# A FRACTIONAL VERSION OF THE HESTON MODEL WITH HURST PARAMETER H [member of] (1/2,1).

1. INTRODUCTION

In classical quantitative finance, it is usual to suppose that risky asset price's dynamics are driven by Brownian motions, as proposed for the first time by Bachelier in 1900 in his Ph.D thesis. Developed by Fisher Black, Robert Merton and Myron Scholes in the seventies, the famous Black and Scholes model remains popular nowadays. Indeed, as closed pricing formulae are provided for European call and put options, the model is easy to implement. Although, the constant volatility assumption of the model  contradicts the empirical observations, i.e. the implied volatility generally depends on time [14, 8]. This led to consider more sophisticated models, e.g. dynamics with local volatilities , but also stochastic volatility models [10, 1].

Despite these improvements, we may observe in practice a long-term correlation between the underlying asset prices, see . To address this issue for the Heston model, a natural idea is to replace the two Brownian motions by fractional Brownian motions (FBM), see [5, 21]. Indeed, heavier tail distributions and long-range dependence are some of the interesting features of the FBM models that confirms their relevance, see [2, 4]. The fractional Black-Scholes (FBS) model, one of the first FBM models, appears to be more efficient and flexible than the classical Black and Scholes model to reproduce the behaviour of the stock dynamics despite its limited capacity to fit the market data, see .

In this paper, we introduce and study the stochastic differential equation defined as the fractional version of the Heston model (FHM) when the Hurst parameter H [member of] (1/2,1). Precisely, we first recall the definition of a fractional Brownian motion and we formulate the existence and uniqueness theorem for the stochastic differential equation in the fractional Heston model. We show that there exists a unique solution which is positive. The proof is based on singular equations driven by rough paths which are studied in Section 3. This section generalises results of  to a larger class of drivers. We then deduce the existence of solutions to singular equations driven by a FBM in Section 4, which is also a generalisation of . Moreover, we show that these solutions are not necessary stationary. For specific drivers, we give an explicit expression of the expectation. This allows to deduce an explicit expression of the expectation of the fractional Cox-Ingersoll-Ross (CIR) process, which may be seen a generalisation of the case H = 1/2 to the case H > 1/2. In particular, the fractional CIR process is mean-reverting.

2. FRACTIONAL HESTON MODEL

2.1. Reminder on the fractional Brownian motion. Fractional Brownian motions were first introduced by Kolmogorov in 1940 .

Definition 2.1. A Gaussian stochastic process [([B.sup.H.sub.t]).sub.t[greater than or equal to]0] of Hurst parameter H [member of] (0,1) is called (standard) fractional Brownian motion, if

1. The paths of [B.sup.H] are continuous and satisfy [B.sup.H.sub.0] = 0.

2. E[BtH] = 0 and Var[BtH] = t2H, for any t > 0.

3. The increments of [B.sup.H] are stationary.

4. The process [B.sup.H] admits the covariance function

(2.1) [mathematical expression not reproducible].

Existence of such Gaussian processes satisfying (2.1) is discussed in . The class of FBM processes may be splitted into two categories apart from the standard Brownian motion, i.e. when H = 1/2. If H [member of] (1/2, 1), the increments of FBM are positively correlated so that the process BH satisfies a long dependence behaviour useful to describe phenomenon with memory and persistence. When H [member of] (0, 1/2), as the increments of [B.sup.H] are negatively correlated, it may be used to model intermittency and anti-persistency, see . In this paper, we only consider FBM processes with Hurst parameter H [member of] (1/2, 1). In 1968, Mandelbrot and Van Ness  gave the following stochastic integral representation of a FBM process:

(2.2)

[mathematical expression not reproducible]

where B is a standard Brownian motion. The first term models the current innovation or shock and the second part contains a moving average of historical shocks, see . We recall the following properties, see :

Theorem 2.2. A fractional Brownian motion BH satisfies the following properties

1. The process BH is self-similar.

2. The trajectories of [B.sup.H] are almost surely nowhere differentiable.

The following definition may be found in .

Definition 2.3. A stationary process [([Y.sub.t]).sub.t[greater than or equal to]0] with finite variance is said to have long range dependance if its autocorrelation function [C.sub.t]([tau]) := cor([Y.sub.t], [Y.sub.t+[tau]]) decays as a power of the lag [tau]: [C.sub.t]([tau]) ~ L([tau])/[[tau].sup.[alpha]], as [tau] [right arrow] [infinity], where [alpha] [member of] (0,1) and L is slowly varying at infinity, i.e. for all a > 0, L(at)/L(t) [right arrow] 1 as t [right arrow] [infinity].

As shown in [15, 19], a fractional Brownian motion [B.sup.H] such that H [member of] (1/2, 1) admits a long-range dependence.

