On the dual problem of utility maximization

Transcription

1 On the dual problem of utility maximization Yiqing LIN Joint work with L. GU and J. YANG University of Vienna Sept. 2nd 2015 Workshop Advanced methods in financial mathematics Angers

2 1 Introduction Basic settings Literature review on convex duality methods 2 Utility maximization without random endowment Duality method The dual optimizer: existing result The dual optimizer: alternative method 3 Utility maximization with bounded random endowment Duality method The dual optimizer Remarks

3 Basic settings The market model Consider a filtered probability space (Ω, F, (F t ) 0 t T, P) satisfying the usual conditions, where T is a finite horizon. The market consists of a bond and a stock, where the bond is of zero interest rate and the stock-price process S is a strictly positive semimartingale. No arbitrage condition : M e (S). For an initial value x and a predictable S-integrable trading strategy H, the value process X = (X t ) 0 t T is given by X t = x + (H S) t, 0 t T. We call H admissible if for 0 t T, the associated terminal value X t M, for some positive M. The agent receives an exogenous random endowment e T F T at time T, satisfying ρ := e T <.

4 Basic settings Utility maximization on the positive half line A utility function U : (0, ) R represents the agent s preferences over the terminal wealth. The function U is assumed to be strictly concave, strictly increasing and continuously differentiable satisfying the Inada conditions: U (0) := lim x 0 U (x) =, U ( ) := lim x U (x) = 0. and the condition of reasonable asymptotic elasticity (RAE): AE(U) = lim sup x xu (x) U(x) < 1. The aim of the agent is to maximize the expected utility from the terminal wealth: u(x) := sup E[U(x + (H S) T + e T )], x > 0, H adm

5 Literature review on convex duality methods The Itô framework: [Karatzas-Lehoczky-Shreve-Xu, 1991], [Cvitanić-Karatzas, 1992], etc...; The general semimartingale framework: U : (0, ) R e T = 0 [Kramkov-Schachermayer, 1999] bounded e T [Cvitanić-Schachermayer-Wang, 2001] unbounded e T [Hugonnier-Kramkov, 2004] U : R R: - locally bounded semimartingale models: [Schachermayer, 2001], [Owen, 2002], [Owen-Žitković, 2009]; - general semimartingale models: [Biagini-Frittelli, 2008], [Biagini-Frittelli-Grasselli, 2011]. Optimal consumption, with constraints, etc...

6 1 Introduction Basic settings Literature review on convex duality methods 2 Utility maximization without random endowment Duality method The dual optimizer: existing result The dual optimizer: alternative method 3 Utility maximization with bounded random endowment Duality method The dual optimizer Remarks

7 Duality method Duality method: e T = 0 Define and X (x) := {X : X = x + (H S), H adm, X T 0} C(x) := { g L 0 +(F T ) : 0 g X T, for some X X (x) }, where the latter is the set of positive terminal values, which can be dominated by some admissible strategies initialed from x > 0. Then, the maximization problem (primal problem) can be rewritten into u(x) := sup E[U(x + (H S) T )] = sup E[U(g)]. H adm g C(x)

8 Duality method Definition (Supermartingale deflators) We call a positive semimartingale Y a supermartingale deflator, if for each X X (1), XY is a supermartingale. Moreover, we denote by Y(y) the collection of all such processes starting from y, namely, Y(y) := { Y 0 : Y 0 = y, XY is a supermartingale, X X (1) }. Note D(y) := {h L 0 +(F T ) : 0 h Y T, for some Y Y(y)}. Then, the dual problem is formulated as where v(y) = inf E[ V (Y T ) ] = inf E[ V (h) ], Y Y(y) h D(y) V (y) := sup{u(x) xy}, y > 0. x>0

9 Duality method Theorem ([Kramkov-Schachermayer, 1999]) Assume Then No arbitrage condition : M e (S), U satisfies the Inada conditions and RAE, u(x) <, for some x > 0. The value functions u, v have the same properties as U and V. For any y > 0, there exists a unique dual optimizer ĥ D(y). Let ŷ := u (x), then there exists a unique primal solution ĝ C(x), which is defined by ĝ := (U ) 1 (ĥŷ). E[ĝĥ] = xŷ.

