namely,we want to show the theorem below. Theorem 4.13. (Chain rule) Let B be the Itˆo fractional Brownian rough path with H ∈ ( , ] , and X be the geometric fBM with parameters σ and µ , G be a function smooth enough.Then (4.43) (cid:90) G ( X, t ) dX = (cid:90) σG ( X, t ) XdB + (cid:90) µG ( X t , t ) X t dt, where the integrals are understood as above.Proof. Let Y = (cid:82) G ( X, t ) dX , H = (cid:82) f ( (cid:101) B ) d (cid:101) B . Then their associated almost rough path are (cid:98) Y , (cid:98) H respectively, where (cid:98) Y s,t = G ( X s , s ) X s,t + D x G ( X s , s ) X s,t , (cid:98) Y s,t = G ( X s , s ) ⊗ G ( X s , s ) X s,t , and (cid:98) H s,t = f ( B s , s ) B s,t + f ( B s , s )( t − s ) + D x f ( B s , s ) B s,t , (cid:98) H s,t = f ( B s , s ) ⊗ f ( B s , s ) B s,t . We should prove that(4.44) | (cid:98) Y is,t − (cid:98) H is,t | ≤ ω ( s, t ) θ , i = 1 , , ∀ s < t, ∃ θ > , for some control ω . If the difference of two quantities is controlled by ω like (4.44), we usethe symbol (cid:39) to represent it. So we should show (cid:98) Y is,t (cid:39) (cid:98) H is,t .Denote Z = (cid:82) σF ( B, t ) dB , h s,t = (cid:82) ts µF ( B t , t ) dt , and (cid:98) Z , (cid:98) h denote their respective almostrough path, that is, (cid:98) Z s,t = σF ( B s , s ) B s,t + σ F ( B s , s ) B s,t , (cid:98) Z s,t = σ F ( B s , s ) ⊗ F ( B s , s ) B s,t , (cid:98) h s,t = µF ( B s , s )( t − s ) . It is easy to verify that the almost rough path associated with X is given by(4.45) (1 , Z s,t + h s,t , Z s,t ) (cid:39) (1 , Z s,t + (cid:98) h s,t , Z s,t ) . Two quantities on both sides in (4.45) are all almost rough paths. So we have the followingrelations: X s,t = Z s,t + h s,t (cid:39) (cid:98) Z s,t + (cid:98) h s,t , (4.46) X s,t (cid:39) Z s,t (cid:39) (cid:98) Z s,t . (4.47)(i) For the first level, (cid:98) Y s,t = G ( X s , s ) X s,t + D x G ( X s , s ) X s,t = G ( F ( B s , s ) , s ) X s,t + ∂ G ( F ( B s , s ) , s ) X s,t (cid:39) G ( F ( B s , s ) , s )( (cid:98) Z s,t + (cid:98) h s,t ) + ∂ G ( F ( B s , s ) , s ) (cid:98) Z s,t = G ( F ( B s , s ) , s )( σF ( B s , s ) B s,t + σ F ( B s , s ) B s,t + µF ( B s , s )( t − s ))+ ∂ G ( F ( B s , s ) , s )( σ F ( B s , s ) ⊗ F ( B s , s ) B s,t ) . T ˆO ROUGH PATH INTEGRALS 23 Since D x f ( x, s ) = σ ∂ G ( F ( x, s ) , s ) F ( x, s ) ⊗ F ( x, s ) + σ G ( F ( x, s ) , s ) F ( x, s ) , so that (cid:98) Y s,t (cid:39) f ( B s , s ) B s,t + f ( B s , s )( t − s ) + D x f ( B s , s ) B s,t = (cid:98) H s,t . Thus we have proved the first part of the claim.(ii) For the second level paths, we have (cid:98) Y s,t = G ( X s , s ) ⊗ G ( X s , s ) X s,t = G ( F ( B s , s ) , s ) ⊗ G ( F ( B s , s ) , s ) X s,t (cid:39) G ( F ( B s , s ) , s ) ⊗ G ( F ( B s , s ) , s ) (cid:98) Z s,t = G ( F ( B s , s ) , s ) ⊗ G ( F ( B s , s ) , s )( σ F ( B s , s ) ⊗ F ( B s , s ) B s,t )= σ G ( F ( B s , s ) , s ) F ( B s , s ) ⊗ G ( F ( B s , s ) , s ) F ( B s , s ) B s,t = f ( B s , s ) ⊗ f ( B s , s ) B s,t = (cid:98) H s,t , which thus completes our proof. (cid:3) We mention that with the Stratonovich rough paths, the chain rule still holds by the sameargument as Itˆo rough paths above.4.6. Zero mean property of Itˆo Integrals. Now we need