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THE FINANCIAL VALUE OF A WEAK INFORMATION ON A FINANCIAL MARKET FABRICE BAUDOIN AND LAURENT NGUYEN-NGOC

Abstract. The results of [4] are extended under weaker assumptions to d-dimensional and possibly discontinuous processes and applied to the modelling of weak anticipations both on complete and incomplete financial markets. In the case of a complete market, we show that there exists a minimal probability measure associated with an anticipation. Remarkably, this minimal probability does not depend on the selected utility function. The approach is also shown to be flexible enough to allow corrections on the initial anticipation. Moreover, at the end of the paper, we study another kind of conditioning: The pathwise conditioning. Throughout the paper, Markovian models are studied in details as canonical examples. Keywords: Financial value of an anticipation, Minimal Markov models, Pathwise conditioning, Portfolio Optimization, Weak information. AMS subject classifications: Primary 60G99, 60J25, 90A09, 90A40; secondary 90A10.

1. Introduction On a given financial market, different agents generally have different levels of information; besides the public information, some of them may possess privileged information, which leads them to make anticipations on some future realizations of functionals of the price process. For a utilitybased agent, the basic question to ask is then: What is the financial value of this information? To answer this question, recent works have proposed some models based on the theory of initial enlargement of filtrations, developed in [19], [20], [21]; the utility maximization problem has been solved in [1], [2] for continuous processes (see also [12], [16], [17]) and the same approach has been taken in [11] in the context of processes with jumps. One drawback of the enlargement approach is that it leads to arbitrage possibilities, whereas the initial market is arbitrage-free. Nevertheless, in a recent paper [8], the authors studied some models constructed from progressive enlargements and proved that if the rate at which the additional noise in the insider’s information vanishes is slow enough, then there is no arbitrage and the additional utility of the insider is finite. We can also note that other approaches to insider’s effects in financial markets are made

Fabrice Baudoin: Department of Financial and Actuarial Mathematics, Vienna University of Technology, Wiedner Hauptstrasse 8-10/105, A - 1040 Vienna, Austria. E-mail: [email protected]. Laurent Nguyen-Ngoc: Laboratoire de Probabilités et Modèles Aléatoires, Universités de Paris VI & VII, F-75252, France. E-mail: [email protected]. 1

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FABRICE BAUDOIN AND LAURENT NGUYEN-NGOC

by [3] and [26] in the context of an equilibrium theory, with different kind of traders acting in the market. In this paper, in the spirit of [4], we propose an alternative modelling of the additional information that privileged investors possess. We assume that their extra knowledge consists of the law of a functional Y of the price process, rather than the exact value of Y as in the enlargement approach. This knowledge is referred to as a weak information on the price process. We then define the financial value of this weak information as the lowest increase in utility that can be gained by the insider from this extra knowledge. In the case of a complete market, we prove that there exists a unique probability measure which realizes the value of the weak information. One of the most remarkable features of this probability is that it does not depend on the utility function chosen at the beginning. Although this approach is already hinted at in [4], our mathematical framework is slightly different since we do not need to be in the classical assumptions of the theory of initial enlargement of filtration about the existence of a regular disintegration of Y (for instance, we can deal with anticipations on last passage times of transient Markov processes). Moreover, we allow d-dimensional discontinuous processes; and finally we deal with incomplete markets. In the case of an incomplete market, by using duality methods introduced in [22], [23], and further developed in [25] for utility maximization, we obtain the fact that in some sense the dual problem associated with the computation of the financial value of the information is also independent of the utility function. Furthermore, we give an interesting lower bound for the financial value of a weak information on the value of the price at a given date. To illustrate our work, we study some examples of minimal markets. Precisely, we study in details the case where the market is complete and the price process Markovian under the martingale measure. We show then that the minimal measure can be characterized as a solution of a martingale problem. Furthermore, we obtain the fact that in this case, the optimal wealth process is itself Markovian. In this paper, we also introduce another kind of weak anticipation. Namely, we study the case of an insider who knows all the conditional laws of the Y . We also show, in the case of a continuous diffusion, that the weak approach is flexible enough to allow both dynamic corrections on the initial anticipation and concatenations of these anticipations. Finally, at the end of the paper, we study another type of conditioning which corresponds to the case of an insider who is in the following position: He knows that with probability one, ∀t > 0, St ∈ O, where O ⊂ Rd is a non-empty, open set with smooth boundary.

