THE PRICING PROBLEM WITH INSIDE INFORMATION: ARBITRARY Ω ♦ 123

ˆ It remains to prove that the rst dual feasibility condition holds for y¯₁, i.e. that

∅S(0)dP trivially, it only remains to show that E[¯y₁S(1)] = E[S(0)] = S(0). From the previous item it

where the nal equality follows becauseQ is a martingale measure w.r.t.

(G_t). But this implies that the rst dual feasibility equation holds as well.

Hence, for each absolutely continuous martingale measure, there is a fea-sible dual solution.

Conversely, assume there exists a feasible dual solutiony1,y¯tfort= 1, ..., T− 1. We want to show that this dual solution corresponds to an equivalent martin-gale measure. Dene Q(F) :=R

Fy1(ω)dP(ω)for all F ∈ F (note thaty1≥0, since it is feasible in the dual problem, and that one may assume E[y1] = 1, since the dual problem is invariant under translation). The problem is to prove that for any y¯t, the dual feasibility conditions can be interpreted as martingale conditions.

is, from the denition of conditional expectation, equivalent to

¯

yT−1S(T−1) =E[y1S(T)|G_T−1] =E[y1|G_T−1]EQ[S(T)|GT−1] where the last equality follows from Lemma 6.8.

From this, it follows thatEQ[S(T)|GT−1] =S(T−1)(which is a martingale equation, so by considering the component corresponding to S0 (the bond), and using that the market is assumed to be normalized, so S0(t, ω) = 1 for all

From the denition of conditional expectation, this is equivalent to From the nal dual feasibility condition, R

Ay¯_T−1S(T −1)dP =R From the denition of conditional expectation, (6.13) is equivalent to

¯

y_tS(t) =E[y₁S(T)|G_t]

=E[y1|Gt]EQ[S(T)|Gt]

where the last equality uses Lemma 6.8. Hence, it suces to show that y¯_t = E[y₁|G_t].

By considering equation (6.13) forS₀ (the bond) Z

which, from the denition of conditional expectation, implies thaty¯_t=E[y₁|G_t]. Hence,E[S(T)|G_t] =S(t)fort= 1, ..., T −1.

From the rst dual feasibility equation, it follows that E[S(T)|G₀] =S(0): The rst dual feasibility condition states that

Z

From the same argument as before, using the second and third dual feasibil-ity equations, and that G0 ⊆ Gt for all t ≥ 0, it follows that R

6.2. THE PRICING PROBLEM WITH INSIDE INFORMATION: ARBITRARYΩ ♦125 so the martingale condition is OK also for t = 0. Note that E_Q[S(T)|GT] =

S(T), sinceS(T)isG_T-measurable.

Finally, to generalize this to the regular martingale condition E[S(t)|G_s] = S(s) for s≤t, s, t ∈ {0,1, ..., T}. Choose s and t such that s≤ t, and s, t ∈ {0,1, ..., T}. Then

EQ[S(t)|Gs] =EQ[EQ[S(T)|Gt]|Gs]

=E_Q[S(T)|Gs]

=S(s).

Here, the rst equation follows from the martingale condition that has already been proved, and the second equality comes from the rule of double expectation, and that Gs⊆ Gtfors≤t.

Hence,Qturns the price processSinto a martingale, and each feasible dual solution corresponds to an absolutely continuous martingale measure w.r.t. (Gt), i.e. Q∈ M^a(S,G).

To summarize, it has been proven that the dual problem of the seller's pricing problem (in a normalized market) for a seller with information corresponding to the ltration (Gt)t can be rewritten

sup_Q∈Ma(S,G)EQ[B], (6.14)

whereM^a(S,G)denotes the family of all absolutely continuous martingale mea-sures with respect to the ltration(Gt)t.

As for proving that there is no duality gap, i.e. that the value of problem (6.14) is equal to the value of problem (6.4), this can be done via Theorem 9 in Pennanen and Perkkiö [30], which will be called Theorem 6.9 in this thesis. This theorem is based on a conjugate duality setting similar to that of Rockafellar, and gives conditions for the value functionϕ(see Section 2.3) to be lower semi-continuous. Hence, from Theorem 2.44 (which is Theorem 7 in Rockafellar [34]), if these conditions hold, there is no duality gap (since ϕ(·) is convex, because the perturbation functionF was chosen to be convex).