check if the first level of our Itˆointegrals defined above have zero mean, that is, we need to verify that(4.48) E (cid:20)(cid:90) ts f ( B ) dB (cid:21) = 0 . One dimension case. We prove it in the case that dimension d = 1. First, we supposethat f has first and second continuous derivatives, by It formula proved above, we can showthat(4.49) E (cid:20)(cid:90) ts f (cid:48) ( B ) ◦ dB (cid:21) = E (cid:20) (cid:90) ts f (cid:48)(cid:48) ( B r ) dr H (cid:21) . The computation is routine, so we omit the details.For the general case, set F ( x ) = (cid:82) x −∞ f ( y ) dy , F ( −∞ ) = 0, so that F (cid:48) ( x ) = f ( x ). By (4.49)we get(4.50) E (cid:20)(cid:90) ts F (cid:48) ( B ) dB (cid:21) = E (cid:20) (cid:90) ts F (cid:48)(cid:48) ( B r ) dr H (cid:21) . Thus the expectation of the first level of the Itˆo integral(4.51) E (cid:20)(cid:90) ts f ( B ) dB (cid:21) = 0 . High dimension case. Now we can prove that eqn (4.49) still holds when dimension d ≥ F : R d → L ( R d , R e ). Let X s,t := (cid:82) ts F ( B ) dB . The i -th componentof first level of this Itˆo integral is X is,t = (cid:34) d (cid:88) j =1 (cid:90) ts F ij ( B ) dB ( j ) (cid:35) . As a remark here, the integral (cid:82) ts F ij ( B ) dB ( j ) is well-defined, which can be understood as (cid:82) ts (cid:101) F ij ( B ) dB , where (cid:101) F ij ( x , · · · , x d )( ξ , · · · , ξ d ) = F ij ( x , · · · , x d ) ξ j . Therefore E (cid:104)(cid:82) ts F ij ( B ) dB ( j ) (cid:105) =0, which yields that(4.52) E (cid:20)(cid:90) ts F ( B ) dB (cid:21) = 0 . Zero mean property of time-dependent functions. In this subsection, we will show thatfor the time-dependent function F ( x, t ) we can still have the mean zero property, i.e.(4.53) E (cid:20)(cid:90) ts F ( B, u ) dB (cid:21) = 0 . As the time independent case, we first show the one dimensional case, then by conditionalexpectation technique we conclude the high dimensional cases. By the It formula (4.20) andRemark 4.12, we know that it is equivalent to F ( B t , t ) − F ( B s , s ) = (cid:90) ts D x F ( B, u ) dB + (cid:90) ts D u F ( B u , u ) du + 12 (cid:90) ts D x F ( B u , u ) du H . (4.54)Then in order to prove the zero mean property, we should verify that E ( F ( B t , t ) − F ( B s , s )) = (cid:90) ts E [ D u F ( B u , u )] du + 12 (cid:90) ts E [ D x F ( B u , u )] du H . For the one dimension case, the left-hand side above can be computed as the following(4.55) E ( F ( B t , t ) − F ( B s , s )) = (cid:90) R ( F ( t H x, t ) − F ( s H x, s )) ϕ ( x ) dx, where ϕ is the standard normal probability density function.On the other hand, (cid:90) ts E [ D u F ( B u , u )] du = (cid:90) ts (cid:20)(cid:90) R ∂ F ( u H x, u ) ϕ ( x ) dx (cid:21) du = (cid:90) R (cid:20) ( F ( t H x, t ) − F ( s H x, s )) − (cid:90) ts x∂ F ( u H x, u ) du H (cid:21) ϕ ( x ) dx. (4.56) T ˆO ROUGH PATH INTEGRALS 25 and 12 (cid:90) ts E [ D x F ( B u , u )] du H = 12 (cid:90) ts (cid:20)(cid:90) R ∂ F ( u H x, u ) ϕ ( x ) dx (cid:21) du H = (cid:90) ts (cid:20)(cid:90) R ∂ F ( u H x, u ) xϕ ( x ) dx (cid:21) du H = (cid:90) R (cid:20)(cid:90) ts ∂ F ( u H x, u ) du H (cid:21) xϕ ( x ) dx. (4.57)Combining (4.55),(4.56) and (4.57), we thus obtain that(4.58) E (cid:20)(cid:90) ts