FINANCIAL VALUE OF A WEAK INFORMATION

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2. Preliminaries and Definitions Let T > 0 be a constant finite time horizon. In what follows, we work on a continuous-time, arbitrage-free financial market. Namely, let Ω, (Ft )0≤t≤T , P be a filtered probability space which satisfies the usual conditions (i.e. the filtration F is complete and right-continuous) and such that moreover the filtration F is quasi left-continuous. We assume that there are d basic tradeable assets, the price process of which is a F–adapted positive local martingale (St )0≤t≤T . In addition we assume that S is square-integrable and that F0 is trivial (which implies that S0 is constant). Remark 1. Of course, since S is a local martingale under P, the market just defined has no arbitrage (precisely, there are no arbitrage opportunities with tame portfolios, see [27]; see also [10] for a general view on the absence of arbitrage property). Moreover, to take a null drift under P, means that we consider discounted prices (or equivalently, prices expressed in the bond numéraire). The key point is that we start from the observable dynamics of S (a model of S under P can be calibrated to market data) and not from the “true” but unobservable dynamics of S under a so-called “historical” measure. In this respect, our approach differs from traditional works on the subject of utility maximization. It is important to note that we could actually start from any semimartingale model for the price. Let us now introduce a few common definitions and notations. e ∼ P such Definition 2. The space M (S) of martingale measures is the set of probabilities P e that (St ) is an F-adapted local martingale under P. 0≤t≤T

Definition 3. The space AF (S) of admissible strategies is the space of Rd –valued and F– predictable processes Θ integrable with respect to the price process S, such that Z t Θu · dSu 0

0≤t≤T

e F) martingale for all P∈M(S). e is a (P, Remark 4. Θit is the number of shares of the risky asset S i held by an investor at time t, and the wealth process associated with the strategy Θ ∈ A (S), with initial capital x, is given by Z t Θu · dSu . Vt = x + 0

In particular, all our strategies are assumed to be self-financing. We shall also very often assume that the financial market Ω, (Ft )0≤t≤T , P, (St )0≤t≤T

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is complete in the sense that the martingale (St )0≤t≤T enjoys the following predictable representation property (in abbreviate PRP) : For each F-adapted local martingale (Mt )0≤t≤T there exists a predictable process Θ locally in L2 such that Z t Mt = M0 + Θu · dSu , t ≤ T. 0

Remark 5. Under the previous assumption, we have M (S) = {P}. In what follows, we use the following notion of utility functions (see [23]): Definition 6. A utility function is a strictly increasing, strictly concave and twice continuously differentiable function U : (0, +∞) → R which satisfies lim U 0 (x) = 0,

x→+∞

lim U 0 (x) = +∞.

x→0+

We use the convention that U (x) = −∞ for x ≤ 0. We shall denote by I the inverse of U 0 , and ˜ the convex conjugate of U : by U ˜ (y) = max (U (x) − xy) U x>0

(that is, the Fenchel-Legendre transform of −U (−x)). For simplicity, we limit ourselves to smooth utility functions, although it is also possible to obtain some results in the non-smooth case (see e.g. [6]). In our examples, we study the cases U (x) = ln(x) and U (x) = xα . The case U (x) = eαx is also interesting; it does not fit into the previous definition, nevertheless our results remain true in this case. Let us now introduce the object on which anticipations will be made and what will be called the financial value of such an anticipation. Let P be a Polish space (e.g. P = Rn , P = C (R+ , Rn ), etc...) endowed with its Borel σ–algebra B (P) and let Y : Ω → P be an FT –measurable random variable. We denote by PY the law of Y under P. Let now ν be a probability measure on B(P), which corresponds to the anticipation of the informed investor: He knows that the law of Y is ν. We assume that ν is equivalent to PY with a bounded density ξ. In financial terms, this means that the informed investor does not have information which is “opposite” to the market. The financial value of the weak information (Y, ν) should satisfy the following rule: The less precise the information, the lower its value. For instance the minimal information is ν = PY because under P the price process (St )0≤t0 Z sup E U x + ν