The theorem is rewritten to suit the notation of this thesis.

Theorem 6.9 Assume there exists a y∈Y and anm∈ L¹(Ω,F, P)such that forP-a.e. ω∈Ω

F(H, u, ω)≥u·y(ω) +m(ω)for all(H, u)∈R^T(N+1)×R^2T, (6.15) where · denotes the standard Euclidean inner product. Assume also that A:=

{H ∈ H_G : F^∞(H(ω),0, ω) ≤ 0 P−a.s.} is a linear space. Then, the value function ϕ(u) is lower semi-continuous on U and the inmum of the primal problem is attained for all u∈U.

In Theorem 6.9, H ∈R^T(N+1) is a vector representing a stochastic process H with N + 1 components at each time t ∈ {0,1, . . . , T −1} and HG denotes the family of all stochastic processes that are adapted to the ltration (Gt)_t.

F^∞ is the recession function of F. In general, if h(x, ω) is a function and dom(h(·, ω))6=∅,h^∞is given by the formula (see Pennanen and Perkkiö [30])

h^∞(x, ω) = sup

λ>0

h(λx+ ¯x, ω)−h(¯x, ω)

λ (6.16)

(which is independent ofx¯).

For the proof of Theorem 6.9, see Pennanen and Perkkiö [30].

Actually, there is one dierence between the frameworks of Rockafellar [34], and Pennanen and Perkkiö [30]. In [30] it is assumed that the perturba-tion funcperturba-tion F is a so-called convex normal integrand (in order to be able to change the order of minimization and integration). Example 1 in Penna-nen [29] considers the same choice of perturbation functionF as we have cho-sen above, and this examples states that F is a convex normal integrand if the objective function and the constraint functions of the primal problem are convex normal integrands. Clearly, for our primal problem (6.7), all of these functions are convex. To check that they are normal integrands, an example from Rockafellar and Wets [36] will be applied. Example 14.29 in [36] states that all Caratheodory integrands are normal integrands, i.e. that all functions f :Rⁿ×Ω→¯

Rsuch thatf(x, ω)is measurable inωfor eachx, and continuous in x for each ω ∈ Ω, is a normal integrand (where x may depend measur-ably on ω). Hence, we check that this holds for one of the constraint func-tions of the primal problem (6.7). Let ft(H, ω) :=S(t, ω)·∆H(t, ω). Choose H ∈ R^(N+1)T. Then, ft(H, ω) = S(t, ω)·∆H(t) (where H(t) denotes the part of the vector H ∈ R^(N+1)T corresponding to time t) is F-measurable, since S(t) is Ft-measurable for all t and Ft ⊆ F. Now, choose ω ∈ Ω, then ft(H, ω) =S(t, ω)·∆H(t), which is continuous inH (since it is linear). Hence, ft is a convex, normal integrand. Since the objective function and constraint functions of the primal problem (6.7) are similar (and B is FT-measurable), the same type of arguments prove that (from Example 14.29 in Rockafellar and Wets [36]) all of these functions are convex normal integrands. Hence (from Ex-ample 1 in Pennanen [29]),F is a convex normal integrand, and therefore, the framework of Pennanen and Perkkiö, in particular Theorem 6.9, can be applied.

Now, assume that the set A in Theorem 6.9 is a linear space. We would like to prove that the inequality (6.15) holds for the perturbation function cor-responding to the primal problem (6.4).

As shown previously, the perturbation function takes the form F(H,(u, v₁, v₂, w)) :=S(0)·H(0)

ifB(ω)−S(T, ω)·H(T −1, ω)≤u(ω)for all ω ∈Ωand S(t, ω)·∆H(t, ω)≤ v₁^t(ω) for all t ∈ {1, ..., T −1}, ω ∈ Ω, −S(t, ω)·∆H(t, ω) ≤ v^t₂(ω) for all t ∈ {1, ..., T −1}, ω ∈ Ω, −S(0)·H(0) ≤ w and F(H,(u, v1, v2, w)) := ∞ otherwise.

Now, choosey(ω) := (0,(0)_t,(0)_t,−1)for allω ∈Ω. Then,y∈ L^q (since P is a nite measure). Also, choose m(ω) =−1for allω ∈Ω. m∈ L¹ sinceP is a nite measure.

6.2. THE PRICING PROBLEM WITH INSIDE INFORMATION: ARBITRARYΩ ♦127