D x F ( B, u ) dB (cid:21) = 0 . By using (cid:101) F ( x, t ) = (cid:82) x −∞ F ( y, t ) dy , we finally get (4.53) as for one dimensional case of oneform above.Now we turn to the high dimension case. Let X is,t := (cid:34) d (cid:88) j =1 (cid:90) ts F ij ( B, u ) dB ( j ) (cid:35) , i = 1 , · · · , d. We need to prove that E (cid:104)(cid:82) ts F ij ( B, u ) dB ( j ) (cid:105) = 0. Let F ij =: f for simplicity. Then E (cid:20)(cid:90) ts f ( B, u ) dB ( j ) (cid:21) = E E (cid:34)(cid:18)(cid:90) ts f ( x , · · · , B ( j ) , · · · , x d , u ) dB ( j ) (cid:19) (cid:35) x i = B ( i ) ,i (cid:54) = j = E E (cid:34)(cid:18)(cid:90) ts (cid:101) f ( B ( j ) , u ) dB ( j ) (cid:19) (cid:35) x i = B ( i ) ,i (cid:54) = j = 0 . Hence E (cid:2) X is,t (cid:3) = 0 for i = 1 , · · · , d . Thus we have proved that the expectation of time-dependent function is also zero, i.e. eqn (4.53) holds.5. Fractional Black-Scholes model We continue the study of Example 4 in section 4.4. We want to study the Itˆo fractionalBlack-Scholes model, fBS for simplicity, and its corresponding market. We want to show themarket is arbitrage free under a restriction of the class of trading strategies. To this end, wefirst give the arbitrage strategy under Stratonovich fractional Black-Scholes market. Arbitrage strategy in Stratonovich fBS market. Since the Stratonovich inte-gral does not have zero mean property, therefore fractional Black-Scholes market based onStratonovich integral suggests the possibility of existence of arbitrage. Shiryayev gave anarbitrage trading strategy in [31] under Stratonovich fBS market but driven by fBM withHurst parameter H > / 2. We adapt this strategy in our case when H ∈ ( , ].The market has a stock (the risky asset) X whose price process is X t := X ,t at time t . Weassume that X satisfies the differential equation driven by Stratonovich fractional Brownianrough path with H ∈ ( , ] as the first part of example 4, i.e.(5.1) dX = µX t dt + σX ◦ dB, X = x, t ∈ [0 , T ] . The solution X of this equation has been constructed in example 4 and X t := X ,t = xe σB t + µt .It is assumed that there is a money market (the risk-less asset) M , that is, an asset whoseprice at time t is not subject to uncertainty. Namely, the price process M t satisfies thefollowing equation(5.2) dM t = rM t dt, M = 1 , t ∈ [0 , T ] , where r > M t = e rt .A portfolio ( γ t , ζ t ) gives the number of units γ t , ζ t held at time t in the money market andstock market, respectively. The value process V t ∈ R of the portfolio is given by(5.3) V t = γ t M t + ζ t X t . The portfolio is called self-financing if(5.4) V t = V + (cid:90) t γ s dM s + (cid:90) t ζ ◦ dX . Note that the second integral on the right hand side is the first level of the Stratonovichintegral against rough path X defined in (5.1).Now consider the following portfolio γ t = 1 − e σB t +2( µ − r ) t , (5.5) ζ t = 2 x − ( e σB t +( µ − r ) t − , (5.6)we will show that this trading strategy is an arbitrage one. First, by (5.5) and (5.6), we getthe value process of the portfolio V t = (1 − e σB t +2( µ − r ) t ) e rt + 2( e σB t +( µ − r ) t − e σB t + µt = e rt (cid:0) e σB t +( µ − r ) t − (cid:1) ≥ . T ˆO ROUGH PATH INTEGRALS 27 By