International Journal of Theoretical and Applied FinanceVol. 1, No. 3 (1998) 331–347c©World Scientific Publishing Company

INSIDER TRADING IN A CONTINUOUS TIME

MARKET MODEL

AXEL GRORUD

C.M.I., Universite de Provence, 39 rue Joliot-Curie,

13453 MARSEILLE cedex 13, OMEGA, INRIA, BP 93, F06902 Sophia-Antipolis, FranceE-mail: agrorud@abel.univ-mrs.fr

MONIQUE PONTIER

U.M.R. CNRS 6628, Batiment de Mathematiques, Universite d’Orleans,B.P. 6759, 45067 ORLEANS cedex 02, France

E-mail: pontier@labomath.univ-orleans.fr

Received 18 November 1997Revised 3 April 1998

This paper uses the enlargement of Brownian filtrations and a probability change formodelling the observation of a financial market by an insider trader. A characterizationof admissible strategies and a criterion for optimization are given. Then a statistical testis proposed to test whether or not the trader is an insider.

1. Introduction

This paper deals with the problem of the insider trading: some financial agent

knows something about the future. Thus, a market model is built on a filtered

probability space (Ω, (Ft, t ∈ [0, T ]), IP), the prices of assets being given by the

equation:

Sit = Si0 +

∫ t

0

Sisbisds+

∫ t

0

Sis(σis, dWs), 0 ≤ t ≤ T, S0 ∈ IRd, i = 1, . . . , d ,

where W is d-dimensional Brownian motion and (., .) denotes the scalar product in

IRd.

From the beginning, t = 0, the investor knows a random variable L ∈L1(Ω,FT ; IRκ), κ ∈ IN∗, (for instance, he knows that some trading will be done

and when it will be done); for two assets of prices S1 et S2, the random variable

could be their ratio at time T : L = lnS1T − lnS2

T . The “natural” filtration known by

the insider trader is Ft∨σ(L). To apply the standard results, we use the associated

right continuous filtration, denoted by Y : Yt = ∩s>t(Fs ∨ σ(L)), t ∈ [0, T ].

331

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332 A. Grorud and M. Pontier

But on the filtered probability space (Ω, (Yt, t ∈ [0, T ]), IP), the process W is

no longer a semi-martingale. Following Follmer and Imkeller [5], an equivalent

probability measure Q is built such that under Q, for all t < T, the σ-algebra Ftis independent of σ(L). Thus W is a (Y, Q)-Brownian motion. Another useful

method is the initial enlargement of filtrations, it allows to find some conditions on

L so that there exist a Y-Brownian motion B and an increasing process A satisfying

Wt = Bt+At. This was studied when L is a Gaussian random variable by Yor [21],

Chaleyat–Maurel and Jeulin [3].

More generally, Jacod [10] did the same when the family of conditional laws

Qt(ω, .) of L given Ft is dominated almost surely by a non-random measure; see

also Song [20]. The Bouleau–Hirsch [2] results give some simple conditions on L so

that these conditional laws are dominated by the Lebesgue measure. With some

extra hypotheses, Imkeller [7], specifies the decomposition of the semi-martingale

W.

Karatzas and Pikovsky [11, 12] studied similar problems on some examples of

real or vectorial random variables: L = W1, L = (λiWi1 + (1 − λi)εi)i=1,d with

a family of independent Gaussian variables ε, L = S1 the price at time 1, and

L = 1IS1<p. All these cases satisfy our hypothesis HJ, so our machinery runs.

An interesting point of their paper is the consideration of an optimization problem

with constraints on the portfolios. But, our contribution, moreover our relatively

general condition HJ, is the statistical test on the hypothesis if the trader is an

insider trader or not. Let us also quote Back [1] and Kyle [13].

The market model notations and hypotheses are given in Sec. 2. In Sec. 3, the

insider trader wealth equation is justified on the new probability space (Ω,Y, Q).

The admissible strategies are characterized in Sec. 4 and an insider trader optimal

strategy is produced on [0, A] when A < T, but some uncertainty remains on the

price assets between times A and T, (cf. [11]).

When A = T, a risk-neutral probability measure is not so easy to find. Perhaps

no such measure exists since an arbitrage strategy can be given with trading at

times 0 and T on the two assets of prices S1 and S2 (L = lnS1T

S2T

): suppose that

S10 = S2

0 = 1, then the strategy θ1 = sgn(L); θ2 = −sgn(L) is Y0-mesurable, and

the initial wealth X0 = π1 + π2 is 0. At time T the wealth: XT = θ1S1T + θ2S2

T =

|S1T − S2

T | is strictly positive. So θ is an arbitrage strategy, but it is not necessarily

admissible (cf. Dana and Jeanblanc-Pique [4]) because Xt = θ1S1t + θ2S2

t could be

non positive on a non-negligible set of [0, T ]× Ω.

