Correlations and bounds for stochastic volatility models

www.elsevier.com/locate/anihpc

Correlations and bounds for stochastic volatility models

P.-L. Lions

, M. Musiela

aCollège de France, 3, rue d’Ulm, 75005 Paris, France

bCEREMADE – UMR C.N.R.S. 7534, Université Paris-Dauphine, place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France cFIRST, BNP-Paribas, 10 Harewood Avenue, London NW1 6AA, United Kingdom

Received 22 May 2005; accepted 26 May 2005 Available online 9 November 2006

Abstract

We investigate here, systematically and rigorously, various stochastic volatility models used in Mathematical Finance. Mathe- matically, such models involve coupled stochastic differential equations with coefficients that do not obey the natural and classical conditions required to make these models “well-posed”. And we obtain necessary and sufficient conditions on the parameters, such as correlation, of these models in order to have integrable orL^psolutions (for 1< p <∞).

©2005 Published by Elsevier Masson SAS.

1. Introduction

This paper is the first of a series devoted to the rigorous mathematical analysis of (some of) the most popular and classical stochastic volatility models in Finance (see [1–4] and [7–9]), some of which are indeed used by financial engineers for practical purposes.

In this paper, we present sharp conditions (most of the time, necessary and sufficient conditions) under which these models are meaningful – in a sense that will be made precise later on. And we shall show that the correlation parameter isthecrucial parameter that makes the model meaningful or not. Let us immediately point out that we shall investigate, in the following articles of this series, various issues such as the long time behaviour of option prices, the well-posedness for fractional power models and semi-explicit formulas, or short time asymptotics. . . .

At this stage, let us detail the type of mathematical information we shall derive. And we begin with the following particular case – that we treat in Section 2 below together with some more general volatility equations. . .

dFt=σ_tF_tdWt, F0=F >0, (1)

dσt=ασtdZt, σ0=σ >0 (2)

whereZ_t=ρW_t+

1−ρ²B_t;(W_t, B_t)is a standard two-dimensional Brownian motion,α >0 andρ∈ [−1,+1]is the correlation parameter.

Although (1), (2) is a very classical stochastic volatility model “used” in Mathematical Finance, we are not aware of much rigorous mathematical analysis of the stochastic system (1), (2). And, surprisingly enough, we show here that

* Corresponding author.

E-mail address:lions@ceremade.dauphine.fr (P.-L. Lions).

0294-1449/$ – see front matter ©2005 Published by Elsevier Masson SAS.

doi:10.1016/j.anihpc.2005.05.007

stringent assumptions on the correlationρ are necessary to make it a meaningful system! More precisely, we show below the following facts (after defining in a straightforward wayFt. . . ):

– F_t is, for allρ, a continuous positive, integrable supermartingale;

– F_t is a martingale if and only ifρ0 (ifρ >0,E[F_t]< F for allt >0!);

– Ifρ=0,F_tis a continuousintegrablemartingale such that sup_t∈[0,T]E[F_t|LogF_t|]<0, sup_t∈[0,T]E[F_t^m] = +∞

for anym >1, for allT ∈(0,+∞);

– Letm >1, then sup_t∈[0,T]E[F_t^m]<∞for allt∈(0,∞)if and only ifρ−√

(m−1)/m;

– Ifρ= −1, thenF_t is bounded for allt0.

Additional informations are provided below. Let us only mention here that the difficulty with the system (1), (2) stems from the fact that the mapping [(σ, F )→(ασ, σ F )] does not satisfy the usual condition of linear growth at infinity. A slightly more precise interpretation consists in recalling the Novikov criterion (which is sufficient to guarantee the fact that “Ft” is a martingale, see for instance [5]. . . ) namely

E

exp 1

2 T 0

σ_t²dt

<∞, for allT >0.

However, one can check that this never holds (for anyρ) ifσ solves (2)!

The conclusions for our study in Section 2 are thus:

(i) the model (1), (2) is not, in general, well-posed and

(ii) in order to use a well-posed model, and if we wish to be able to manipulate standard objects such as the variance ofF for instance, one needs to assume (at least) that−1ρ−1/√

2 (takem=2 in the above results. . . ).

The proof of the above results is given in Section 2 (2.1–2.4). The final Section 2.5 is devoted to the extension of these results to the case of a general stochastic equation forσt in face of (2).

Section 3 is devoted to the study of general models of the following type

dFt=σ_t^δF_t^βdWt, (1)

dσt=ασ_t^γdZt+b(σ_t)dt (2)

whereδ,β,γ are positive parameters (with some natural restrictions that we do not detail here) andbrepresents a trend (=drift) that could be for instance:b(σ )= −b(σ−σ_∗)for some givenb, σ_∗>0.

We shall then give and prove necessary and sufficient conditions for the well-posedness of (1), (2) in terms ofδ, β,γandρ(the correlation betweenW_tandZ_t). Once more, we shall also briefly mention extensions to more general expressions than power laws in (1), (2).