10 The dual optimizer: existing result In general, the optimal element Ŷ from Y(ŷ) associated with the dual optimizer ĥŷ is not a martingale. An example can be found in [Kramkov-Schachermayer, 1999] even with an one-period model. However, under certain condition, Ŷ is proved a local martingale. Theorem ([Larson-Žitković, 2007]) In addition to the conditions for the above theorem, suppose that S is continuous, then the dual optimizer ĥŷ is attained by a local martingale from the set of supermartingale deflators. Outline of the proof: That S is continuous and M e (S) implies the following representation ([Delbean-Schachermayer, 1995]): S t = 1 + M t + t 0 λ u d M u, 0 t T, where M is a local martingale and λ is a predicable M-integrable process.

11 The dual optimizer: existing result Proposition For any Y Y(y), we have the following multiplicative decomposition Y = ye( λ M)E(L)D, where L is a càdlàg local martingale satisfying M, L 0, and D is a predictable, non-increasing, strictly positive, càdlàg process with D 0 = 1. It can be verified by Itô s formula that ye( λ M)E(L) Y(y). Then, from the fact that V is strictly decreasing, one can deduce that Ŷ = ŷe( λ M)E( L), which is a local martingale, namely, D 1.

12 The dual optimizer: alternative method We would like to provide an alternative method to prove the same theorem [Larson-Žitković, 2007]. Based on the same idea, we could generalize this theorem to the case of bounded random endowment in the next section. The idea is as follows: we first stop the process X by a sequence of stopping times {τ k } k N, such that before each τ k, X is bounded away from 0. Precisely, define a localizing sequence τ k := inf{t : X t < 1/k} T. Since X is continuous, Xτk 1/k and P(lim k τ k = T ) = 1. Then, we shall construct a process Ŷ, such that ŶT = ĥŷ and prove that the stopped process Ŷ τ k is a martingale by means of that XŶ is a uniformly integrable martingale. Finally, we prove that Ŷ D(ŷ).

13 The dual optimizer: alternative method From the result in [Kramkov-Schachermayer, 1999], one can find a sequence {Q n } n=1 from Me (S) such that ŷ dqn dp ĥŷ, a.s.. We denote by Y n the associated density process, which is a martingale, i.e., Yt n := ŷ dqn dp. Ft Then, we construct a process Ŷ in terms of {Y n } n=1, such that Ŷ T = ĥŷ. To this end, we need the following lemma.

14 The dual optimizer: alternative method Lemma ([Czichowsky-Schachermayer, 2014]) Let {Y n } n=1 be a sequence of non-negative optional strong supermartingales Y n = {Yt n } 0 t T starting at Y0 n = y. Then there is a sequence {Ỹ n } n=1 of convex combinations Ỹ n conv(y n, Y n+1, ) and a non-negative optional strong supermartingale Ŷ = {Ŷt} 0 t 1 such that for every [0, T ]-valued stoppting time τ, we have convergence in probability, i.e., Ỹ n τ Ŷτ. WLOG, we may choose a subsequence such that for each k, Ỹ n τ k Ŷτ k, a.s..

15 The dual optimizer: alternative method In our case, Ỹ n are all true martingales associated with the equivalent local martingale measure Q n defined by ŷ d Q n dp = ỸT. Obviously, Ỹ T n ŶT = ĥŷ. Fixing k, by super-replication theorem, we have E[ X τk Ŷ n τ k ] xŷ. On the other hand, by applying the above lemma again, one can see that XŶ is an optional strong supermatingale. Furthermore, X 0 Ŷ 0 = E[ X T Ŷ T ] = E[ĝĥ] = xŷ, then XŶ is a true martingale. Therefore, E[ X τk Ŷ τk ] = xŷ.

16 The dual optimizer: alternative method Lemma Let X L 0 (Ω, F, P), X a > 0, a.s., and {Y n } n=1 L1 +(Ω, F, P), Y n Y, a.s.. If E[XY ] lim inf n E[XY n ]. Then, {Y n } n=1 is uniformly integrable. By the above lemma, we know from { E[ X τk Yτ n k ] xŷ; E[ X τk Ŷ τk ] = xŷ. that {Yτ n k } n=1 is uniformly integrable and thus, the stopped process Ŷ τ k is a true martingale. Thus, Ŷ is a local martingale and has a càdlàg version. Moreover, by the lemma in [Czichowsky-Schachermayer, 2014] again, we can verify that for each X X (1), XŶ is a supermatingale.

17 1 Introduction Basic settings Literature review on convex duality methods 2 Utility maximization without random endowment Duality method The dual optimizer: existing result The dual optimizer: alternative method 3 Utility maximization with bounded random endowment Duality method The dual optimizer Remarks

18 Duality method Duality method: e T bounded Recall the primal problem of the utility maximization u(x) := sup E[U(x+(H S) T +e T )] = sup E[U(x+g+e T )], x > 0, H adm g C 0 where C 0 := {g : g = (H S) T, H adm}.