In Sec. 5 we show other general examples, and finally, in Sec. 6 a statistical test

is proposed: the hypothesis H0 is “the financial agent is not an insider trader”; the

alternative hypothesis H1 is “the financial agent is an insider trader”.

2. Market Model

Let W be a d-dimensional Brownian motion, let (Ω,F , (Ft, t ∈ [0;T ]), IP) a

filtered probability space, with Ω = C([0, T ]; IRd). Consider a financial market with

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Insider Trading in a Continuous Time Market Model 333

d assets, where prices are described by

Sit = Si0 +

∫ t

0

Sisbisds+

∫ t

0

Sis(σis, dWs), 0 ≤ t ≤ T , (1)

and the bond evolves according to the equation: S0t = 1 +

∫ t0S0srsds.

The parameters b, σ, r are supposed to be bounded on [0, T ], to be adapted, and

to take their values in IRd, IRd×d, IR; the matrix σt is invertible dt⊗dP almost surely

and let ηt = σ−1t (bt − rt1I), for t ∈ [0;T ].

We denote, for a probability measure P, H1(P) the following hypothesis:

H1(P) : ηt verifies

∃A ∈]0;T [, ∃C ∈ IR, ∃k > 0, ∀s ∈ [0, A], EP [exp k‖ηs‖2] ≤ C .

This hypothesis is a Novikov’s type criterion (cf. [19, p. 323]) to obtain an equivalent

probability using the Girsanov’s transform.

Remark 2.1. Along this paper, we shall define hypotheses which suppose the

existence of a “final” time A ∈]0;T [. We can suppose, without loss of generality that

the final time A is the same in each hypothesis, if not we could take the minimum

of the finite number of times.

A financial agent has a positive amount X0 at time t = 0 and he wants to

optimize his consumption investment strategy; (Yt = ∩s>tFs ∨ σ(L))t∈[0;T ] is the

filtration of his knowledge. His consumption rate is c, a Y-adapted non-negative

process such that∫ T

0 csds < ∞, IP a.s. He has also θi units of the ith asset. His

wealth at time t is then Xt =∑di=0 θ

itSit.

Consider the standard hypothesis:

H2 “self-financing”: dXt =d∑0

θitdSit − ctdt , (2)

i.e. the consumption is only financed by profits from portfolio, and not by external

endowments (as wages, for instance). Then, the wealth of the insider should satisfy

the equation:

dXt = θ0tS

0t rtdt+

d∑i=1

θitSitbitdt+

d∑i=1

θitSit(σ

it, dWt)− ctdt ,

with initial wealth X0 which is a positive and Y0-measurable variable. So the wealth

equation should be an anticipative one, and the next calculus is only formal.

Denote πit = θitSit the amount invested on the ith asset, for i = 1, . . . , d, and

remark that θ0tS

0t = Xt −

∑d1 π

it. Let π = (πit, i = 1, . . . , d) denote the portfolio.

Thus, the wealth is the solution of the following stochastic differential equation:

dXt = (Xtrt − ct)dt+ (πt, bt − rt1I)dt+ (πt, σtdWt), X0 ∈ L0(Y0) .

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334 A. Grorud and M. Pontier

Let Rt = (S0t )−1 be the discounting factor; then, the discounted wealth satisfies

the equation:

XtRt +

∫ t

0

Rscsds = X0 +

∫ t

0

Rs(πs, bs − rs1I)ds+

∫ t

0

Rs(πs, σsdWs) . (3)

By an admissible consumption investment strategy (π, c) we mean a process such

that π is Y-predictable and c is Y-adapted, c ≥ 0,∫ T

0 csds < ∞ and σ∗π is in

L2[0;T ] IP-almost surely; and so that the wealth Xπ,c associated to this strategy is

non-negative dt⊗ dIP almost surely.

If the process (π, c) is Y-adapted, the stochastic integral in the last equation is

anticipating. To give a meaning to Eq. (3), our main hypothesis will be:

H3: There exist A ∈]0;T [ and a probability measure Q equivalent to IP on the

σ-algebra FA ∨ σ(L) such that under Q, for any t ≤ A, the σ-algebras Ft and σ(L)

are independent.

3. Cancellation of the Anticipation

In this section, we give a set of hypotheses which allows us to get a classical

framework for the insider trader wealth equation.

Lemma 3.1 (T. Jeulin). Assume:

HJ: There exist A ∈]0;T [ and a FA⊗B(IRκ)-measurable function q(A, .) such that

for any bounded borelian function f on IRκ :

EIP[f(L)/FA] =∫

IRκ f(x)q(A,ω, x)IPL(dx), and q(A,ω, x) > 0 IP ⊗ IPL almost

surely, where IPL is the probability law of L.

Then, if we denote Q = 1q(A,L) IP, for any t ≤ A, Q|Ft = IP|Ft and the probability

measure Q satisfies the hypothesis H3.