2. Log-normal like models 2.1. Preliminaries

We first wish to derive explicit formulae forσ andF. As is well-known, the solution σ of (2) is a continuous martingale (integrable to all powers. . . ) given by

σt=σexp

αZt−α²

2 t , fort0. (3)

Then, at least formally, we expect “the” solution of (1) to be given by F_t=Fexp

t 0

σ_sdWs−1 2

t 0

σ_s²ds

, fort0, (4)

which is a continuous positive process. This, however, needs to justified. And we do so by a classical localisation argument. We introduce, for instance, for alln1

τ_n=inf{t0|σ_t> n}. (5)

Of course,τ_n→

n +∞a.s. and solving (1) “up to timeτ_n” yields F_t∧τn=F_tⁿ=Fexp

t 0

σ_s1(s<τn)dWs−1 2 t 0

σ_s²1(s<τn)ds

, ∀t0. (6)

AndF_tⁿconverges a.s. toF_t given by (4) (for allt0)asngoes to+∞.

This allows to show the following easy (and general) fact

Proposition 2.1.F_t is a continuous, positive, integrable supermartingale.

Proof. The integrability is a trivial consequence of Fatou’s lemma and of the fact thatF_tⁿis a martingale. Indeed, we have for allt0

E[Ft]lim

n

E F_tⁿ

=F.

Similarly,F_t is a supermartingale since, for allM >0,F_tⁿ∧Mis a supermartingale andF_tⁿ∧Mconverges a.s.

and thus inL¹toF_t∧M. 2 2.2. The independent case

Proposition 2.2.Ifρ=0, then we have for allT >0 sup

t∈[0,T]E

Ft|LogFt|

<∞ (7)

andF_t is a continuous integrable martingale.

Proof. (i) Since the function(x→xLogx)is bounded when negative, it is enough to prove (7) replacing|LogFt|by LogF_t and the above proof of Proposition 2.1 will yield (7) as soon as we show for allt >0

sup

n

E

F_tⁿlogF_tⁿ

<∞ (8)

(observe indeed that (x→xLogx) is convex and thusE[F_tⁿLogF_tⁿ] is nondecreasing int sinceF_tⁿ is a martin- gale. . . ).

Then, we observe that we have E

F_tⁿLogF_tⁿ

=E

LogF+ t 0

σ_s1(s<τn)dWs−1 2 t 0

σ_s²1(s<τn)ds

F_tⁿ

=FLogF +FE t

0

σs1(s<τ_n)

dWs+σs1(s<τ_n)ds

−1 2 t 0

σ_s²1(s<τ_n)ds

where we used a Girsanov transform (“changingW_tintoW_t+_t

0σ_s1(s<τn)ds).

Hence, we have for alln1,t0 E

F_tⁿLogF_tⁿ

=FLogF +F 2E

t 0

σ_s²1(s<τn)ds

FLogF+F 2E

t 0

σ_s²ds

=FLogF +F σ² 2

e^α2t−1 α²

.

(ii) The bound (7) combined with the proof of Proposition 1 now immediately yields the factF_tⁿconverges inL¹ toF_t and, thus,F_tis a martingale. 2

Remarks.

(i) As is well known, (7) yields the integrability of sup_t∈[0,T]|F_t|for allT >0;

(ii) A similar argument to the one used in the proof of Proposition 2.2 yields the following bounds for allT >0 sup

t∈[0,T]E

F_t|LogF_t|^m

<∞, for allm1; (9)

sup

t∈[0,T]E

σ_t^pFt|LogFt|^m

<∞, for allm, p1; (10)

(iii) As we shall see below, whenρ=0,F_t isnotintegrable to any power larger than 1.

2.3. The positive correlation case

We want to show now thatE[F_t]< Fifρ >0,t >0. In fact, the argument shown below yields the strict monotonic- ity ofE[F_t]with respect tot.

Indeed, we go back to the proof of Proposition 2.1 and write F=E[F_t∧τn] =E[F_t1(tτn)] +E[F_τn1(τn<t )].

SinceE[F_t1(tτn)]^↑_nE[F_t], our claim will be shown if we prove that we have for allt >0 lim

n

E[F_τn1(τn<t )]>0. (11)

In order to do so, we write E[F_τn1(τn<t )] =F E

1(τn<t )exp τn

0

σ_sdWs−1 2

τn

0

σ_s²ds

and we use Girsanov formula to obtain a new measurePunder whichσ solves (up toτ_n)

dσt=ασ_tdZt+αρσ_t²dt. (12)

And we have

E[F_τn1(τn<t )] =FP (τ _n< t ).