2.2. Main result. Let us consider the following stochastic differential equation:

(2.3) d[S.sub.t] = [mu][S.sub.t]dt + [square root of [V.sub.t][S.sub.t]d[B.sup.1,H.sub.t], t [member of] [0, [infinity]),

(2.4) d[V.sub.t] = k([theta] - [V.sub.t])dt + [sigma][square root of [V.sub.t]] d[B.sup.2,H.sub.t], t [member of] [0,[infinity]),

where ([B.sup.1,H], [B.sup.2,H]) is a two dimensional FBM process with the Hurst parameter H [member of] (1/2,1) and [kappa] > 0, [theta] > 0, [mu] [member of] R and [sigma] > 0 are constants. We also suppose that [V.sub.0] > 0 and [S.sub.0] > 0 are given. Recall that, as in the classical Heston model, we could suppose that there exists a constant [rho] [greater than or equal to] 0 satisfying

(2.5) [mathematical expression not reproducible].

This implies that the correlation between two increments of [B.sup.1,H] and [B.sup.2,H] per unit of time is the constant [rho]. Nevertheless, we do not need this assumption is this paper.

The following theorem, which is the main goal of this paper, states that the system of SDEs above admits a unique solution. By definition, we call it the price dynamics of the risky asset S in the fractional Heston model. Moreover, Equation (2.4) defines a fractional version of the CIR model. We shall prove below that it is a mean- reverting stochastic process, i.e. [lim.sub.t[right arrow][infinity]] [EV.sub.t] = [theta].

The proof of existence and uniqueness is based on the study of singular equations driven by rough paths given in Section 3. This generalizes results of  that allow to consider singular equations driven by a FBM as done in Section 4.

Theorem 2.4. The S.D.E's (2.3) and (2.4) admit unique positive solutions.

Proof. Let us introduce W = [[sigma].sup.-2]V. Then, (2.4) reads as

d[W.sub.t] = k([[sigma].sup.-2] [theta] - [W.sub.t])dt + [square root of [W.sub.t]] d[B.sup.2,H.sub.t], t [member of] [0,T].

This is the singular equation (4.2) of Section 4 with g(t,x) = [kappa][[sigma].sup.-2] ([theta] - [[sigma].sup.2]x). This function satisfies the required Condition [C.sub.g] hence the S.D.E. (2.4) admits a unique positive solution by Theorem 4.6. Notice that the integral with respect to [B.sup.2,H] is a pathwise Young integral, see . Moreover, V is almost surely continuous and admits finite moments of all orders.

In order to show that (2.3) admits a unique positive solution, we shall apply the results of [16, Section 5.3.3] to the S.D.E.

(2.6) d[Y.sub.t] = [mu]dt + [square root of [V.sub.t]] d[B.sup.1,H.sub.t], [Y.sub.0] = 0.

It suffices to verify that the two functions [sigma](t,x) = [square root of [V.sub.t]] and b(t,x) = [mu] satisfy [16, Conditions [H.sub.1], [H.sub.2], Section 5.3.2] for some constants that may depend on [omega] [member of] [OMEGA]. The only difficulty is to show that the process [square root of V] is Holder continuous of order H. As V is positive and a.s. bounded on every interval [0, T], T > 0, this is equivalent to show that V or W are Holder continuous of order H. By the proof of Theorem 4.6, W = [X.sup.2]/4 where X is the positive process

(2.7) [mathematical expression not reproducible].

As V is positive and a.s. bounded, the process s [??] [X.sup.-1.sub.s] g(s, [X.sup.2.sub.s]/4) is a.s. bounded on [0, T] hence the integral process in the expression of X is Lipschitz. We deduce that X is Holder continuous of same order H than [B.sup.2,H]. Since X is a.s. bounded on [0,T], we get that W is Holder continuous of same order H. We conclude that the S.D.E. (2.6) admits a unique solution Y on each interval [0,T] hence it is possible to conclude on [0, [infinity]).

Therefore, by the change of variable formula [16, Section 5.2.2], we deduce that the process S := [S.sub.0][e.sup.Y] satisfies (2.3) and is positive. Reciprocally, suppose that S is a.s. positive and satisfies (2.3). Consider [omega] [member of] [OMEGA] such that [alpha]([omega]) := [min.sub.t[member of][0,T]] [S.sub.t]([omega])/[S.sub.0] > 0 where T > 0. We construct a function [gamma] on R which is twice differentiable and satisfies [gamma](x) = log(x) for all x [greater than or equal to] a(u). Applying the change of variable formula to the deterministic function [gamma](S([omega])/[S.sub.0]), i.e. for [omega] fixed, we deduce that the function t [??] Y([S.sub.t](u)/[S.sub.0]) satisfies the same S.D.E. (2.6) than Y in the pathwise sense hence [gamma](S([omega])/[S.sub.0]) = Y([omega]) by [16, Theorem 5.3.1] and [16, Section 5.3.3]. As [gamma](S([omega])/[S.sub.0]) = log(S([omega])/[S.sub.0]), we deduce that S = [S.sub.0][e.sup.Y] a.s. We then conclude as T [right arrow] [infinity].