u (x, ν) =

T

Θu dSu

0

Θ∈A(S)

dPY (U ◦ I) Λ (x) (y) ν (dy) dν P

Z = where Λ (x) is defined by

dPY I Λ (x) (y) ν(dy) = x. dν P

Z

Proof. Let Q = DP ∈ E ν . We first proceed to rewrite Z T sup EQ U x + Θu · dSu Θ∈A(S)

0

differently. In fact, from classical results on complete markets, the Lagrangian associated with the optimization problem above is known to be given by ˜ y L(y) = xy + EP DU D

(y > 0)


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˜ is the convex conjuguate of U. where we recall that U Moreover Z U x+ sup E

T

Q

Θu · dSu

= inf L(y). y>0

0

Θ∈A(S)

Hence we have P

u(x, ν) = inf

xy + inf E

y>0

D

where D runs through the densities dQ/dP, Q ∈

Eν.

˜ DU

y D

˜ Now the function z 7→ z U

y z

is convex

for fixed y, and by proposition 9, we obtain y P ˜ u(x, ν) = inf xy + E ξ(Y )U . y>0 ξ(Y ) y ˜ ˜ the properties to be strictly convex, inherits from U The function y 7→ xy + EP ξ(Y )U ξ(Y ) continuously differentiable and to tend to +∞ as y → +∞; hence there exists Λ(x) that realizes the inf, and Λ(x) is given by P

x−E

ξ(Y )I

Λ(x) ξ(Y )

= 0.

Remark 12. From this proof, it follows immediately that if we denote by V the optimal wealth process under Pν then we have: dPY VT = I Λ (x) (Y ) . dν It is interesting to note that we can give explicit formulas for the most commonly used utility functions. Example 13. (See [4]) (1) Let α ∈ (0, 1) and U (x) =

xα α

xα u (x, ν) = α

then

"Z P

#1−α 1 1−α dν (y) P (Y ∈ dy) . dPY

(2) Let U (x) = ln x then Z u (x, ν) = ln x + P

dν dν (y) ln (y) P (Y ∈ dy) . dPY dPY

3.3. Examples: Minimal Markov Markets. In order to illustrate the previous results, we now study the particular case when S is also a Markov process. Diffusions driven by Brownian motion and/or Poisson process enter this category. Hence in this section we assume that under P, S is both a local martingale and a Feller process with values in Rd and that moreover Y = ST . Under this assumption, we will characterize

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Pν as the solution of a martingale problem, and also study the optimal wealth and proportion processes. We denote by L the (extended) generator of S under P. We assume that D(L) contains C 2 functions with compact support and that for such functions φ, Z d ∂2φ 1 X ai,j (x) + (φ(x + y) − φ(x) − y · ∇φ(x)) N (x, dy) (3.1) Lφ(x) = 2 ∂xi ∂xj Rd \{0} i,j=1

where a(x) is a smooth function with values in nonnegative definite symmetric d × d matrices, and N (x, dy) is the Lévy kernel of S (see [9]). In the following, for a matrix M, M ∗ will denote the transpose of M and M k the k-th column of M ; for a vector v ∈ Rd , diag (v) denotes the d × d matrix 0 ... 0 . .. ..  . . ..  0  . .  .. .. ... 0 0 . . . 0 vd Sometimes, it will also be convenient to note [vj ]j=1,...,d 

v1

   .  instead of v.

Example 14. (1) If we take S as a continuous diffusion, given by (3.2)

dSt = diag (St ) σ(St )dWt where W is a standard d–dimensional Brownian motion, then we have Lφ(x) =

d 1 X ∂2φ ai,j (x) (x) 2 ∂xi ∂xj i,j=1

where a =

diag (x) σσ ∗ diag (x).