Of course, it is well-known that the local solution of (12) blows up and thus P (τ _∞< t ) >0 for t >0 large enough, whereτ_∞=limn↑τ_n is the maximal existence time. This would, of course, be sufficient to yield the fact that E[F_t]< F for t large enough. We are not aware, however, of more precise blow-up results which show that P (τ _∞< t ) >0 for allt >0. And we do so by the following relatively easy argument where we fixt=t₀>0. First of all, all the differential equations we write below are understood to hold up toτ_∞. Then, we writex_t=Logσ_t and we have

dxt=αdZt+

αρe^xt −α² 2

dt, x₀=x=Logσ. (13)

Next, we recall that for eachλ >0, there exists a positive constantν(=ν(λ))such thatP (sup_t∈[0,t0]|Z_t−λt|<1)ν.

Hence, we have, on the setA= {supt∈[0,t₀]|Z_t−λt|<1}, the following inequalities

x_tα(λt−1)+αρ t 0

e^xsds−α² 2 t,

(x_t+α)

αρe^−αt

0

e^xs+α+e^α ρ

λ−α

2

ds, xt+αyt

wherey_t solves:

˙ y_t=

αρe^−α e^y+e^α

ρ

λ−α 2

, y₀=Logσ+α.

We then chooseλlarge enough such that the solutiony_t of this ODE blows up fort < t₀. Thus,τ_∞< t₀onAand P (τ_∞< t₀)ν(λ).

2.4. The negative correlation case

Theorem 2.3.Ifρ <0,F_t is a continuous integrable martingale and, for anym >1,sup_t∈[0,T]E[F_t^m]<∞for all T >0if and only ifρ−√

(m−1)/m.

Remarks.

(i) The proof below may be refined to yield the following fact: ifρ >−√

(m−1)/m, thenE[F_t^m] = +∞for all t >0.

(ii) Also, the proof below allows to show the following bounds for all T ∈(0,+∞), p 1: if −1ρ

−√

(m−1)/m, then sup

t∈[0,T]E F_t^m

1+σ_t^p

<∞.

In particular, ifm=2 and−1ρ−1/√

2, then, sup_t∈[0,T]E[F_t²(1+σ_t²)]<∞and Eq. (1) holds!

(iii) Of course, ifρ= −1, Theorem 2.3 yields the finiteness of all moments ofF_t. In that case, however,F_t is even bounded. Indeed, we have

F_t=Fexp t

0

σ_sdWs−1 2 t 0

σ_s²ds

andσ_t=σ−α_t

0σ_sdWs. Hence, F_t=Fexp

1

α(σ−σ_t)−1 2 t 0

σ_s²ds

Fexp

σ α

.

(iv) The proof below also shows the following monotonicity inρproperty of certain expectations. More precisely, if we consider for anym1ϕ(ρ)=E[F_t^mΦ(σ )]for some functionalΦ overC([0, t])such thatΦ(σ₁)Φ(σ₂) ifσ₁σ₂on[0, t], thenϕis nondecreasing with respect toρ.

Proof of Theorem 2.3. (i) We begin with the case whenρ−√

(m−1)/m. In that case, the arguments introduced above show that it is enough to prove the following bound for allt >0

sup

n

E F_t^m_∧τn

<∞. Next, we write

F_t^m_∧τ

n

=F^mE

exp

m²−m 2

t∧τn

0

σ_s²ds

exp

m

t∧τn

0

σ_sdWs−m² 2

t 0

σ_s²ds

.

Hence, using once more Girsanov formula, we have E

F_t^m_∧τ

n

=F^mE

exp

m²−m 2

t∧τn

0

σ_s²ds

wherePis a new measure under whichσ solves

dσt=ασ_tdZt+αρmσ_t²dt. (14)

We then introduce the functionw_n(σ, t )defined by E

exp

m²−m

2

t∧τn

0

σ_s²ds

and, as is well known,wnis the unique smooth solution of

⎧⎨

⎩

∂w_n

∂t −α²

2 σ²∂²w_n

∂σ² −αρmσ²∂w

∂σ −m²−m

2 σ²w_n=0 for 0σn, t0, w_n|t=0=1 for 0σn, w_n(n, t )=1 fort0.

(15) We finally obtain a bound onwnby building an explicit supersolution of (15) and using the maximum principle (or if one prefers to avoid the use of differential equations, one may just apply Itô’s formula to that supersolution . . . ).

Indeed, letw=exp(μσ )forμ0. Obviously,w1 ifσ0 and

−α² 2 σ²∂²w

∂σ² −αρmσ²∂w

∂σ −m²−m

2 σ²w=σ²w

−α²

2 μ²−αρmμ−m²−m

2 .

This quantity is nonnegative as soon as we can findμ0 such that−α²/2μ²−αρmμ−(m²−m)/20 and this is possible if and only ifρ²m²m²−mi.e.ρ²(m−1)/m.