applying the basic principle/Itˆo formula for Stratonovich integral to V t =: f ( B t , t ), wehave V t = V + (cid:90) t re rs (cid:0) e σB s +( µ − r ) s − (cid:1) ds + (cid:90) t µ − r ) e σB s +( µ − r ) s e rs (cid:0) e σB s +( µ − r ) s − (cid:1) ds + (cid:90) t σe rs (cid:0) e σB s +( µ − r ) s − (cid:1) e σB s +( µ − r ) s ◦ dB s = (cid:90) t rγ s e rs ds + (cid:90) t µζ s X s ds + (cid:90) t σζ s X s ◦ dB s = (cid:90) t γ s dM s + (cid:90) t ζ s ◦ dX s . The last equality is by the chain rule of Stratonovich integral in section 4.5.Hence, the portfolio (5.5), (5.6) is self-financing in this financial market. Note that theinitial payment at t = 0 is V = 0, but after that the value of this portfolio is positive almostsurely. This means one gets free lunch with no risk.5.2. Arbitrage free under a class of trading strategies. Now we consider the Itˆo frac-tional Black-Scholes market. As for the Stratonovich fBS market, we suppose that the markethas a stock X (the risky asset) whose price process is X t := X ,t but now it satisfies the dif-ferential equation driven by Itˆo fractional Brownian rough path.(5.7) dX = µX t dt + σXdB, X = x, t ∈ [0 , T ] . The solution is also constructed in example 4. The risk-less asset money market M satisfiesthe equation (5.2), i.e. M t = e rt .Suppose a portfolio ( γ t , ζ t ) gives the value process V t ∈ R by(5.8) V t = γ t M t + ζ t X t . In this Itˆo fBS market, we restrict the class of trading strategies. We call a portfolio is admissible if γ t = γ ( X t , t ), and ζ t = ζ ( X t , t ). Besides, a portfolio is called self-financing if V t = V + (cid:90) t γ s dM s + (cid:90) t ζdX , where the second integral on the right hand side is the first level of the Itˆo integral againstrough path X defined in (5.7).Then by the chain rule of Itˆo fractional Brownian rough path, we have V t = V + (cid:90) t γ s dM s + (cid:90) t ζdX = V + (cid:90) t re rs γ s ds + (cid:90) t µζ s X s ds + (cid:90) t σζ s X s dB By (5.8) we also have(5.9) γ t = e − rt ( V t − ζ t X t ) , plugging it into last equality, we get(5.10) V t = V + (cid:90) t rV s ds + (cid:90) t σζ s X s (cid:18) µ − rσ ds + dB (cid:19) . In order to prove there is no arbitrage in this case, we first introduce the Girsanov theoremfor fBM with Hurst parameter H ≤ / . Girsanov’s theorem. The following version of Girsanov theorem for the fBM has beenobtained in ([13], Theorem 4.9), and we also suggest reader to see [7], Theorem 4.1 and prooftherein. In our case, we would like to show that there is a new probability measure (cid:98) P suchthat(5.11) (cid:98) B t = B t + µ − rσ t, which is still an fBM under this measure (cid:98) P . This is what Girsanov theorem says in usual.Now let K H ( t, s ) be a square integrable kernel given by(5.12) K H ( t, s ) = C H (cid:34) H − (cid:18) t ( t − s ) s (cid:19) H − − (cid:90) ts (cid:18) u ( u − s ) s (cid:19) H − duu (cid:35) (0 ,t ) ( s ) . Define the operator K H on L ([0 , T ]) associated with the kernel K H ( t, s ) as(5.13) ( K H f )( s ) = (cid:90) T f ( t ) K H ( t, s ) dt. Given an adapted and integrable process u = { u t , t ∈ [0 , T ] } , consider the transformation(5.14) (cid:98) B t = B t + (cid:90) t u s ds, since fBM B can be