(2) If we take S as a jump-diffusion, given by (3.3)

dSt = diag (St− ) ζ(St− )dMt where M is a d–dimensional compensated Poisson process with constant intensity λ ∈ Rd , then we have Lφ(x) = λ ·

h

i φ(x + ζ (x))

h

˜j

j=1,...,d

i − φ(x)1 − ∇φ(x) · ζ (x)

˜j

j=1,...,d

˜ where 1 is the d–dimensional vector filled with 1’s, and ζ(x) = diag (x) ζ(x). 3.3.1. The Minimal Probability as a Solution of a Martingale problem. With the same assumptions as in the previous paragraph, we now show that Pν solves a martingale problem. This approach concerns only the law of S and should be contrasted to the approach developed in sections 5.2 and 5.3 which are more pathwise oriented.


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Recall that we are studying the case Y = ST and thus Pν = ξ(ST )P. Hence, there is a function d such that the density process Dt = E[ξ(ST )|Ft ] = d(t, St ). Note that we assumed that ξ is a.s. bounded and strictly positive, so d(t, x) is strictly positive for almost all t and x. Theorem 15. Pν solves the martingale problem associated to Lν and the initial distribution δs0 , where 1 ¯ L(φd)(t, x) d(t, x) 1 1 ∂(φd) (t, x) + L(φ(t, .)d(t, .))(x) = d(t, x) ∂t d(t, x) ¯ and φd ∈ D(L), ¯ where L¯ is the generator of the space-time process for any φ such that φ ∈ D(L) Lν φ(t, x) =

(t, St ) which is given by: ∂φ ¯ Lφ(t, x) = (t, x) + L(φ(t, .))(x). ∂t Rt Proof. We check that φ(t, St ) − 0 Lν φ(u, Su )du is a Pν –martingale, with Lν defined above. For this, let 0 ≤ u ≤ t and let ψ : [0, ∞[×Rd → R a bounded function. We have Eν (ψ(u, Su ) (φ(t, St ) − φ(u, Su ))) = E (ψ(u, Su )d(t, St )(φ(t, St ) − φ(u, Su ))) = E (ψ(u, Su )d(t, St )φ(t, St ) − ψ(u, Su )d(u, Su )φ(u, Su ))) . Rt

¯ L(φd)(r, Sr )dr is a P–martingale, we have Z t ¯ E (ψ(u, Su )d(t, St )φ(t, St )) = E ψ(u, Su ) φ(u, Su )d(u, Su ) + E L(φd)(r, Sr )dr|Fu

But since φ(t, St )d(t, St ) −

0

u

so that, since

1 d(t,St )

is a

Pν

martingale,

Z t 1 ¯ L(φd)(r, Sr )dr E (ψ(u, Su ) (φ(t, St ) − φ(u, Su ))) = E ψ(u, Su ) d(t, St ) u Z t 1 ν ¯ = E ψ(u, Su ) L(φd)(r, Sr ) dr d(r, Sr ) u Z t ν ν = E ψ(u, Su ) L (φd)(r, Sr )dr ν

ν

u

which completes the proof.

Remark 16. With additional assumptions, typically Lipschitz conditions on the coefficients defining L, one could even conclude that Pν is the unique solution to the martingale problem for Lν . We refer to [13] for more details on the relationship between Markov processes and martingale problems. The above result can also be stated as follows: Under Pν , S is a inhomogeneous Markov process whose semigroup is characterized by the operator Lν . Specializing to the models introduced in example 14, we obtain the following results.