(ii) Next, ifρ∈(−√

(m−1)/m,0), we want to show thatE[F_t^m] = +∞(at least fortlarge enough, see the above remarks). In order to do so, we observe that the proof made in step (i) yields the following inequality

E F_t^m

E

F_t^m1(t <τ_n)

=F^mE

exp

m²−m 2

t 0

σ_s²ds

1(t <τ_n)

=F^mz_n(σ, t )

and,z_nis the (smooth) solution of

⎧⎨

⎩

∂z_n

∂t −α² 2 σ²∂²z_n

∂σ² −αρmσ²∂z_n

∂σ −m²−m

2 σ²z_n=0 for 0σn, t0, z_n|t=0=1 for 0σn, z_n(n, t )=0 fort >0.

(16)

It is then possible to show, by a rather technical argument detailed in Appendix A, thatz_n(σ, t )^↑_n + ∞ for each σ >0 and for allt >0. 2

Remarks.The monotonicity property stated in Remark (iv) after Theorem 2.3 is easily deduced from the preceding proof once we observe that the solutionσ of (14) is nondecreasing with respect toρby classical results on stochastic differential equations.

2.5. More general volatility equations

In this section, we consider the case of a general equation for the volatilityσ in place of (2) namely

dσt=μ(σ_t)dZt+b(σ_t)dt, σ₀=σ0 (17)

where we assume thatμ, bare smooth (for instance) functions on[0,∞)such that

μ(0)=0, b(0)0, (18)

μ(ξ ) >0 forξ >0, μis Lipschitz on[0,∞), (19)

b(ξ )C(1+ξ ) on[0,∞),for someC0. (20)

These conditions, that we assume throughout this section and do not recall, insure the existence and uniqueness of a nonnegative solution of (17) (such thatE[supt∈[0,T]|σ_t|^p]<∞for all 1p <∞, T ∈(0,∞)). Let us immediately mention that it is straightforward to adapt the arguments and thus the results below to situations where (19) or (20) do not hold anymore (assuming, for example, that (19), (20) are replaced by a local Lipschitz on[0,∞)condition forμ. . . ). We shall not detail here such easy adaptations.

First of all, Proposition 2.1 and its proof are still valid. Next, the analogues of Proposition 2.2 and of the facts stated and shown in Section 2.2 are given by the following

Theorem 2.4.

(i) If the following condition holds lim sup

ξ→+∞

ρμ(ξ )ξ+b(ξ )

ξ⁻¹<∞, (21)

thenE[F_t|LogF_t|]<∞,E[sup0st|F_s|]<∞for allt0andF_t is an integrable nonnegative martingale.

(ii) If the following condition holds lim inf

ξ→+∞

ρμ(ξ )ξ+b(ξ )

ϕ(ξ )⁻¹>0 (22)

for some smooth, positive, increasing functionϕ such that _∞1

ϕdξ <∞, thenF_t is not a martingale and we have:

E[F_t]< F₀ for allt >0.

Remarks.

(i) Notice that, ifb=0 andμ(ξ )=αξ(α >0), in which case (17) reduces to (2) and (21) is equivalent toρ0, while (22) is equivalent toρ >0 (takeϕ(ξ )=ξ²). And we recover, as a very particular case, the results contained in the preceding sections that concern the integrability ofF_t.

(ii) If we assume thatμandbsatisfy

ξ→+∞lim μ(ξ )

ξ =μ_∞0, (23)

ξ→+∞lim b(ξ )

ξ² =b_∞∈ [−∞,0], (24)

then (21) holds ifρμ_∞+b_∞<0, while (22) holds ifρμ_∞+b_∞>0 (again, takeϕ(ξ )=ξ²).

(iii) In the case whenρ=0 (the independent case) orρ <0 (the negative correlation case), then (20) implies obvi- ously that (21) holds.

Proof of Theorem 2.4. Proof of part(i): We only sketch it since it is almost the same as the one of Proposition 2.1.

It suffices to observe that, if (21) holds, then we have for someC0 ρμ(ξ )ξ+b(ξ )C(1+ξ ) on[0,∞)

and this allows as to obtain some bounds onE(σ _t²)for allt >0 by a simple application of Itô’s formula. The rest of the proof is then exactly the same as the one of Proposition 2.1.

Proof of part(ii): Once more, we only sketch it since it is very similar to the arguments made in Section 2.3: indeed, we only need to show that the blowup time ofσ_t solution of

dσt=μ(σ_t)dZt+

ρμ(σ_t)σ_t+b(σ_t)

dt, σ₀=σ >0 (25)

may be made, with positive probability, as small as we wish. In order to do so, we introduceψ (ξ )=ξ 1

1

μ(η)dηand considerηt=ψ (σt)which solves (up to the first blowup time)

dηt=dZt+ β(σ_t)

μ(σ_t)−1 2

μ(σ_t) μ²(σt) dt whereβ(ξ )=ρσ (ξ )ξ+b(ξ ).