In the classical Heston model, the volatility process satisfies the stochastic differential equation (2.4) with H =1/2. We then deduce that t [??] [EV.sub.t] satisfies an o.d.e. of first order and finally

[EV.sub.t] = ([V.sub.0] - [theta])[e.sup.-Kt] + [theta], t [greater than or equal to] 0,

so that [theta] = [lim.sub.T[right arrow][infinity]] [EV.sub.t] . We say that V is mean-reverting. Note that the increments of V are not weakly stationary by Lemma 4.2. By Corollary 4.8, we deduce the following result:

Proposition 2.5. The fractional volatility process V, solution to Equation (2.4), satisfies

(2.8) [EV.sub.t] = ([V.sub.0] - [theta])[e.sup.-[kappa]t] + 9 + [e.sup.- [kappa]t] [[delta].sub.H] (t),

where [[delta].sub.H] is a differentiable function which satisfies [[delta].sub.H](t) G (0, [[sigma].sup.2][t.sup.2H]/2)) for all t [greater than or equal to] 0.

This implies that the variance V of the fractional Cox-Ingersoll-Ross (CIR) process (2.4) is larger when H > 1/2 than it is when H = 1/2. Nevertheless, we still have [lim.sub.t[right arrow][infinity]] [EV.sub.t] = [theta].

We leave for further research a deeper study of the FHM regarding long range dependence, as well as discretization of the process and pricing in finance with this model.

3. SINGULAR EQUATIONS DRIVEN BY ROUGH PATHS

For any s [less than or equal to] t, we consider the Banach space of continuous functions C([s,t]) equipped with the topology of the supremum norm we denote by [[parallel]f[parallel].sub.[s,t]], f [member of] C([s, t]). When a continuous function is defined on a subset I of R, we naturally extend the notation by denoting its supremum by [parallel]f[[parallel].sub.I]. The space of Holder continuous functions of order [beta] > 0 is denoted by [C.sup.[beta]] ([s,t]) and its norm is

[mathematical expression not reproducible]

We consider the deterministic differential equation driven by a rough path [phi] of :

(3.1) [x.sub.t] = [x.sub.0] + [[integral].sup.t.sub.0](s, [x.sub.s])ds + p(t), t [member of] [0, [infinity]),

where [x.sub.0] > 0 is a constant, [phi] is continuous and [phi](0) = 0. We impose conditions on f.

Definition 3.1. We say that a function f = f (t,x), (t,x) [member of] [R.sub.+] x (0, to) is locally Lipschitz with respect to the space variable x on x [member of] (0, [infinity]) if for all c, C > 0 such that c < C, and T > 0, we have [absolute value of (f (t, x) - f (t, y))] [less than or equal to] [L.sub.T,c,C] [absolute value of (x - y)] whatever x,y [member of] [c, C] and t [less than or equal to] T for some Lipschitz constant [L.sub.T,c,C] depending on T, c, C.

Condition [C.sup.1.sub.f]:

1. f is a locally Lipschitz function with respect to the space variable x on x [member of] (0, [infinity]).

2. There exists constants m, [gamma] [greater than or equal to] 0 such that f (t,x) [greater than or equal to] - m - [gamma]x for all x > 0 and t [greater than or equal to] 0.

3. The mapping t [??] f (t,x) is continuous for all x > 0.

Note that the conditions we impose on f are weaker than the conditions of . In particular, f is not necessarily differentiable with respect to x [member of] (0, [infinity]) and x [??] f (t,x) is not necessarily non increasing. The following condition is also considered when [phi] [member of] [C.sup.[beta]] ([0,T]) for all T [greater than or equal to] 0, where [beta] [member of] (0,1), to obtain a solution to (3.1) on the whole interval [0, [infinity]).

Condition [C.sup.[beta].sub.f]: For every T > 0, there exists [epsilon] > 0, [[gamma].sub.0] > 0, [alpha] [member of] ([[beta].sup.-1] - 1, [[beta].sup.-1]) and positive constants c, d, such that f (t,x) [greater than or equal to] [cx.sup.-[alpha]] - d for all t [member of] [T - [epsilon], T] and x [member of] (0, [[gamma].sub.0]].

Theorem 3.2. Let f = f (t,x), (t,x) [member of] [R.sub.+] x(0, [infinity]), be a function satisfying condition [C.sup.1.sub.f]. Then, there exists a unique positive solution to (3.1) on some maximal interval [0,T*) such that [mathematical expression not reproducible] exists if T* < [infinity]. Moreover, if [phi] [member of] [C.sup.[beta]] ([0,T]) for all T > 0, then T* = [infinity] under Condition [C.sup.[beta].sub.f].