Proof. Let f a bounded borelian function on IRκ and B ∈ FA:

EQ[f(L)1IB] = EIP

[f(L)

q(A,L)1IB

]

= EIP

[EIP

[f(L)

q(A,L)1IB

/FA]]

= EIP

[∫IRκ

f(x)

q(A, x)q(A, x)IPL(dx)1IB

],

using that B ∈ FA and the definition of Q we obtain

EQ[f(L)1IB] = EIP[f(L)]IP(B) . (4)

If f = 1, we get Q(B) = IP(B); if B = Ω, we get EQ[f(L)] = EIP[f(L)]. So the

lemma is proved.

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Insider Trading in a Continuous Time Market Model 335

Proposition 3.2. Under the hypothesis HJ, the process (Wt, t ≤ A) is a

(Y, Q)-Brownian motion.

The first step of the proof is the following lemma.

Lemma 3.3. Let N ∈ L∞(Ω,FA, Q). Then, for any t ≤ A:

EQ[N/Ft ∨ σ(L)] = EIP[N/Ft] .

Proof. Let B a Ft-measurable bounded random variable and f ∈ L1(IRκ; IPL),

the independence of the σ-algebras Ft and σ(L) and the fact that Q|Ft = P|Ft(cf. Lemma 3.1) imply

EQ[NBf(L)] = EQ[NB]EQ[f(L)]

= EQ[EIP[NB/Ft]]EQ[f(L)]

= EQ[EIP[N/Ft]Bf(L)] .

Proof of the Proposition. Compute the characteristic function of an incre-

ment of W given the past, ∀u ∈ IRd, ∀s > 0, ∀s′ ∈]s, t+ s[:

EQ[eiu.(Wt+s−Ws)/Ys] = EQ[EQ[eiu.(Ws+t−Ws′+Ws′−Ws)/Fs′ ∨ σ(L)]/Ys]

since Ys ⊂ Fs′ ∨ σ(L) for any s′ > s. Using the lemma and the Fs′-measurability

of Ws′ −Ws, we get

EQ[eiu.(Wt+s−Ws)/Ys] = EQ[eiu.(Ws′−Ws)EIP[eiu.(Ws+t−Ws′ )/Fs′ ]/Ys]

= e− ‖u‖2

2(t+s−s′)EQ[eiu.(Ws′−Ws)/Ys] .

When s′ decreases to s, the result follows from the dominated convergence Lebesgue

theorem.

Then, the discounted wealth is well defined, with a Y-predictable portfolio, under

the probability measure Q:

XtRt +

∫ t

0

Rscsds = X0 +

∫ t

0

Rs(πs, bs − rs1I)ds+

∫ t

0

Rs(πs, σsdWs) .

Lemma 3.4. The hypothesis H3: “There exists A ∈]0;T [ and a probability

measure P ′ equivalent to IP on FA ∨ σ(L) such that for any t ≤ A and under P ′

the σ-algebras Ft and σ(L) are independent” implies HJ.

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336 A. Grorud and M. Pontier

Obviously, this lemma and Lemma 3.1 prove the equivalence between H3 and

HJ.

Proof. Let M = dIPdP ′ |FA∨σ(L)

. So there exists a FA ⊗ B-integrable strictly

positive function g such that M(ω) = g(ω,L(ω)). Moreover, under P ′, the σ-

algebras Ft and σ(L) are independent; thus, the probability measure P ′ is the

product P ′|FA ⊗ P′|σ(L). Let f be a bounded borelian function, then

EIP[f(L)/FA] =EP ′ [g(., L)f(L)/FA]

EP ′ [g(., L)/FA]=

∫g(., x)f(x)dP ′L(x)∫g(., x)dP ′L(x)

. (5)

The strict positivity of g(.,x)∫g(.,x)dP ′

L(x)

shows that the conditional law of L given FAunder IP is equivalent to the law of L under P ′. But IP and P ′ are equivalent, so

the law of L under IP is equivalent to the conditional law of L given FA under

IP.

The hypotheses HJ or H3 are not so easy to verify. But, with the enlargement

of filtration techniques, more tractable sufficient conditions can be obtained with

the following propositions.

Proposition 3.5 (Jacod [10]). Assume H’: ∃A ∈]0;T [ such that for t ≤ A,

the conditional law Qt of L given Ft is absolutely continuous with respect to a σ-

finite measure ν on IRκ. Then

— for t ≤ A, there exists a measurable version of the conditional density p(t, x) =dQtdν

(x) such that for all x ∈ IRκ, (ω, t) 7→ p(ω, t, x) is a martingale and can be

written as

p(t, x) = p(0, x) +

∫ t

0

(α(s, x), dWs) ;

moreover p(s, L) > 0 dIP almost surely, for all s ≤ A,— if Mt is a F-continuous local martingale M0 +

∫ t0(βs, dWs), for t ≤ A, then

the quadratic variation d〈M,p〉t is equal to 〈α, β〉tdt and the process

Mt = Mt −∫ t

0

〈α(., x), β〉u|x=L

p(u, L)du, 0 ≤ t ≤ A

is a Y-continuous local martingale.