We may then conclude as in Section 2.3: observe indeed that we have for someν >0,C0 and for allλ0 β(ξ )νϕ(ξ )−C,

∞>

+∞

η

ϕ(ψ⁻¹(η)) μ(ψ⁻¹(η))+λ

₋1

dη= +∞

σ

ϕ(ξ )+λμ(ξ )₋1

dξ→0 asλ→ +∞.

We next investigate conditions under whichFt∈L^mfor somem∈(1,∞). And we begin with sufficient conditions for the finiteness ofE[F_t^m](or equivalently ofE[sup_0stF_s^m])for allt >0. We thus follow the approach introduced in Section 2.4 above and write, at least formally, using once more a Girsanov transform

E F_t^m

=F₀^mE

exp

m t 0

σ_sdWs−m 2

t 0

σ_s²ds

=F₀^mE

exp

m²−m 2

t 0

σ_s²ds

wherePis a probability under whichσt solves

dσt=μ(σ_t)dZt+β(σ_t)dt (26)

andβ(ξ )=mμ(ξ )ξ+b(ξ ).

Still arguing formally, we need to obtain some upper bounds on the solutionwof

⎧⎪

⎨

⎪⎩

∂w

∂t −1

2μ²(ξ )∂²w

∂ξ² −β(ξ )∂w

∂ξ −m²−m

2 ξ²w=0 forξ0, t0, w|t=0=1 forξ0.

(27)

If we follow the argument introduced in Section 2.4 above and look for an exponential supersolution of the form

w(ξ, t )=exp(Aξ+Bt )

whereA0,B0, we are led to the following condition lim inf

ξ→+∞

−1

2A²μ²(ξ )−Aβ(ξ )−m²−μ 2 ξ²

>−∞ for someA0. (28)

Example.If we assume (23), (24), (28) holds if there existsA0 such that

−1

2A²μ²_∞−A(mρμ_∞+b_∞)−m²−m 2 >0 or equivalently if

ρ <− m−1

m − b_∞

mμ_∞. (29)

And, ifρmμ_∞+b_∞= −μ_∞√

m²−m, (28) holds if we have lim inf

ξ→+∞

−1 2A²₀

μ²−μ²_∞ξ²

−A₀

β−(ρmμ_∞+b_∞)ξ²

−m²−m

2 ξ² >−∞

whereA₀=√

m²−m/μ_∞.

In particular, if we haveμ(ξ )≡μ_∞ξ,b(ξ )=b_∞ξ²then the condition (28) is equivalent to ρ−

m−1

m − b_∞

mμ_∞ (29)

(and in the case whenb_∞=0, we recover the condition obtained in Section 2.4 above). 2 If (28) holds, we can findA0,B0 such that we have on[0,∞)

−1

2A²μ²(ξ )−Aβ(ξ )−m²−m

2 ξ²−B.

Then,w(ξ, t ) =exp(Aξ+Bt )is indeed a supersolution of (26). From there on, one can justify as in Section 2.4 that we have

E F_t^m

w(σ ₀, t ) for allt0.

and we obtain the following

Theorem 2.5.If (28)holds, and thus in particular if(23),(24)and(29)hold, then we have for allT >0 E

sup

0tT

F_t^m

<∞.

Remark.Instead of assuming (23), (24) and (29), we may simply assume that (23) and (29) hold, denoting by b_∞=lim sup

ξ→+∞

b(ξ )ξ⁻²

∈ [−∞,0].

We conclude this section with a brief study of necessary conditions. More precisely, we are going to give conditions onμ, bthat insure thatE[F_t^m] = +∞for allt >0,F0>0,σ0>0. In order to simplify the presentation, we assume that (μ, b) satisfies (23), (24) (we could, in fact ignore (24) and simply denote by b_∞=lim infξ→+∞{b(ξ )ξ⁻²r} whereb_∞∈ [−∞,0]in view of (20)). Next, we introduce the following condition (to be compared with (29) or (29))

ρ >− m−1

m − b_∞

mμ_∞. (30)

And we have the

Theorem 2.6.If (30)holds, then we have for allF₀>0,σ₀>0 E

F_t^m

= +∞.

We skip the proof of Theorem 2.6 since it follows step by step the one made in Section 2.4 (and in Appendix A).

3. General models 3.1. Preliminaries

We consider throughout Section 3 the extension of the results and arguments introduced in the previous sections to a (more) general stochastic volatility model of the following type

dFt=σ_t^δF_t^βdWt, F₀=F 0, (1)

dσt=ασ_t^γdZt+b(σ_t)dt, σ₀=σ0 (2)

whereα, β, γ , δ >0, b(0)0, b is locally Lipschitz on[0,∞),b satisfies (20) andρ∈ [−1,+1] stands for the correlation between the Brownian motionsW_tandZ_t as before.

We next discuss some natural restrictions upon the parametersβ andγ. We first assume that β1 andγ 1.