Proof. By the assumptions on f, note that for all c, C such that 0 < c [less than or equal to] C, and for all [DELTA] [greater than or equal to] 0, we have [[parallel]f[parallel].sub.[0,A]x[c,C]] < [infinity]. To see it, we use the local Lipschitz property of f with respect to the space variable as well as the continuity of f with respect to the time variable. Note that we may reformulate the problem if we replace f by f + m [disjunction] d and [phi] by [??](t) = [phi] - (m [disjunction] d)t so that we may assume without loss of generality that m = d = 0. Let us consider a fixed constant C such that C [greater than or equal to] 3[x.sub.0]. It is also possible to find [T.sub.0] [member of] (0,1) small enough so that, by uniform continuity, [mathematical expression not reproducible] and, as [T.sub.0] [right arrow] 0, we have [mathematical expression not reproducible] in the case where [gamma] > 0 in Condition [C.sup.1.sub.f]. Consider the following iterative scheme:

(3.2) [mathematical expression not reproducible]

where [x.sup.0.sub.t] = [x.sub.0] for all t [member of] [0, [T.sub.0]]. Let us show by induction that [x.sup.n.sub.t] [member of] [[2.sup.-1] [x.sub.0], C] for all t [member of] [0, [T.sub.0]]. This is the case with n = 0. Suppose that this holds with [x.sup.n]. As [x.sup.n] > 0, we deduce by assumption with m = 0 that f (s, [x.sup.n.sub.s]) [greater than or equal to] -[gamma][x.sup.n.sub.s] for all s [member of] [0,[T.sub.0]] hence

[mathematical expression not reproducible].

Moreover, [phi](t) [greater than or equal to] -[4.sup.-1][x.sub.0] by construction of [T.sub.0] hence

[x.sup.n+1.sub.t] [greater than or equal to] [x.sub.0] - [4.sup.-1] [x.sub.0] + [phi](t) [greater than or equal to] [2.sup.-1] [x.sub.0].

Let us show that [mathematical expression not reproducible]. We have

[mathematical expression not reproducible].

The last inequality is deduced from the conditions imposed on C and [T.sub.0]. Let us define

[g.sub.n](t) = [absolute value of ([x.sup.n+1.sub.t] - [x.sup.n.sub.t])], t [member of] [0, [T.sub.0]], n [greater than or equal to] 0.

[g.sub.n](t) = [absolute value of ([x.sup.n+1.sub.t] - [x.sup.n.sub.t])]. t [member of] [0, [T.sub.0]], n [greater than or equal to] 0.

By continuity, note that [mathematical expression not reproducible] < [infinity]. Since f is locally Lipschitz with respect to the space variable on (0, [infinity]), let us consider a constant k = k([T.sub.0], [x.sub.0], C) such that [absolute value of (f (s, x) - f (s, y))] [less than or equal to] k[absolute value of (x - y)] for all x, y [member of] [[2.sup.-1] [x.sub.0], C] and s [member of] [0, [T.sub.0]]. By (3.2), we get that

[g.sub.n](t) = [absolute value of ([x.sup.n+1.sub.t] - [x.sup.n.sub.t])], t [member of] [0, [T.sub.0]], n [greater than or equal to] 0.

By induction, we then deduce that [mathematical expression not reproducible] for all t [member of] [0,[T.sub.0]]. Therefore, the sequence [g.sub.n+1] = [g.sub.1] + [[summation].sup.n.sub.i=0] ([g.sub.i+1] - [g.sub.i]) is absolutely convergent with respect to the supremum norm hence uniformly converges to [mathematical expression not reproducible] on [0, [T.sub.0]]. Moreover, it is trivial that [x.sup.(T)] satisfies (3.1) on [0,[T.sub.0]].

Let us now prove that the equation (3.1) admits a unique positive solution on every compact [0,T], T > 0. To see it, consider two solutions x and y and let us consider [mathematical expression not reproducible]. We then deduce that t

[mathematical expression not reproducible],

where k = k(T, c, C) is a constant such that [absolute value of (f (s, x) - f (s, y))] [less than or equal to] k [absolute value of (x - y)] for all x,y [member of] [c, C] and s [member of] [0,T]. By induction, we deduce that [[parallel]x - y[parallel].sub.[0,t]] [less than or equal to] 2C[k.sup.n][t.sup.n]/n! for all t [less than or equal to] T. As n [right arrow] [infinity], we get that [parallel]x - y[parallel] [0,T] = 0 hence x = y.

Consider the set [LAMBDA] [contains as member] [T.sub.0] of all T > 0 such that (3.1) admits a unique positive solution [x.sup.(T)] on [0,T]. Note that if [T.sub.1], [T.sub.2] [member of] [LAMBDA] satisfy [T.sub.1] [less than or equal to] [T.sub.2], then [mathematical expression not reproducible] coincides on [0,[T.sub.1]] by uniqueness. Therefore, we may define the function [x.sub.t] := [x.sup.(t).sub.t] on t [member of] [0,[T.sup.*]) where [T.sup.*] = sup [LAMBDA]. This function satisfies the equation (3.1) on [0,T*) and is uniquely defined and positive.