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Let us first turn to the case of a continuous diffusion given by (3.2). We assume moreover that σσ ∗ is a positive definite, symmetric, C ∞ function, with bounded partial derivatives, such that: inf kσ(x)σ ∗ (x)k ≥ a > 0.

x∈Rd

Under this assumption, the transition function pt , 0 ≤ t ≤ T is known to exist and to be differentiable, with pt (x, y) =

d P (St ∈ dy|S0 = x) dy

In this setting, we obtain: Theorem 17. Under the probability Pν , the process (St )0≤t −1, and that ξ is C 1 and bounded together with its partial derivatives. We denote by Pt the semigroup of S: Pt (x, dy) = P (St ∈ dy|S0 = x) ˜ and recall that ζ˜ is the matrix ζ(x) = diag (x) ζ(x). We then have:


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Theorem 19. Under Pν , S is a semimartingale with canonical decomposition "R # Z t ˜j (Su ))PT −u (Su , dy) ξ(y + ζ d ˜ u )diag (λ) R R −1 du St = s0 + ζ(S 0 Rd ξ(y)PT −u (Su , dy) j=1,...,d Z t fu , + diag (Su− ) ζ(Su− )dM 0≤u0 e P∈M(S)

.

0

Θ∈A(S)

uQ (x, ν) = inf

"

e xy + inf ν EQ U Q∈E

e dP y dQ

!#)

e dP y dQ

.

!#) .

e )P, e is given by ξ(Y e where ξe = Now, the optimal Q associated with a fixed P ( " !#) y e )U e ξ(Y e u(x, ν) = inf inf xy + E e ) y>0 P∈M(S) e ξ(Y 1 ˜ e = inf inf xy + E U (yπ(Y )) y>0 π∈D π(Y ) and we conclude with straightforwards computations.

dν eY dP

. Hence,


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Remark 23. We always have u (x, ν) ≥ U (x) . e is a convex function, for ξ ∈ D and y > 0 Indeed, since U Z e (yξ (u)) ν (du) ≥ U e (y) U P

and hence, e (y) + xy = U (x) . u (x, ν) ≥ inf U y>0

e ∈ M (S) such that Moreover, if 1 ∈ D, i.e. there exists P e (Y ∈ dy) = ν (dy) P then u (x, ν) = U (x) . Remark 24. It would be really interesting to know more about the quantity Z e (yπ (u)) ν (du) inf U π∈D

P

which seems to be hard to evaluate, even in the simplest examples of incomplete markets, such as stochastic volatility models. As just noticed, we do not know the value of u(x, ν) explicitly, but we have the following Proposition 25. Assume that P =Rd and Y = ST . Then we have for x > 0 Z u (x, ν) ≥ U (α + β · u) ν (du) d (R∗+ ) where α ∈ R+ and β ∈ Rd+ are defined by R x R∗ d U 0 (α + β · u) ν (du) ( +) R = 1, 0 d (α + β · u) U (α + β · u) ν (du) (R∗+ ) R x R∗ d u U 0 (α + β · u) ν (du) ( +) R = S0 . 0 (α + β · u) ν (du) (α + β · u) U d ∗ (R+ ) Proof. Let y > 0. Let us consider the convex functional L : V → R+ ∪ {+∞} R e (yξ (u)) ν (du) ξ → R∗ d U +

defined on the convex set Z V = {ξ ≥ 0,

Z ξ (u) ν (du) = 1,

R∗+ d

R∗+ d

uξ (u) ν (du) = S0 }.

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Since D ⊂ V, we have Z

Z e (yξ (u)) ν (du) ≥ inf U

inf

ξ∈D

ξ∈V

R∗+ d

R∗+ d

e (yξ (u)) ν (du) . U

Now, because L is convex, in order to find the minimum of L on V, it suffices to find a critical point. An easy computation shows that such a critical point ξe must satisfy Z 0 e e U y ξ (u) η (u) ν (du) = 0 R∗+ d

for all η such that Z

Z η (u) ν (du) = 0,

R∗+ d

uη (u) ν (du) = 0. R∗+ d

This implies 1 ξe(u) = U 0 (α (y) + β (y) · u) y d where α (y) ∈ R+ and β (y) ∈ R+ are defined by Z U 0 (α (y) + β (y) · u) ν (du) = y R∗+ d

Z R∗+ d

u U 0 (α (y) + β (y) · u) ν (du) = y S0

(see Remark 26 below). Hence !