Indeed, ifβ >1, orγ >1, it is easy to check that (1) or (2) becomes in general (see for instance [6]) an explosive

stochastic differential equation. Then, as is well-known, in order to have a unique local solution of (1), (2) one needs to assume thatγ 1/2 andβ1/2. However, as we shall prove in another article of this series, one can solve (1), (2) for allβ, γ∈(0,1]provided one restrictsF_t, ifβ <1/2, andσ_t, ifγ <1/2, to be nonnegative. This is why, with this further restriction, all the results we prove below are in fact valid for allβ, γ∈(0,1]although the arguments may need to be adapted ifβ orγ∈(0,¹₂)(through the use of techniques that we shall introduce for such fractional powers in a future work). . . .

With these restrictions, there exists a unique solutionσt of (2) which remains nonnegative and we have for all T >0

E

sup

0tT

σ_t^p

<∞, for allp∈(1,∞). (31)

At this point, we observe that the case whenβ <1 is easily handled since, in that case, (1) yields a unique solution whose moments (to an arbitrary high order) remains finite. Indeed, we have for anym∈(1,∞)and for anyn1

E F_t^m_∧τ

n

=F^m+E t 0

mm−1

2 σ_s^2δF_s^2β_∧τ^+m−2

n 1(τ_n>s)ds

whereτ_n=inf(s0/Fsn). Therefore, ifT ∈(0,∞)is fixed, we have for allt∈ [0, T] E

F_t^m_∧τ

n

F^m+mm−1 2p E

t 0

σ_s^pds

+m(m−1)

2q ·E

t 0

F_s^m_∧τ

nds

whereq=m/(2β+m−2)∈(1,∞)and 1/p+1/q=1. We then deduce from (31) a bound independent ofn1 on sup_t∈[0,T]E[F_t^m_∧τn]and we conclude (lettingngo to+∞) that, ifβ <1, then (1) defines uniquely a martingale F_t which belongs toL^mfor all 1< m <∞.

This is why we consider from now on the caseβ=1 i.e.

dFt=σ_t^δF_tdWt, F₀=F >0,

dσt=ασ_t^γdZt+b(σ_t)dt, σ₀=σ >0 (32)

(ifb(0) >0, we may consider as well the case whenσ0=0. . . ), with the above conditions onα,γ,δ,bandρ. 3.2. When isFt a martingale?

We begin with the study of the integrability ofF_t and the related issue of the existence of a martingaleF_t solving the first equation of (32).

Theorem 3.1.

(i) Ifρ >0and ifγ+δ >1, we assume thatbsatisfies lim sup

ξ→+∞

b(ξ )+ραξ^γ+δ

ξ <∞ (33)

(and we make no restriction onb,ρin the other parameter cases). Then,F_t is an integrable nonnegative martin- gale and

E

F_t|LogF_t|

<∞, E

sup

0st

|F_s|

<∞ for allt0.

(ii) Ifρ >0,γ+δ >1andbsatisfies lim inf

ξ→+∞

b(ξ )+ραξ^γ+δ

ϕ(ξ ) >0 (34)

for some smooth, positive, increasingϕsuch that_∞1

ϕdξ <∞, thenF_t is not a martingale and we have:

E[F_t]< F₀ for allt >0.

Remarks.

(i) It is worth comparing (33), (34) with (21), (22): those are obviously the “same” conditions withμ(ξ )replaced by αξ^γ andξ byξ^δ.

(ii) Ifρ0 or ifγ+δ1, (33) obviously holds.

Sketch of proof. The proof of the above result is once more entirely similar to the one made in Sections 2.2 or 2.5.

And we shall not repeat it. Let us only mention that we have E[F_tLogF_t] =1

2E t 0

σ_s^2δds whereσt, underP, solves

dσt=ασ_t^γdZt+

ρασ_t^γ+δ+b(σt)

dt, σ0=σ <0. (35)

And this equation then leads naturally to conditions (33) and (34). 2 3.3. Sufficient conditions

Let 1< m <∞. We state in the following result conditions that insure theL^mintegrability ofF_t or equivalently E

sup

0st

F_s^m

<∞ (36)

for some (or any. . . )t >0.

Theorem 3.2.

(i) Ifδ < γ 1/2orγ < δ1/2orγ=δ <1/2, then(36)holds for allt0.

(ii) Ifγ < δ,δ >1/2 (resp.δ < γ,γ >1/2)andγ+δ <1, then(36)holds for allt0.

(iii) Ifγ < δ,δ >1/2 (resp.δ < γ,γ >1/2)andγ+δ=1, we setb_∞=lim sup_ξ→+∞(b(ξ )/ξ )andq=1+δ−γ= 2δ. Then, ifρ−√

(m−1)/m−b_∞/(αm),(36)holds for allt0and, ifρ >−√

(m−1)/m−b_∞/(αm), we consider the solutionA(t )of

A˙=α²

2 q²A²+q(ραm+b_∞)A+m²−m

2 , A(0)=0, (37)

then(36)holds for allt < T0whereT0∈(0,∞)is the blow-up time ofA.