If [T.sup.*] = [infinity], we may conclude about the lemma. Otherwise, we show that [mathematical expression not reproducible] exists and [mathematical expression not reproducible]. Indeed, in the contrary case, there exists l > 0 and a sequence [([t.sub.n]).sub.n[greater than or equal to]1] such that [mathematical expression not reproducible] as n [right arrow] to. We may assume without loss of generality that [mathematical expression not reproducible]. Fix a constant C [greater than or equal to] 6l and [T.sup.*.sub.0] > 0 small enough such that [T.sup.*.sub.0] [less than or equal to] l[(4[gamma]C).sup.-1] if [gamma] > 0, such that [mathematical expression not reproducible]. Note that [T.sup.*.sub.0] does not depend on n. Consider an arbitrary n [greater than or equal to] 1. Let us introduce the following scheme: [mathematical expression not reproducible] and for m [greater than or equal to] 1:

(3.3) [mathematical expression not reproducible].

As above, we may show by induction that [x.sup.m.sub.t] [member of] [[2.sup.- 1]l, C] for all t [member of] [[t.sub.n], [t.sub.n] + [T.sup.*.sub.0]] and all m [greater than or equal to] 1. Then, we also deduce that the sequence [([x.sup.m]).sub.m[greater than or equal to]1] uniformly converges to some function [z.sup.n] on [[t.sub.n], [t.sub.n] + [T.sup.*.sub.0]] such that

(3.4) [mathematical expression not reproducible].

We then define [mathematical expression not reproducible] on [0,[t.sub.n] + [T.sup.*.sub.0]] where we recall that x is the solution to Equation (3.1) on [0,[T.sup.*]). Note that [[??].sup.n] satisfies Equation (3.1) on [0, [t.sub.n] + [T.sup.*.sub.0]]. Therefore, by definition of [T.sup.*], we deduce that [t.sub.n] + [T.sup.*.sub.0] [less than or equal to] [T.sup.*] for all n. As [T.sup.*.sub.0] does not depend on n, we deduce as n [right arrow] [infinity] that [T.sup.*.sub.0] [less than or equal to] 0 hence a contradiction.

Let us now suppose that Condition [C.sub.[beta]] holds and [T.sup.*] < [infinity]. Then, [mathematical expression not reproducible] since we assume without loss of generality that d = 0. We deduce a contradiction by repeating the arguments in the proof of [11, Theorem 2.1].

We may reproduce the proof of [11, Theorem 2.1] under our assumptions and under a weaker assumption than [11, (iii)]:

Condition [C.sup.2.sub.F]: For all T > 0, there exists positive constants [h.sub.T] and [[epsilon].sup.1.sub.T], [[epsilon].sup.2.sub.T] > 0 such that [absolute value of (f(t,x))] [less than or equal to] [h.sub.T] ([x.sup.-1] + 1) for all x [member of] (0, [[epsilon].sup.1.sub.T]) and [absolute value of (f(t,x))] [less than or equal to] [h.sub.T] (x + 1) for all x [member of] ([[epsilon].sup.2.sub.T], [infinity]) and t [less than or equal to] T.

Theorem 3.3. Let f be a function defined on [R.sub.+] x (0, [infinity]) which satisfies Conditions [C.sub.[beta]] and [C.sup.i.sub.f], i = 1, 2. Then, for any [gamma] > 2 and for all T > 0, the unique solution x to (3.1) satisfies

(3.5) [mathematical expression not reproducible],

where [C.sub.1,[gamma],[beta],T] and [C.sub.1,[gamma],[beta],T] are constants depending on [gamma], [beta], T and f.

Proof. We exactly follow the proof of [11, Theorem 2.1] but we do not use the assumption (iii) of . With y = [x.sup.[gamma]], recall that (iii) is used in  to obtain the following inequality: [mathematical expression not reproducible] where c is a constant depending on the function h of (iii). Instead, since / is locally Lipschitz, there exists under Conditions [C.sup.i.sub.f] = 1, 2, a constant [L.sub.T,f] depending on f and T such that [absolute value of (f(t,x))] [less than or equal to] [L.sub.T,f] (x + 1) for all x [member of] [[[epsilon].sup.1.sub.T], [infinity]) and t [less than or equal to] T. Therefore, we deduce that [mathematical expression not reproducible]. Otherwise, if [y.sup.1/[gamma].sub.u] [less than or equal to] [[epsilon].sup.2.sub.T], we have the inequality [mathematical expression not reproducible]. Finally, for all u [less than or equal to] T, we deduce the inequality [mathematical expression not reproducible] for some constant L(f, T) depending on f and T. This implies that, in the proof of [11, Theorem 2.1], we replace [[parallel]y[parallel].sup.1-2/[gamma].sub.[s,t]] by the sum [mathematical expression not reproducible]. This substitution does not change the desired inequality because, as in the proof of [11, Theorem 2.1], we use the inequality [x.sup.[alpha]] [less than or equal to] 1 + x for all [alpha] [member of] (0,1) to bound from above the three terms [mathematical expression not reproducible].