represented by the integral along standard Brownian motion W , we canwrite (5.14) into (cid:98) B t = B t + (cid:90) t u s ds = (cid:90) t K H ( t, s ) dW s + (cid:90) t u s ds = (cid:90) t K H ( t, s ) d (cid:102) W s , where W t is a standard Brownian motion and(5.15) (cid:102) W t = W t + (cid:90) t K − H (cid:18)(cid:90) · u r dr (cid:19) ( s ) ds. By the standard Girsanov theorem for Brownian motion applied to (5.15), as a consequence,we have the following version of the Girsanov theorem for the fBM with Hurst parameter H ≤ , which has obtained in [13], [28] and [7]. Theorem 5.1. (Girsanov theorem for fBM with H ≤ )( [13] , Theorem 4.9; [28] , Theorem2; [7] , Theorem 4.1) Let B be a fBM with Hurst parameter H ∈ (0 , ] , and v ( s ) := K − H (cid:18)(cid:90) · u r dr (cid:19) ( s ) . Consider the shifted process (5.14). Assume that(i) (cid:82) T u t dt < ∞ , almost surely. T ˆO ROUGH PATH INTEGRALS 29 (ii) E ( Z T ) = 1 , where Z T = exp (cid:18) − (cid:90) T v ( s ) dW s − (cid:90) T ( v ( s )) ds (cid:19) , Then the shifted process (cid:98) B is an F Bt -fBM with Hurst parameter H under the new probabilitymeasure (cid:98) P defined by d (cid:98) P d P = Z T . Remark 5.2. Here when u satisfies the condition (i) in Theorem 5.1 with H ≤ , then v = K − H (cid:0)(cid:82) · u r dr (cid:1) is well-defined, and K − H (cid:0)(cid:82) · u r dr (cid:1) ∈ L ([0 , T ]) , where K − H is the inverseof the operator K H . In our case, (cid:98) B t = B t + µ − rσ t, so we can apply the Girsanov Theorem 5.1 to it. Let P be the distribution of fBM B , and (cid:98) P be the distribution constructed from P by Girsanov theorem. In terms of (cid:98) B t , we can write(5.10), under (cid:98) P , as(5.16) V t = V + (cid:90) t rV s ds + (cid:90) t σζ s Xd (cid:98) B . Transforming this equation, we have(5.17) e − rt V t = V + σ (cid:90) t e − rs ζ s Xd (cid:98) B , t ∈ [0 , T ] . Taking expectation under the measure (cid:98) P , we have(5.18) e − rT E (cid:98) P [ V T ] = V + E (cid:98) P (cid:20) σ (cid:90) T e − rs ζ s Xd (cid:98) B (cid:21) . By Girsanov’s theorem and our zero mean property of integrals, we get that(5.19) E (cid:98) P (cid:20)(cid:90) T e − rs ζ s Xd (cid:98) B (cid:21) = 0 . Thus we may conclude that(5.20) e − rT E (cid:98) P [ V T ] = V . Hence the probability measure (cid:98) P defined in Theorem 5.1 is a risk neutral measure. Thenthis Itˆo fBS market has no arbitrage under the class of admissible trading strategy.5.4. Option pricing formula. Moreover, we want to give a pricing formula under risk-neutral measure (cid:98) P for the financial derivative F at time t = 0. Our Itˆo fBS market isarbitrage free under this risk-neutral measure (cid:98) P in Theorem 5.1. Hence, the price under thisrisk-neutral measure (cid:98) P defined in section 5.3 is(5.21) V = e − rT E (cid:98) P [ V T ] . So we can work out explicitly the price, then get the theorem below. Theorem 5.3. (Fractional Black-Scholes pricing formula) The price of claim F ( X T ) underfractional Black-Scholes model and risk-neutral measure (cid:98) P in Theorem 5.1 is (5.22) V = e − rT (cid:90) R F ( X e σT H y + rT − σ T H ) ϕ ( y ) dy, where ϕ ( x ) = √ π e − x is the standard normal density function.Proof. Since the price