(iv) Ifγ < δ,δ >1/2 (resp.δ < γ,γ >1/2)andγ+δ >1, then(36)holds for allt >0ifρ <−√

(m−1)/m− b_∞/(αm)whereb_∞=lim sup_ξ→+∞(b(ξ )/ξ^γ+δ).

(v) Ifγ=δ >1/2, we setb_∞=lim sup_ξ→+∞(b(ξ )/ξ^2γ). Then, ifρ <−√

(m−1)/m−b_∞/(αm),(36)holds for allt >0.

(vi) If γ =δ=1/2, we set b_∞=lim sup_ξ→+∞(b(ξ )/ξ ) and we denote by A the solution of (37) with q =1.

Then, (36) holds for all t 0 if ρ −√

(m−1)/m−b_∞/(αm) while (36) holds for all 0t < T0 if ρ >−√

(m−1)/m−b_∞/(αm).

Remarks.

(i) We shall see in the next section that this result is essentially optimal.

(ii) The case studied in Section 2 above corresponds to part (v) of the above result(γ =δ=1. . .).

Sketch of proof of Theorem 3.2. Exactly as in the proof of Theorem 2.3 (see Section 2.4), we need to obtain upper bounds (independent ofn >1) onz_n(t, σ )=E(exp(^m2₂^−m _t∧τn

0 σ_s^2δds))whereσsolves dσt=ασ_t^γdZt+

ραmσ_t^γ+δ+b(σ_t)

dt, σ₀=σ >0, (38)

whereτn=inf(t0, σt> n). Obviously,znsolves uniquely (in viscosity sense for instance)

⎧⎪

⎪⎨

⎪⎪

⎩

∂zⁿ

∂t −α²

2 ξ^2γ∂²zn

∂ξ² −

ραmξ^γ+δ+b(ξ )∂zn

∂ξ −m²−m

2 ξ^2δz_n=0 for 0< ξ < n, t >0,

z_n|t=0≡1, z_n(n, t )≡1.

(39)

And we shall obtain those upperbounds by consideringw=exp(A(t )ξ^q+B(t ))whereA(0)=0,B(0)=0,A0, B0 andq >0 are to be determined in such a way thatwis a supersolution of (39) that is

0∂w

∂t −α²

2 ξ^2δ∂²w

∂ξ² −

ραmξ^γ+δ+b(ξ )∂w

∂ξ −m²−m 2 ξ^2δw

or ⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩ 0w

Aξ˙ ^q+ ˙B−α²

2 Aq(q−1)ξ^2γ+q−2−α²

2 q²A²ξ^2(γ+q−1)

−

ραmξ^γ+δ+b

qAξ^q−1−m²−m 2 ξ^2δ .

(40)

We first chooseq=1. We begin with case (i) (expect for the subcaseδ < γ =1/2 that we shall study separately below). We use the fact that b(ξ )C₀(1+ξ ) on[0,∞)for some C₀0. And we check easily that (40) holds provided we chooseA=e^Mt −1 for someM >0 large enough and thenB=K(e^2MT −1)for someK >0 large enough.

We next turn to case (ii). We chooseq=1+δ−γ. Notice that, sinceγ+δ <1,q >2δwhileγ+δ+q−1= 2(γ+q−1)=2δ. Hence, (40) holds forξlarge provided we chooseA=e^Mt−1 for someM > qC₀. It is then easy to chooseBin such a way that (40) holds for allξ0.

We complete the study of case (i) by considering the subcase when 1/2=γ > δ. In that case, we chooseq <1, close to 1 so thatq >2δ,q > γ+δ+q−1,q >2(1/2+q−1). And we conclude choosingA=e^Mt−1 for some M > qC₀. . . .

We now consider the case (iii) and we choose q=1+δ−γ. In that case, the leading order terms (as ξ goes to+∞)in (40) are given (or can be estimated from below) by

ξ^2δ

A˙−α²

2 q²A²−(ραm+b)qA−m²−m 2 for anyb> b_∞. We next observe that, ifρ <−√

(m−1)/m−b_∞/(αm), there exists forbclose tob_∞a positive rootx¯of

α²

2 x²+(ραm+b)x+m²−m 2 =0.

And, we choose, for such ab,Ato be the solution of A˙=α²

2 q²A²+(ραm+b)qA+m²−m

2 , A(0)=0, (41)

we deduce that A(t )x/q¯ for all t 0. It is then easy to complete the construction of w. Finally, if ρ

−√

(m−1)/m−b_∞/(αm), we solve (37) with b_∞ replaced by b> b_∞ i.e. we solve (41), and we build a su- persolution for allt < TwhereTis the blow-up time ofA(solution of (41)). And we conclude sinceT→T₀as b→b_∞(andT₀= +∞ifρ= −√

(m−1)/m−b_∞/(αm)).