4. SINGULAR EQUATIONS DRIVEN BY A FRACTIONAL BROWNIAN MOTION WITH HURST PARAMETER H [member of] (1/2,1)

Let us consider the singular stochastic differential equation

(4.1) [X.sub.t] = [x.sub.0] + [[integral].sup.t.sub.0] (s, [X.sub.s])ds + [B.sup.H.sub.t], t [greater than or equal to] 0,

where [x.sub.0] > 0, [B.sup.H] is a fractional Brownian motion with Hurst parameter H [member of] (1/2,1) and f is function which satisfies Conditions and [C.sub.[beta]], [C.sup.i.sub.f] =1, 2. Repeating the proof of [11, Theorem 3.1] with the results of Section 3, we obtain the following:

Theorem 4.1. Suppose that [x.sub.0] > 0 and f is a function which satisfies Conditions [C.sub.[beta]] and [C.sup.i.sub.f], i = 1,2. Then, there exists a unique positive pathwise solution X to Equation (4.1) such that E([[parallel]X[parallel].sup.p.sub.[0,T]]) < [infinity] for all p > 0 and T [greater than or equal to] 0.

The following result is classical; we use it to show Theorem 4.3, which implies that the increments of X are not stationary at least when f (0, [X.sub.0]) < 0.

Lemma 4.2. Let X be an integrable process on [R.sub.+] and consider the function [phi](t) = E[X.sub.t] - [X.sub.0], t [greater than or equal to] 0. If the increments are weakly stationary, then [phi] is additive. Therefore, if [phi] is continuous on [R.sub.+], [phi](t) = at for all t [greater than or equal to] 0 where [alpha] is a constant.

Proof. As the increments are weakly stationary, we easily deduce that [phi] is additive, i.e. [phi](t + h) = [phi](t) + [phi](h). By induction, we deduce that [phi](nt) = n[phi](t) for all n [member of] N. If a,b [member of] N, with b > 0, we get that b[phi] ((a/b)t) = [phi](at) = a[phi](t). This implies that [phi](qt) = q[phi](t) for all non negative rational numbers q. We then conclude by density and continuity with [phi](1) = [alpha].

Theorem 4.3. Suppose that [x.sub.0] > 0 and f is a function which satisfies Conditions [C.sub.[beta]] and [C.sup.i.sub.f], i = 1, 2. Consider the positive solution X to Equation (4.1). If the increments of X are weakly stationary and, if the mapping t [??] Ef (t,[X.sub.t]) is continuous at 0, then E[X.sub.t] = [X.sub.0] + f (0,[X.sub.0])t for all t [greater than or equal to] 0. In particular, f (0, [X.sub.0]) [greater than or equal to] 0.

Proof. The function [phi] defined by

[mathematical expression not reproducible]

is differentiable at zero and we have [phi]'(0) = f (0, [X.sub.0]). Moreover, E[X.sub.t] = [X.sub.0] + [phi](t). As E ([[parallel]X[parallel].sub.[0,T]]) < [infinity] for all T [greater than or equal to] 0, we deduce that the mapping t [??] E[X.sub.t] is continuous and so is [phi]. Suppose that the increments of X are weakly stationary. By Lemma 4.2, [phi] is linear hence we have [phi](t) = f(0, [x.sub.0])t for all t [greater than or equal to] 0.

Note that the mapping t [??] Ef (t, [X.sub.t]) is continuous at 0 in the following case.

Corollary 4.4. Suppose that f (t,x) = f (x) = a[x.sup.-1] - bx where a > 0 and b [member of] R. Then, the mapping t [??] Ef (t, [X.sub.t]) is continuous at 0. Therefore, when b > 0 and [x.sup.2.sub.0] > [ab.sup.-1], the increments are not weakly stationary.

Proof. As E([[parallel]X[parallel].sub.[0,T]]) < [infinity] for all T [greater than or equal to] 0, we deduce that the mapping t [??] E[X.sub.t] is continuous at zero. Moreover, following the arguments used in the proof of [11, Proposition 3.4], we get that E[X.sup.-1.sub.0] [less than or equal to] [X.sup.- 1.sub.0] if t [member of] [0, [t.sub.0]] where [t.sub.0] > 0 is small enough. Indeed, it suffices to notice that f'(x) is bounded from above, which implies that the Malliavin derivative [([D.sub.s][X.sub.t]).sub.0[less than or equal to]s[less than or equal to]t] is a bounded process so that Proposition 3.4 is still valid in our more general case. Finally, by the Jensen inequality, we get that [(E[X.sub.t]).sup.-1] [less than or equal to] E[[X.sub.t].sup.-1] [less than or equal to] [X.sup.-1.sub.0] if t [member of] [0,[t.sub.0]]. Since the mapping t [??] E[X.sub.t] is continuous at zero, we finally deduce that [lim.sub.t[right arrow]0] E[X.sup.-1.sub.t] = [X.sup.-1.sub.0]. This implies that the mapping t [??] Ef (t, [X.sub.t]) is continuous at 0. When b > 0 and [x.sup.2.sub.0] > [ab.sup.-1], we get that f ([x.sub.0]) < 0 so that the increments are not weakly stationary by Theorem 4.3.

Remark 4.5. In the case where t [??] E[X.sup.-1.sub.t] is continuous and f (t,x) = f (x) = a[x.sup.-1] - bx, we may show that

[mathematical expression not reproducible].