As a corollary, the vectorial process

Bt = Wt −∫ t

0

ludu, 0 ≤ t ≤ A, where lis =αi(s, L)

p(s, L), i = 1, . . . , d , (6)

is a Brownian motion on the filtered probability space (Ω,Y, IP).

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Insider Trading in a Continuous Time Market Model 337

Proposition 3.6. Let HN be the following hypothesis:

∃A ∈]0;T [, ∃k > 0, ∃C,∀s ∈ [0, A], EIP[expk‖ls‖2] < C . (7)

Assume H’ and HN, then H3 holds.

Proof. Processes l and B do exist, thanks to hypothesis H’. The hypothesis

HN implies the existence of a (Y, IP)-uniformly integrable martingale M1t , t ≤ A,

such that

dM1t = −M1

t (lt, dBt),M10 = 1 ,

thus there is an equivalent probability measure IP1 = M1AIP on YA ⊃ FA ∨ σ(L).

Then, Wt = Bt +∫ t

0 lsds for t ≤ A is a (Y, IP1)-Brownian motion. Thus, under IP1,

W is independent of Y0 and so Y0 and Ft are independent for any t ≤ A.

To end this section, we get a sufficient condition for H’. Let Ω = C([0, T ]; IRd)

and H = h ∈ Ω/h ∈ L2([0;T ], IRd). Let w(h) =∫ 1

0 (hs, dWs), for h ∈ H, and let

S be the set of real Wiener functionals (cf. P. Malliavin [14] or D. Nualart [16]):

S = F ∈ L2(Ω)/∃n ∈ IN, f ∈ C∞b (IRn), such that

F = f(w(h1), . . . , w(hn)), with h1, . . . , hn ∈ H .

Let F ∈ S and DF ∈ L2(Ω× [0;T ]; IRd) be defined by

DtF =i=n∑i=1

∂f

∂xi(w(h1), . . . , w(hn))hit .

D is the usual stochastic gradient associated to the Wiener process W. D∗ denotes

its dual operator. Let ID2,1([0;T ]) be the Sobolev space closure of S with respect

to the norm ‖F‖22,1 = ‖F‖22 +E[∫ T

0‖DsF‖2ds].

Let HC denotes the hypothesis

L ∈ ID2,1([0;T ]) such that

∫ T

t

‖DsL‖2ds > 0, IP almost surely for any t ∈ [0, T [ .

(8)

Proposition 3.7. Under HC we have that ∀t < T, the conditional law Qt of

L given Ft is absolutely continuous with respect to the Lebesgue measure.

Proof. (i) ID2,1 is actually the standard Dirichlet space on Wiener space Ω

(cf. [2]). Let X be the set W is , 0 ≤ s ≤ t; i = 1, . . . , d and ID(X ) the Dirichlet

subspace of ID generated by this family (cf. [2]). Let H be the orthogonal subspace

in L2(Ω;L2([0;T ]); IP) of the set

ZDU ; U ∈ ID(X ); Z ∈ L∞(Ω) ,

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338 A. Grorud and M. Pontier

and let DX be the operator PHD defined on ID2,1, where PH is the orthogonal

projection on H.Let us remark that for any f ∈ ID2,1, DXf = Df1I[t;T ]. Then the condition

(b) in the Proposition 5.2.5 of [2] is easily verified (i.e. gPHh ∈ Dom D∗ when g

(resp. h) belongs to a dense subset of L2(Ω) (resp. L2([0;T ]))). Thus the operator

(ID2,1, DX ) is closable as an operator from L2(Ω) to L2(Ω;L2([0;T ]); IP).

Then, the hypothesis HC, and Theorem 5.2.7 (a) [2] give the result.

Example. We can see that L = lnS1T − lnS2

T verifies the set of hypotheses H1,

H2, HJ in a market with continuous and deterministic coefficients. Actually, L is

then a Gaussian variable and we can use the results of [3]; if the vector σ1−σ2 6= 0

in [t, T ] for all t:

L =

∫ T

0

βsds+

∫ T

0

(γs, dWs) ,

where βs = (b1s − b2s)− 12 (‖σ1

s‖2 − ‖σ2s‖2) and γs = σ1

s − σ2s .

In this way we get the result given by Chaleyat–Maurel and Jeulin [3]:

Bt = Wt −∫ t

0

γr(∫ TrγsdWs)∫ T

r‖γu‖2du

dr

is a (Y, IP)-Brownian motion.

Moreover the distribution of ls is N(

0, ‖γs‖2∫T

s‖γu‖2du

). See also [11].