The argument for case (vi) is the same than the one we just made provided we chooseq=1 (and solve (41) with q=1. . . ).

We next study case (iv). And we chooseq=1+δ−γ. Sinceq <2δ, the leading order terms (asξ goes to+∞) in (40) can be estimated from below by

ξ^2δ

−α²

2 q²A²−(ραm+b)qA−m²−m

2 .

And, ifρ <−√

(m−1)/m−b_∞/(αm), we may findA >0 such that the quantity between brackets is strictly positive.

We then chooseA(t )≡A(andB(t )≡B >0 large enough) to complete the construction ofw.

There only remains to treat case (v). We may then simply takeq=1 and follow the argument made in the preceding case. . . . 2

Remark. In cases (iv) and (v), the borderline value ρ = −√

(m−1)/m−b_∞/(αm) is slightly more delicate since the result depends on the behaviour of “b(ξ )−b_∞ξ^γ+δ”. Indeed, in case (v), one may prove that, for ρ= −√

(m−1)/m−b_∞/(αm), (36) holds for allt0 ifb−b^∞ξ^2γ is bounded from above on[0, m).

Similarly, in case (iv), one may check that, still forρ= −√

(m−1)/m−b_∞/(αm), (36) holds for allt 0 if {(b−b^∞ξ^γ+δ)ξ^δ−γ+α²(δ−γ )ξ^γ+δ−1/2}is bounded from above on[0,∞).

3.4. Necessary conditions

We consider in this section some necessary conditions for the boundedness of E[F_t^m]. As we shall see, these conditions are very close to the ones derived in the preceding section. They are summarised in the following

Theorem 3.3.

(i) Ifγ < δ,δ >1/2 (resp.δ < γ,γ >1/2)andγ+δ=1, we letb_∞=lim infξ→+∞(b(ξ )/ξ )andq=1+δ−γ= 2δ. Then, ifρ >−√

(m−1)/m−b_∞/(αm),E[F_t^m] = +∞for allt > T0whereT0∈(0,∞)is the blow-up time ofAsolution of(37).

(ii) Ifγ < δ,δ >1/2 (resp.δ < γ,γ >1/2)andγ+δ >1, we setb_∞=lim infξ→+∞(b(ξ )/ξ^γ+δ). Then, ifρ >

−√

(m−1)/m−b_∞/(αm),E[F_t^m] = +∞for allt >0.

(iii) Ifγ =δ >1/2, we setb_∞=lim infξ→+∞(b(ξ )/ξ^2γ). Then, ifρ >−√

(m−1)/m−b_∞/(αm),E[F_t^m] = +∞

for allt >0.

(iv) Ifγ=δ=1/2, we setb_∞=lim infξ→+∞(b(ξ )/ξ ). Then, ifρ >−√

(m−1)/m−b_∞/(αm),E[F_t^m] = +∞for allt > T₀whereT₀∈(0,∞)is the blow-up time ofAsolution of(37)withq=1.

Remarks.

(i) Comparing Theorems 3.3 and 3.2 (and the remarks made afterwards) shows that the conditions introduced in Theorem 3.2 are sufficient and “almost” necessary.

(ii) As an example, we mention the so-called Heston model which corresponds to the choiceγ=δ=1/2 andb(ξ )= b(σ¯−ξ )for someb,σ >¯ 0. Obviously, we haveb_∞= −b. And Theorems 3.2 and 3.3 imply the following facts:

– ifρ−√

(m−1)/m+b/(αm), thenE[F_t^m]<∞for allt0;

– ifρ >−√

(m−1)/m+b/(αm), then E[F_t^m]<∞for allt < T0andE[F_t^m] = +∞for alltT0, where T0∈(0,∞)is the blow-up time of the solutionAof (37) withq=1.

Sketch of proof of Theorem 3.3. We begin with the proof of case (iii). In that case, the argument is almost the same as the one presented in Appendix A, replacing in (A.1) by(νexp(νt σ^2γ)).

We now turn to the proof of case (ii). Again, one can follow the argument introduced in Appendix A, replacingσ byσ^q(observe indeed that, at least formally,η_t=σ_t^qsatisfies

dηt=qσ_t^q−1dσt+q(q−1) 2 σ_t^q−2

ασ_t^γ2

dt

=αqσ_t^δdZt+

ραmσ_t^2δ+b(σ_t)σ_t^δ−γ

dt+q(q−1)

2 α²σ_t^δ+γ−1dt

=αqη^δ/q_t dZt+

ραmη_t^2δ/q+b η^1/q_t

η_t^{((δ−γ )/q)}

dt+q(q−1)

2 α²η^((δ_t ^+γ−1)/q)dt andδ+γ−1<2δ).

Cases (i) and (iv) are of a slightly different nature since the momentE[F_t^m]becomes infinite only after timeT₀. In order to keep the presentation as simple as possible, we simply consider the case whenb(ξ )=b_∞ξ (the general