Let us now consider the singular stochastic differential equation

(4.2) [mathematical expression not reproducible]

where [y.sub.0] > 0 and g is a function which satisfies the following conditions denoted by Condition [C.sub.g]. Notice that the conditions imposed on g in  are not correctly formulated as the authors made a small error when defining f in terms of g. When corrected, our conditions remain weaker.

Condition [C.sub.g]:

1. The mapping t [??] g(t, x) is continuous on [0, [infinity]) for all x > 0.

2. The function g is locally Lipschitz with respect to the space variable x on x [member of] (0, [infinity]) and there exists constant m, [alpha] [greater than or equal to] 0 such that g(t, x) [greater than or equal to] -m[x.sup.1/2] - [alpha]x for all x > 0 and t [greater than or equal to] 0.

3. For every T > 0, there exists [epsilon] > 0, [[gamma].sub.0] > 0, [alpha] [member of] ([[beta].sup.-1] - 1, [[beta].sup.-1]) and positive constants c, d, such that g(t,x) [greater than or equal to] c[x.sup.1-[alpha]/2] - d[x.sup.1/2] for all t [member of] [T - t,T] and x [member of] (0, [[gamma].sub.0]].

4. For all T > 0, there exists positive constants [h.sub.T] and [[epsilon].sup.1.sub.T], [[epsilon].sup.2.sub.T] such that we have [absolute value of (g(t,x))] [less than or equal to] hT(1 + [x.sup.1/2]) for all x [member of] (0,[[epsilon].sup.1.sub.T]), and [absolute value of (g(t,x))] [less than or equal to] [h.sub.T](x + [x.sup.1/2]) for all x [member of] ([[epsilon].sup.2.sub.T], [infinity]) and t [less than or equal to] T.

Theorem 4.6. Let g be a function satisfying Condition [C.sub.g]. Then, Equation (4.2) admits a unique positive pathwise solution Y such that E([[parallel]Y[parallel].sup.p.sub.[0,T]]]) < [infinity] for all p > 0.

Proof. Let us consider the function [f.sup.g](t, x) = 2[x.sup.-1]g(t, [x.sup.2]/4) for all t [greater than or equal to] 0 and x > 0, i.e. g(t, y) = [f.sup.g](t, 2[y.sup.1/2])[y.sup.1/2]. Using the change of variable x = 2[y.sup.1/2], the chain rule for young integrals yields that Y is a positive solution to (4.2) if and only if X = 2[Y.sup.1/2] is a positive solution to (4.1) with the driver function [f.sup.g]. As [f.sup.g] satisfies the conditions of Theorem 3.2 as well as Conditions [C.sub.[beta]] and [C.sup.f] if and only if g satisfies Condition [C.sub.g], we deduce that Equation (4.2) admits a unique positive pathwise solution Y given by Y = [X.sup.2]/4 where X is the unique positive solution to (4.1) with the driver function [f.sup.g].

Note by [16, Proposition 5.2.3], under Condition [C.sub.g], we may estimate the expectation of Y as follows:

(4.3) [mathematical expression not reproducible],

(4.4) [mathematical expression not reproducible],

where D is the Malliavin derivative operator, X = 2[square root of Y] is the solution of Equation (4.1) with f = [f.sup.g] and

[f.sup.g](t,x) = 2[x.sup.-1]g(t,[x.sup.2]/4), t [greater than or equal to] 0, x > 0.

Moreover, if fg(t, x) = f (x) is differentiable, we have

(4.5) [mathematical expression not reproducible].

We deduce the following:

Theorem 4.7. Let g = g(x) be a function satisfying Condition [C.sub.g] and let Y be the unique positive solution to Equation (4.2). Suppose that the derivative of [f.sup.g] is bounded from above and the mapping t [??] Eg([Y.sub.t]) is continuous at zero. Then, if the increments of Y are weakly stationary, we necessarily have E[Y.sub.t] = [y.sub.0] + vt for all t [greater than or equal to] 0, where 0 [less than or equal to] v [less than or equal to] g([Y.sub.0]).

Proof. As E([[parallel]Y[parallel].sup.p.sub.[0,T]]) < [infinity] for every T [greater than or equal to] 0, the function [phi](t) = E[Y.sub.t] - [X.sub.0], t [greater than or equal to] 0 is linear hence E[Y.sub.t] = [y.sub.0] + vt for some v [greater than or equal to] 0 by Lemma 4.2. Moreover, by (4.3) and (4.5), we deduce a constant C > 0 such that E[Y.sub.t] [less than or equal to] [y.sub.0] + [[integral].sup.t.sub.0]Eg([Y.sub.r])dr + [Ct.sub.2H] for all t [less than or equal to] 1. We divide this inequality by t > 0 so that

[mathematical expression not reproducible].

As t [right arrow] 0, we deduce that v [less than or equal to] g([y.sub.0]).

The following result implies that the increments of Y are not weakly stationary if g is an affine function such that g(0) > 0.