4. Admissible and Optimal Strategies

We want now to make a risk neutral change of probability measure on which we

could compute a “fair” price for an option in an insider trader setting.

Proposition 4.1. Assume H1(Q) and let Mt = exp[−∫ t

0(ηs, dWs) −

12

∫ t0‖ηs‖2ds], t ∈ [0, A]. Then, M is a (Y, Q)-uniformly integrable martingale and,

under Q1 = MAQ, the process Bt = Wt+∫ t

0ηsds, for t ≤ A, is a (Y, Q1)-Brownian

motion and the discounted prices are (Y, Q1)-local martingales.

The proof is quite standard (Girsanov theorem).

Let us remark that Q1 = MA

q(A,L) IP and under Q1, the discounted wealth can be

written, with a Y-predictable portfolio:

XtRt +

∫ t

0

Rscsds = X0 +

∫ t

0

Rs(πs, σsdBs), t ∈ [0, A] . (9)

Under H1(Q) and HJ we have defined the risk-neutral probability Q1 on the

probability space (Ω,Y, Q) and under H’, H1(IP) and HN we can define the

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Insider Trading in a Continuous Time Market Model 339

probability Q on the probability space (Ω,Y, IP) by

dQ

dIP= MA = exp

[−∫ A

0

(ls + ηs, dBs)−1

2

∫ A

0

‖ls + ηs‖2ds].

We have supposed, without loss of generality that the final time A is equal in each

hypothesis. It is easy to prove that Q is also a risk-neutral probability.

Define q(t, x) as the the conditional density of L given Ft with respect to the

law of L; hypothesis HJ insures that for all t ≤ A, q(t, x) > 0 IP⊗IPL almost surely,

where IPL is the probability law of L.

Let Mt = exp[−∫ t

0 (ls + ηs, dBs)− 12

∫ t0 ‖ls + ηs‖2ds], for t ≤ A, we get:

Proposition 4.2. Assume H’, H1(IP), H1(Q) and HN, then Q1 = Q on YAand the two (Y, Q)-uniformly integrable martingales M and q(A,L)M coincide on

[0, A].

Proof. First, on the probability space (Ω,Y, Q) we have by the previous propo-

sition:

dMs = −Ms(ηs, dWs), M0 = 1 .

Second, using the definition of M just above, and writing it on (Ω,Y, Q) with respect

to W instead of (Ω,Y, IP) with respect to B, yields

dMs = −Ms[(ls + ηs, dWs)− (ls + ηs, ls)ds], M0 = 1 .

Moreover, the hypothesis H’ allows us to define the conditional density p(t, x) of L

given Ft with respect to the measure ν(dx) and p satisfies (cf. Proposition 3.5):

p(t, x) = p(0, x) +

∫ t

0

(α(s, x), dWs) .

On the probability space (Ω,Y, Q), L and W are independent (σ(L) ⊂ Y0), so we

can replace x by L in the equation above, then p(t, L) = p(0, L) +∫ t

0 (α(s, L), dWs).

The conditional density q(t, x), of L given Ft with respect to the law of L,

satisfies p(t, x) = q(t, x)p(0, x) for x ∈ IRκ. Since we have p(t, L) > 0 almost surely

[10], Corollary (1.11) and recalling that lt = α(t,L)p(t,L) , we get for t ≤ A:

q(t, L) = 1 +

∫ t

0

q(s, L)(ls, dWs) .

So, (q(t, L))(t≤A) is the martingale (EQ[q(A,L)/Yt])(t≤A). Then, using Ito formula,

for t ≤ A, we get

d(Mtq(t, L)) = Mtdq(t, L) + q(t, L)dMt + d[M, q(., L)]t = dMt .

So Mt = q(t, L)Mt and Q1 = Q on YA.

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340 A. Grorud and M. Pontier

To characterize the admissible strategies, we need a martingale representation

theorem which is not a classical one because the filtration is not the natural filtration

of the Brownian motion B.

Theorem 4.3. Suppose that H1(Q) and HJ hold. Let Z ∈ L1(Ω,YA, Q1);

then there exists a unique Y-predictable ψ such that

Z = EQ1 [Z/Y0] +

∫ A

0

(ψs, dBs) .

Proof. W is a (Y, Q)-Brownian motion. The process W is also F-adapted,

so it is also a (F , Q)-Brownian motion, and any local (F , Q)-martingale has the

representation property with respect to W. Then, Theorem 4.33, p. 176 of [8] shows

that any local (Y, Q)-martingale has the representation property with respect to

W. Finally, the equivalence of the probability measures Q1 and Q implies that any

local (Y, Q1)-martingale has the representation property with respect to B: indeed,

we have

EQ1 [Z/Yt] =EQ[MAZ/Yt]EQ[MA/Yt]

,

and Mt = EQ[MA/Yt] satisfies the equation Mt = 1−∫ t

0 Ms(ηs, dWs).

Moreover the martingale representation property with respect to W gives

EQ[MAZ/Yt] = EQ[MAZ/Y0] +

∫ t

0

(φs, dWs) ,

and Ito formula yields

EQ1 [Z/Yt] = EQ1 [Z/Y0] +

∫ t

0

(φs

Ms+EQ1 [Z/Ys]ηs, dBs

).