Z

0

u (x, ν) ≥ inf

y>0

Rd+

e U (α (y) + β (y) · u) ν (du) + xy U

but for all z > 0 e U 0 (z) = U (z) − zU 0 (z) U hence !

Z u (x, ν) ≥ inf

y>0

U (α (y) + β (y) · u) ν (du) − (α (y) + β (y) · u) y + xy . Rd+

The previous infimum is clearly attained for y > 0 such that α (y) + β (y) · S0 = x which gives the expected result.

Remark 26. α(y) and β(y) are well defined by the formulas above. Indeed, according to a classical theorem of Hadamard (see for instance [15]), the application ! Z Z 0 0 (α, β) 7→ U (α + β · u) ν(du), u U (α + β · u) ν(du) d d (R∗+ ) (R∗+ ) induces a diffeomorphism from (R∗+ × (R∗+ )d )\(0, 0) onto itself because it is proper with an everywhere non-singular differential. Moreover, since the function y 7→ α(y)+β(y)·S0 maps intervals into intervals, there exists y such that α (y) + β (y) · S0 = x


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5. Modelling of a Weak Information Flow Until now, we assumed that the pair (Y, ν) composing the weak information is chosen by the insider once and for all at time 0. In this last section, we propose a framework to model a weak information flow. Precisely, we study three cases: The situation in section 5.1, corresponds to a more precise information about the functional Y than in the static case. We present it first, because it is not so close to the other two. In fact, the whole dynamics of the anticipation is built in one block, while in the next cases, it is built piecewise on successive time intervals. In sections 5.2 and 5.3, we study the following situation, which is what one would tend to do in practice. Given a subdivision 0 = t0 < t1 < · · · < tn = T , the insider reviews his anticipation at each time ti according to his information. He will maintain this anticipation until time ti+1 and at this time he makes a new anticipation, an so on. We show then the consistency of these operations when the mesh of the subdivision t1 , ..., tn tends to 0. Notice that in the setting of a strong information modelling (i.e. by using the theory of initial enlargement of filtration), such operations would no be consistent. 5.1. Dynamic Conditioning. We consider here an insider who has a knowledge of all the conditional laws P (Y ∈ dy | Ft ), 0 ≤ t < T . It means that he knows the “deterministic” evolution of his anticipation. For instance in a Markov model, νt will be a deterministic function of St . Precisely, with Y we associate a continuous F-adapted process (νt )0≤t≤T of probability measures on P (assumed to be the conditional laws of Y under the effective probability of the market) such that νT = δY . We shall denote by G the filtration F enlarged with Y , i.e. Gt is T the P–completion of ε>0 (Ft+ε ∨ σ (Y )), t < T. The predictable σ–field associated with F, will be denoted P(F). We assume that for 0 ≤ t < T, νt admits an almost surely strictly positive bounded density ξt with respect to P (Y ∈ dy | Ft ), i.e. νt (dy) = ξt (y) P (Y ∈ dy | Ft ) . Remark 27. The terminal condition νT = δY is equivalent to ξT = 1. Let E ν be the set of probability measures Q on Ω such that: (1) Q is equivalent to P (2) Q (Y ∈ dy | Ft ) = νt (dy), t < T . In order to ensure that E ν is non empty we have to make the following additional assumption on νt .

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Assumption: There exist a P(F) ⊗ B(P) measurable process (λyt )0≤t 0) and a P-standard Brownian motion (Bt )0≤t≤T whose filtration is F such that dSt = St σ (St ) dBt , 0 ≤ t ≤ T. Let us consider an insider who is in the following position: At time t, he receives a weak information about the price St+dt , this anticipation being only valid for the infinitesimal time dt and at t + dt the investor receives some new information and makes a new anticipation, and so on.... Such an insider tries hence to construct a probability measure P∗ on Ω such that P∗ (St+dt ∈ dy | Ft ) = ξt (y) P (St+dt ∈ dy | Ft ) , y → ξt (y) being the Ft −measurable function, corresponding to the weak information that the investor receives at time t. A canonical way to construct P∗ is the following : At time t, he learns the weak information (St+dt , ξt ) and constructs in the sense of Section 2 a probabilistic bridge in the time interval [t, t + dt) which forces St+dt to follow conditionally to the past filtration Ft (which is not trivial contrary to the static case) the law νt (dy) = ξt (y) P (St+dt ∈ dy | Ft ) . Let us see what it implies on P∗ . Let us denote by σ e the function x → xσ (x) . It is easily seen that we have ξt (St+dt ) = 1 + ξt0 (St ) σ e (St ) dBt (assuming that ξt is differentiable) because of the normalization E (ξt (St+dt ) | Ft ) = 1.