is V = e − rT E (cid:98) P [ V T ] = e − rT E (cid:98) P [ F ( X T )] , by the Girsanov theorem for fBM, E (cid:98) P [ F ( X T )]= E (cid:98) P (cid:20) F (cid:18) X exp (cid:18) σB HT + µT − σ T H (cid:19)(cid:19)(cid:21) = E (cid:98) P (cid:20) F (cid:18) X exp (cid:18) σ (cid:98) B HT + rT − σ T H (cid:19)(cid:19)(cid:21) = E P (cid:20) F (cid:18) X exp (cid:18) σB HT + rT − σ T H (cid:19)(cid:19)(cid:21) = (cid:90) R F ( X e σT H y + rT − σ T H ) ϕ ( y ) dy, which completes our proof. (cid:3) For the European call option , F ( X T ) = ( X T − K ) + , where K > Corollary 5.4. (European call) The pricing formula of the European call option under frac-tional Black-Scholes model and risk-neutral measure (cid:98) P in Theorem 5.1 is (5.23) V = X (1 − Φ( c − )) − Ke − rT (1 − Φ( c + )) , where c − = 1 σT H log (cid:18) KX (cid:19) − rσ T − H − σT H ,c + = 1 σT H log (cid:18) KX (cid:19) − rσ T − H + 12 σT H , and Φ( x ) = (cid:82) x −∞ ϕ ( y ) dy is standard normal distribution function. We should point out that the above option pricing formula is derived based on the no-arbitrage pricing technique but with a serve restriction on the trading strategies where onlyMarkovian type portfolios are allowed. Probably in practice this is the case – tradings aredone based on current market prices. This looks not so reasonable in contrast with fractionalBS markets which are not Markovian, and therefore is not so satisfactory theoretically. How-ever, at the current technology, this is best we can do within the fractional BS markets dueto lack of a flexible stochastic integration theory beyond the semi-martingale setting. Stillthe rough path theory provides the only way forward with a fractional BS market where theHurst parameter is less than half.Finally let us comment on the pricing formula itself, which is similar to that of the classicalBlack-Scholes model. The values are exactly the same if the maturity time T = 1, which T ˆO ROUGH PATH INTEGRALS 31 Figure 1. Price of European call at time 0 for different exercise time T andvarying Hurst parameter H = 0 . , . , . , . 35, where we take σ = 2, K = 3, X = 3 . r = 0 . 05 as an example.seems strange at the first glance. The Hurst parameter H comes in to play a role onlythrough the exercise time T , and appears as a power in the exercise time T through c ± andto modify the volatility σ T (for Black-Scholes’ market) into σ T H (for the fractional BSmarket case), so that the intensity of the volatility is reduced as H < due to the long timememory. In fact the numerics c ± can be rewritten as c ± = 1 σT H log (cid:18) KX (cid:19) − rTσT H ± σT H . Of course one has to understand the scale of the maturity T in time unit has no economicmeaning, and its scale is in fact fixed by the interest rate through e rT , and therefore it looksnatural that H should appear in the power of T to have its effect on the option pricing. SeeFigure 1, we show prices of European call option at time 0 for different exercise time T andHurst parameter H . References [1] E. Alos, D. Nualart. Stochastic integration with respect to the fractional Brownian motion, Stoch. Stoch.Reports , , 3, 129-152 (2003).[2] C. Bender. 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