We have then ψs = φsMs

+EQ1 [Z/Ys]ηs, ∀s ≤ A.

We have as a corollary a usual characterization of admissible strategies:

Proposition 4.4. We suppose that H1(Q), H2 and HJ hold. Let X0 be a

positive Y0-measurable variable. Then for an “admissible” strategy (π, c) and the

associated final wealth Xπ,cA , we have

EQ1

[Xπ,cA RA +

∫ A

0

Rtctdt/Y0

]≤ X0 .

Conversely, given an initial wealth X0 ∈ L1(Y0), a consumption process c, Y-

adapted positive and such that∫ A

0csds <∞ Q1 almost surely, and a random variable

Z ∈ L1(YA, Q1) such that

EQ1

[ZRA +

∫ A

0

Rtctdt/Y0

]= X0 ,

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Insider Trading in a Continuous Time Market Model 341

there exists a Y-predictable portfolio π = (πt, t ∈ [0;A]) such that (π, c) is admissible

and Xπ,cA = Z.

Proof. The first part is standard, under the probability Q1, Eq. (9) is

dXtRt +Rtctdt = Rt(πt, σtdBt), X0 ∈ L0(Y0), 0 ≤ t ≤ A .

It is a positive (Y, Q1)-local martingale, therefore it is a supermartingale with initial

value X0, and we have the result.

For the converse, we use the precedent theorem, let

Nt = EQ1

[ZRA +

∫ A

0

Rtctdt/Yt

], 0 ≤ t ≤ A ,

we have Nt = EQ1 [NA/Y0] +∫ t

0 (ψs, dBs) for a Y-predictable process ψ. For

0 ≤ s ≤ A, let πs = R−1s (σ′s)

−1ψs where σ′s denotes the transposed matrice of

σs, the process π is then Y-predictable. With the strategy (π, c) we get under the

probability measure Q1 the discounted wealth equation:

dXπ,ct Rt +Rtctdt = Rt(πt, σtdBt), X0 ∈ L1(Y0), 0 ≤ t ≤ A .

Then Xπ,ct Rt +

∫ t0 Rscsds is a uniformly integrable (Y, Q1)-martingale which is

equal to the conditional expectation of its terminal value, and, thus

Xπ,ct Rt = EQ1

[ZRA +

∫ A

t

Rscsds/Yt

]

which is positive and shows the admissibility of the strategy.

Optimization of the insider trader’s strategy. We use a couple of utility

functions (U1, U2) such that U1 and U2 are increasing, concave, non-negative C1

functions verifying limx→∞ U ′i(x) = 0. (Usually x 7→ log x is taken, or x 7→ xα,

α ∈]0, 1[).

Now, we have to maximize

(π, c) 7→ J(X0, π, c) = EIP

[∫ A

0

U1(ct)dt+ U2(Xπ,cA )/Y0

]

in the set of admissible strategies under the constraint

EQ1

[Xπ,cA RA +

∫ A

0

Rtctdt/Y0

]≤ X0 .

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342 A. Grorud and M. Pontier

Recall that Q1 = MA

q(A,L) IP. So the constraint becomes

EIP

[(Xπ,cA

MA

q(A,L)RA +

∫ A

0

RtEIP

[MA

q(A,L)

/Yt]ctdt

)/Y0

]

≤ X0EIP

[MA

q(A,L)

/Y0

].

But using Proposition 4.2, MA

q(A,L) is the terminal value of a (Y, IP)-martingale

(Mt)(t≤A) starting at 1, thus we have EIP[ MA

q(A,L)/Y0] = 1.

This maximization is obtained by means of the Lagrange multipliers; the La-

grangian of this constrained problem is

EIP

[∫ A

0

U1(ct)dt+ U2(Xπ,cA ) + λ

(∫ A

0

RtMtctdt+Xπ,cA RAMA −X0

)/Y0

],

where λ is a Y0-measurable random variable. Let

Ii = (U ′i)−1 (10)

and

X (y)(ω) = EIP

[∫ A

0

RtMtI1(yRtMt)dt+RAMAI2(yRAMA)/Y0

](ω) . (11)

It is easy to verify that I and y 7→ X (y)(ω) (for ω fixed) are surjective monotone

functions on IR+ and there exists a Y0-measurable solution of the implicit equation

X (y) = X0; indeed, y 7→ X (y)(ω) (for ω fixed) is strictly decreasing on IR+ from

+∞ to 0, thus there exists a unique λ∗(ω) such that X (λ∗(ω))(ω) = X0(ω). More

precisely,

λ∗(ω) = supy ∈ IR+ : X (y)(ω) ≥ X0(ω) ,

and λ∗(ω) is Y0-measurable. We obtain:

Under hypothesis H1(Q), H2 and HJ, there exists an optimal strategy (π∗, c∗) such

that

J(X0, π∗, c∗) = supJ(X0, π, c), (π, c) admissible .