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Hence, for all test function f, we must have thanks to Itô’s formula (notice that ξt (St ) = 1) E∗ (f (St+dt ) | Ft ) = E (ξt (St+dt ) f (St+dt ) | Ft ) e2 (St ) dt = f (St ) + f 0 (St ) ξt0 (St ) σ e (St )2 dt + 21 f 00 (St ) σ which implies that the semimartingale decomposition of S under P∗ is dSt = ξt0 (St ) σ e (St )2 dt + σ e (St ) dβt where β is a standard Brownian motion under P∗ . Remark 33. The above construction is of course heuristic, but it can be made rigorous using once again Theorem 30. 6. Pathwise Conditioning Before we conclude this paper, we would like to present briefly another kind of conditioning which could be of interest. Precisely, let us assume that an insider is in the following position: He knows that with probability one ∀t > 0,

St ∈ O

where O ⊂ Rd is a non-empty, open, simply connected, and relatively compact set with smooth boundary (the boundary of O shall be denoted by ∂O). Our question is: Can we give to this insider a minimal model which takes into account this information ? We shall answer this question in the case where, under the martingale measure P, S is an homogeneous diffusion with elliptic generator L (the general case where S is only a local martingale seems to need more works). For this, we proceed in two steps: (1) First, we give a sense to the following probability measure P∗ = P(· | ∀t > 0, St ∈ O) (it has to be made, since in general P(∀t > 0, St ∈ O) = 0). (2) Secondly, we compute the semimartingale decomposition of S under P∗ . To fulfill this program, we first define P∗ , as being the weak limit when t → +∞ (it is shown just below that it exists under suitable assumptions) of the following sequence of probabilities Pt = P(· | T∂O = t),

t>0

where T∂O = inf{t ≥ 0, St ∈ / O}.

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It is easily seen, by the strong Markov property of S under P that we have: Pt/Fu =

g(Su , t − u) P/Fu , g(s0 , t)

u < T∂O

where g is defined by P(T∂O ∈ dt | S0 = x) = g(x, t)dt,

(x, t) ∈ O × R+ .

Of course, implicitly, we assume that g such characterized is well defined and positive. Moreover, we shall assume that it is smooth. Proposition 34. We have, for all bounded and F∞ -measurable random variable F , g(Su , t − u) lim E F = E eλ1 u ψ1 (Su ) F t→+∞ g(s0 , t) where λ1 is the smallest (positive) eigenvalue and ψ1 the corresponding eigenfunction of the Dirichlet problem: 1 Lψ + λψ = 0 on O, 2 and ψ/∂O = 0,

ψ1 (s0 ) = 1.

Proof. In fact, this is a direct consequence of the theory of Dirichlet problems for elliptic operators on smooth domains (these problems are widely discussed in Chap. 3 of [28]). Indeed, from this theory, g which solves the following problem: ∂g 1 = Lg on O × R+ and g(x, t) = δ0 on ∂O × R+ , ∂t 2 can be expanded on O × R+ : g(x, t) =

+∞ X

e−λi t ψ˜i (x)

i=1

where 0 < λ1 < ... < λn < ... are the eigenvalues and ψ1 , ..., ψn , ..., corresponding eigenfunctions of the Dirichlet problem 1 Lψ + λψ = 0 on O, and ψ/∂O = 0. 2 The proof follows then almost immediately after straightforward considerations.