Lemma 4.8. Suppose that g(x) = [alpha] - [beta]x, x > 0, where [alpha] > 0 and [beta] [member of] R. Then,

(4.6) [mathematical expression not reproducible],

with the convention (1 - [e.sup.-[beta]t])/[beta] = t when [beta] = 0. Moreover, when [beta] [greater than or equal to] 0, 0 [less than or equal to] [delta](t) [less than or equal to] [2.sup.-1][t.sup.2H] for all t [greater than or equal to] 0.

Proof. Note that the derivative of [f.sup.g] (x) = a[x.sup.-1] - bx, a = 2[alpha] and b = 2-1[beta] is bounded from above on (0, [infinity]). We deduce that [([D.sub.s][X.sub.t]).sub.s[less than or equal to]t] is bounded on any interval [0,T], T > 0. Moreover, note that t [??] [D.sub.s][X.sub.t] is continuous except at the point t = s. By the dominated convergence theorem, we deduce that the mapping p is continuous on any interval [0,T]. Therefore, [delta] is differentiable and [delta]' = p. Moreover, by (4.3), [phi](t) = E[Y.sub.t] satisfies the o.d.e. [phi]'(t) + [beta][phi](t) = p(t) + [alpha]. We then conclude. When [beta] [greater than or equal to] 0, 0 [less than or equal to] [D.sub.s][X.sub.t] [less than or equal to] 1 hence we deduce that 0 [less than or equal to] [delta](t) [less than or equal to] [2.sup.-1][t.sup.2H] for all t [greater than or equal to] 0.

REFERENCES

 G. S. Bakshi, C. Cao and Z. Chen. Empirical performance of alternative option pricing models. Journal of Finance, 52, 5, 2003-2049 (1997).

 H. Bessaih, M.J Garrido-Atienza and B. Schmalfuss. Stochastic shell models driven by a multiplicative fractional Brownian-motion. Physica D: Nonlinear Phenomena, 320, 38-56 (2016).

 F. Black, M. S. Scholes. The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637-654 (1973).

 B. L. S. Prakasa Rao. Pricing geometric Asian power options under mixed fractional Brownian motion environment. Physica A: Statistical Mechanics and its Applications, 446, 92-99 (2016).

 A. Chronopoulou and F. G. Viens. Estimation and pricing under long-memory stochastic volatility. Annals of Finance, 8, 2, 379-403 (2012).

 R. Cont. Long range dependence in financial markets. Fractals in Engineering, 4, 159-179, Springer-Verlag London (2005).

 M. Contreras, R. Pellicer, M. Villena and A. Ruiz. A quantum model of option pricing: When Black-Scholes meets Schrodinger and its semi-classical limit. Physics A: Statistical Mechanics and its Applications, 389, 23, 5447-5459 (2010).

 Fouque J. P., Papanicolaou G. and Sircar K. R. Derivatives in financial markets with stochastic volatility. Cambridge University Press, UK (2000).

 M. Gubinelli, A. Lejay and S. Tindel. Young integrals and SPDEs. Potential Analysis, 25, 4, 307-326 (2006).

 S. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6 (2) 327-343 (1993).

 Y. Hu, D. Nualart and X. Song. A singular stochastic differential equation driven by fractional Brownian motion. Statistics & Probability Letters, 78, 14, 2075-2085 (2008).

 A. N. Kolmogorov. Wienersche Spiralen und einige andere interessante Kurven in Hilbertschen Raum. Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS (26), 115- 118 (1940).

 E. Lepinette and T. Tran. Approximate hedging in a local volatility model with proportional transaction costs. Applied Mathematical Finance, 21, 4, 313-341 (2014).

 A. Lewis. Option valuation under stochastic volatility with Mathematica code. Finance Press,

Newport Beach, California USA (2000).

 B. B. Mandelbrot and J. Van Ness. Fractional Brownian motion, fractional noises and applications. SIAM Review, 10 (4), 422-437(1968).

 D. Nualart. The Malliavin calculus and related topics. Probability and its Applications. Springer Berlin Heidelberg (2006).

 L. C. G Rogers and D. Williams. Diffusions, Markov processes and martingales. Cambridge Mathematical Library. Cambridge University press, Cambridge. Vol. 1: foundations. Second edition (2000).

 S. Rostek. Option pricing in fractional Brownian markets. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag Berlin Heidelberg. Vol. 622 (2009).

 G. Samorodnitsky and M. Taqqu. Stable non-gaussian random processes. Stochastic modelling. Chapman & Hall, Boca Raton, London, New York, Washington, D.C. (1994).

 W. Willinger, M. S. Taqqu and V. Teverovsky. Stock market prices and long- range dependence. Finance and Stochastic, 3, 1, 1-13 (1999).

 Yuliya S. Mishura. Stochasic calculus for fractional brownian motion and related process. Lecture Notes in Mathematics. Springer-Verlag Berlin Heidelberg (2008).

EMMANUEL LEPINETTE AND FARSHID MEHRDOUST

Ceremade, CNRS 7534, PSL National Research, Paris-Dauphine University, France Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P. O. Box: 41938-1914, Rasht, Iran