It is of the form: c∗t = I1(λ∗MtRt); Xπ∗,c∗

A = I2(λ∗MARA), where λ∗ is a Y0-

measurable random variable solution of the implicit equation: X (λ∗) = X0. We then

have the optimal value of the maximization problem:

EIP

[U2 I2(λ∗MARA) +

∫ A

0

U1 I1(λ∗MtRt)dt/Y0

].

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Insider Trading in a Continuous Time Market Model 343

5. Examples

5.1. Let U2 = log and U1 = 0. Under H’, H1(IP) and HN, we obtain RAX∗A =

X0M−1A , therefore

U2(X∗A) = log(R−1A X0) +

∫ A

0

(ls + ηs, dBs) +1

2

∫ A

0

‖ls + ηs‖2ds .

The optimal value of this problem is

J(X0, π∗, c∗) = log(R−1

A X0) +1

2EIP

(∫ A

0

‖ls + ηs‖2ds/Y0

)for all A < T, (12)

since B is a (Y, IP) Brownian motion, so it is independent of Y0.

If L is a Gaussian random variable, the optimal value tends to infinity when

A→ T, and thus is greater than the optimal value of the portfolio of a non-insider

trader with the same optimization function: logx+ 12E[

∫ T0‖ηs‖2ds]: see [11] or the

example after Proposition 3.7, where the process l is given by

lir =γir∫ Tr

(γs, dWs)∫ Tr‖γu‖2du

.

Using the first formalism under HJ, the optimal wealth can be written as X∗A =

X0q(A,L)MA

, and the optimal value is

J(X0, π∗, c∗) = log(R−1

A X0) +EIP

(log

(q(A,L)

MA

)/Y0

).

This is equal to the non-insider optimal value plus the term EIP[log q(A,L)/Y0].

5.2. Under H’, H1(IP), H1(Q) and HN, for Ui(x) = log(x), i = 1, 2, we have

λ∗ = A+1X0

and the optimal strategy:

RAX∗A =

X0

A+ 1M−1A ; Rtc

∗t =

X0

A+ 1M−1t ,

where the anticipating feature is seen in Mt = Mt

q(t,L) , for t ≤ A.It is interesting to note that we can then obtain the expression of the optimal

portfolio, from the proof of Proposition 4.4,

X∗tRt +

∫ t

0

Rsc∗sds = EQ1

[X∗ARA +

∫ A

0

Rsc∗sds/Yt

]. (13)

Let ξs = −(ls + ηs) and Nt = M−1t , then Nt verifies the Ito equation:

dNt = −Nt(ξt, dBt), N0 = 1 . (14)

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344 A. Grorud and M. Pontier

Then

X∗tRt +

∫ t

0

Rscsds

=X0

A+ 1EQ1

[1−

∫ A

0

Ns(ξs, dBs) +A−∫ A

0

[∫ s

0

Nu(ξu, dBu)

]ds/Yt

].

Commuting the stochastic integral and the time integral and conditioning the

stochastic integral with respect to Yt, we obtain

X∗t Rt +

∫ t

0

Rscsds = X0 −X0

A+ 1

[∫ t

0

(1 +A− s)Ns(ξs, dBs)],

and identifying with the expression of the B-martingale we get

Rsσ′sπ∗s =

X0(1 +A− s)A+ 1

Ns(−ξs). (15)

Now, making the same computation from (13) and substracting the consumption

from 0 to t, we have

X∗tRt = EQ1

[X∗ARA+

∫ A

t

Rsc∗sds

/Yt

]

=X0

A+ 1EQ1

[1−∫ A

0

Ns(ξs, dBs)+A−t−∫ A

t

[∫ s

0

Nu(ξu, dBu)

]ds

/Yt

]

=X0

A+ 1EQ1

[1−∫ A

0

Ns(ξs, dBs)+(A−t)[1−∫ t

0

Nu(ξu, dBu)

]/Yt

]. (16)

Using now the differential expression of Nt we get

X∗tRt = X0A+ 1− tA+ 1

Nt, 0 ≤ t ≤ A ,

and this expression used in (15) gives an explicit expression of the optimal portfolio

function of wealth:

π∗s = X∗s (σ′s)−1(ls + ηs) ,

where σ′ is the transpose of σ. This gives a proof of a result conjectured in [11].