From this, we deduce first that the following absolute continuity relationship holds: P∗/Fu = eλ1 u ψ1 (Su ) P/Fu ,

u < T∂O

and secondly from Girsanov’s theorem the semimartingale decomposition of S under P∗ (up to T∂O , but notice that actually T∂O = +∞, P∗ a.s.).


23

Example 35. As a consequence of this, we can deduce for instance that a one-dimensional standard Brownian motion B conditioned by the event {∀t ≤ 0, Bt ∈ (a, b)} with a < 0 < b is a Jacobi diffusion. Example 36. A one-dimensional Brownian motion started from a > 0 and conditioned by the event {∀t ≤ 0, Bt > 0} is a 3-dimensional Bessel process (this example does not fit exactly in our setting, but it is proved exactly in the same way). 7. Conclusion In the present paper, we have shown how to model anticipations on a financial market, in a way which does not suffer the flaws and heavy assumptions of enlargement techniques. Moreover, it is shown in [4] and [5] that from a theoretical point view all the results related to theory of initial enlargement can be recovered from the weak approach by taking for ν the anticipative measure δY . The approach we have taken is can be applied to both complete and incomplete markets. In the complete case, we have obtained the explicit value of the anticipation. It would be very interesting to study more thoroughly some examples of incomplete markets, such as stochastic volatility models. On the other hand, it would also be interesting to apply the present approach to equilibrium price models; for instance, given two agents acting in the same market S with different weak anticipations, what is the equilibrium price of S ? Appendix A. Proof of Theorem 30 For notational convenience, we make the proof in dimension d = 1 but it immediately extends to the d dimensional case. Since for x, y ∈ R2 (x + y)2 ≤ 2 x2 + y 2

(A.1)

we have for t ∈ [0, T ] and n ∈ N (Xtn − Xt )2 ≤

i2 − a (u, Xu )) du hR i2 t +2 0 (σ (u, Xun ) − σ (u, Xu )) dWu . 2

hR

t n 0 (an (u, Xu )

Now, from Cauchy-Schwarz inequality and (A.1) hR i2 Rt t n ) − a (u, X )) du ≤ 2T 0 (an (u, Xun ) − an (u, Xu ))2 du (a (u, X n u u 0 Rt +2T 0 (an (u, Xu ) − a (u, Xu ))2 du.

24


Thus, hR

t n 0 (an (u, Xu )

− a (u, Xu )) du

i2

Rt ≤ 2T K12 0 (Xun − Xu )2 du Rt +2T 0 (an (u, Xu ) − a (u, Xu ))2 du.

On the other hand, from Burkholder-Davis-Gundy inequality, for 0 ≤ t ≤ τ Z t 2 ! Z τ 2 n 2 n E sup (σ (u, Xu ) − σ (u, Xu )) dWu ≤ 4K1 E (Xu − Xu ) du . 0≤t≤τ

0

0

Putting things together, we deduce the following estimation R τ E sup0≤t≤τ (Xtn − Xt )2 ≤ (2T + 8) K12 E 0 (Xun − Xu )2 du R (A.2) t +2T E 0 (an (u, Xu ) − a (u, Xu ))2 du . We apply now Gronwall’s lemma to obtain 2 Rτ 2 E (Xτn − Xτ )2 ≤ 2T (2T + 8) K12 e(2T +8)K1 τ 0 e−(2T +8)K1 u Gn (u) du +2T Gn (τ ) where Z

τ

2

(an (u, Xu ) − a (u, Xu )) du .

Gn (τ ) := E 0

The uniform linear growth assumption on an allows to use the dominated convergence theorem which ensures first the following pointwise convergence Gn →n→+∞ 0. A new use of the dominated convergence theorem in (A.2) gives the expected result, i.e. for 0≤τ ≤T E

sup 0≤t≤τ

(Xtn

2

− Xt )

→n→+∞ 0.

This completes the proof. Remark 37. In fact, with the same kind of proof it is possible to show that under the above assumptions, for p ≥ 2, ! lim E

n→+∞

sup kXtn − Xt k

p

= 0.

0≤t≤T

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