5.3. In a very general context (cf. [7], for precise hypotheses), it is possible to

get a formula for the conditional density p(t, x) with respect to Lebesgue measure

of L given Ft:

p(t, x) = E

(1IL>xD

∗(

DL 1I[t,T ]

|DL 1I[t,T ]|2H

)/Ft),

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Insider Trading in a Continuous Time Market Model 345

and we could get also, using the Ocone–Clark formula, an expression for ls (cf. [7]):

ls(x) =Dsp(s, x)

p(s, x).

So we can give a general expression of the hypotheses HC, H1(IP), H1(Q) and

HN. In the Gaussian case, using these formulae, we get the same results as in [3].

But for a little harder example, like L =∫ T

0

∫ t0 (γsdWs)dWt, where γ is deterministc,

ls becomes rather complicated.

5.4. To get another general example in which HJ is verified, let L = YT where Y

is a strong solution of a usual stochastic differential equation, for which we know that

the probability law L(Yt) has a strictly positive density with respect to Lebesgue

measure (cf. [15]); it is not hard to see that the process (Yt+s − Yt)s is a F-Markov

process (and not only a σ(Y )-Markov process) so the conditional law of YT given Ftis the law of the solution at time T − t of the same stochastic differential equation

with Yt as initial condition and then p(t, x) > 0 almost surely.

6. Statistical Test

We take in that section a logarithmic utility function and deterministic bounded

coefficients. The statistical test is given by: the nul hypothesis is L ∈ F0.

H0 : L ∈ F0, against H1 : L /∈ F0 .

We construct the statistical test on the example L = lnS1T − lnS2

T . L is then a

Gaussian variable and if the vector σ1 − σ2 6= 0 in [t, T ] for all t:

L =

∫ T

0

βsds+

∫ T

0

(γs, dWs) ,

where βs = (b1s − b2s)− 12 (‖σ1

s‖2 − ‖σ2s‖2) and γs = σ1

s − σ2s .

We have lr =γr(∫T

r(γs,dWs))∫

T

r‖γu‖2du

, that Bt = Wt −∫ t

0lrdr is a (Y, IP)-Brownian

motion.

We can compare the optimal strategies for an insider trader and for a non-insider

trader with same initial wealth x > 0. For the latter, the strategy is:

RAX∗A = yM−1

A ; Rtc∗t = yM−1

t , where y =x

A+ 1

with Mt = exp(−∫ t

0(ηs, dWs)− 1

2

∫ t0‖ηs‖2ds), where ηs = σ−1

s (bs − rs1I).We then have to compare the optimal consumptions under H0:

logRtc∗t = log y +

∫ t

0

(ηs, dWs) +1

2

∫ t

0

‖ηs‖2ds ,

to the optimal consumptions under H1:

logRtc∗t = log y +

∫ t

0

(ηs, dWs) +1

2

∫ t

0

‖ηs‖2ds+ log q(t, L) .

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346 A. Grorud and M. Pontier

Let 0 ≤ t0 < t1 < · · · < tn = T be a partition of [0;T ], under the hypothesis H0

we denote, for 0 ≤ i ≤ n− 1,

Yi = logRti+1cti+1 − logRticti =

∫ ti+1

ti

ηsdWs +1

2

∫ ti+1

ti

‖ηs‖2ds .

We have supposed that the coefficients b, r, σ are deterministic, thus Y is a se-

quence of Gaussian independent random variables with expectation 12

∫ ti+1

ti‖ηs‖2ds

and variance∫ ti+1

ti‖ηs‖2ds. Under hypothesis H1, there is an additional term log

q(ti+1,L)q(ti,L) . We could then construct a statistical test with critical region given by

RCi =

ω :

∣∣∣∣Yi(ω)− 1

2

∫ ti+1

ti

‖ηs‖2ds∣∣∣∣ > C

,

and, for instance, a statistical test with level 0.05 is the test with critical region:

RCi =

ω :

∣∣∣∣Yi(ω)− 1

2

∫ ti+1

ti

‖ηs‖2ds∣∣∣∣ > 1.96

√∫ ti+1

ti

‖ηs‖2ds.

Note that, under H1, for 0 ≤ i ≤ n− 2, the random variable

Yi(ω)− 1

2

∫ ti+1

ti

‖ηs‖2ds =

∫ ti+1

ti

ηsdWs + logEQ[q(ti+1, L)/Yti+1 ]

EQ[q(ti, L)/Yti ].

In the case where L is Gaussian, this random variable is a Gaussian, plus a non inde-

pendent combination of chi-2 law random variables plus the constant 12 log

∫ Ttiγ2sds∫

T

ti+1γ2sds

.

Thus the power of the Neyman–Pearson test H ′0 = (L = L0) (L0 is a constant)

against H ′1 = (L = lnS1T − lnS2

T ) can be computed.

Acknowledgement

We are very grateful to L. Denis, F. Hirsch and T. Jeulin for useful discussions.

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