19 Duality method The dual problem is formulated by { [ )] } v(y) := inf E V (y dqr + y Q, e T, y > 0, Q D dp where { D := Q (L ) + : Q (L ) = 1, Q, g x, } for all g C(x), for all x > 0. From the result in [Yosida-Hewitt, 1952], for any Q (L ) +, Q can be uniquely decomposed into Q = Q r + Q s, where Q r is countably additive and Q s is purely finitely additive.

20 Duality method Theorem ([Cvitanić-Schachermayer-Wang, 2001]) Assume Then No arbitrage condition : M e (S), U satisfies the Inada conditions and RAE, u(x) <, for some x > ρ. The value functions u, v have the same properties as U and V. The dual solution Q y D exists for all y > 0 and Q r y is unique. ( ) For all x > x 0 := sup Q D Q, e T, ĝ := I ŷ d Q r ŷ dp x e T C 0 is the solution to the primal problem, where ŷ = u (x). Denote by Ĥ the corresponding optimal strategy, then Q r ŷ, x + (Ĥ S) T + e T = Qŷ, x + (Ĥ S) T + e T = x + Qŷ, e T.

21 The dual optimizer Theorem (Main result) In addition to the conditions for the above theorem, we assume that the filtration is Brownian, then the regular part of the dual optimizer Q r ŷ can be attained by some local martingale Ŷ Y(1). Outline of the proof: for simplicity of notation, we drop the subscript ŷ in Qŷ. We prove that the dual optimizer Q can be approximated by a sequence {Q n } n N from M e (S) such that dq n dp d Q r dp, a.s., and Qn, e T Q, e T.

22 The dual optimizer For each n, denote by Y n the density process associated with Q n. We choose a sequence {Ỹ n } n=1 of convex combinations Ỹ n conv(y n, Y n+1, ), and a non-negative optional strong supermartingale Ŷ, such that Ỹ n τ Ŷτ in probability, for any finite stopping time τ. For each n, denote by Q n the equivalent martingale measure determined by Ỹ n. We define a fictional wealth process by W n t := x + (Ĥ S) t + E Q n [e T F t ] = x + (Ĥ S) t + ẽ n τ. Then, the fictional optimal wealth process can be construct in a similar way as the step above: Ŵ t := x + (Ĥ S) t + ê t, where for any finite stopping time τ, ẽ n τ ê τ in probability.

23 The dual optimizer The process Ŵ Ŷ can be proved a martingale. Thanks to the assumption on the filtration, one can find a sequence of stopping times {τ k } k N such that on 0, τ k, Ŵ stays above 1/k. Consider a cluster point Q of { Q n } n N, which is still a dual optimizer and Q r Q r by the uniqueness. We can prove that Ŷ = d(q F ) r dp. Fixing k, (Q Fτk ) r, x + (Ĥ S) τ k + ê τk = x + Q, e T.

24 The dual optimizer It also can be shown that (Ĥ S) and ê is a martingale under the finitely additive measure Q, namely, Q Fτk, x + (Ĥ S) τ k + ê τk = x + Q, e T. Because x + (Ĥ S) τ k + ê τk 1/k, we can compare the above equality with (Q Fτk ) r, x + (Ĥ S) τ k + ê τk = x + Q, e T, and deduce that (Q Fτk ) s 0, which implies that E[Ŷτ k ] = 1. By Scheffé s lemma, we conclude that {Ỹ τ n k } n N is uniformly integrable and thus, Ŷ is a local martingale from Y(1).

25 Remarks In the case that e T 0, we need a condition on the filtration instead of only assuming that S is continuous. That is because we do not have enough information on the fictional process ê so that it is difficult to stop the fictional optimal wealth process Ŵ and let it stay away from 0. if we could do better? namely, could we find a martingale associated with the dual optimizer? - If e T = 0, [Kramkov-Weston, 2015] have a positive answer under some (A p ) condition over the dual domain. - If e T is uniformly bounded, [Larsen-Soner-Žitković, 2015] have a counterexample with a geometric brownian motion stock price process.

26 Remarks In the case that U supports the whole real line and S is locally bounded, Bellini, Frittelli, Owen, Schachermayer, Žitković, observe that the dual optimizer does not lose any mass. However, it may not be equivalent to P (only absolutely continuous). If we consider the numeraire based model in a market with proportional transaction cost, we can deduce a similar result when e T = 0, i.e., if S is continuous and satisfying (NUPBR), the dual optimizer is attained by some local martingale from the set of supermatingale deflators. The case that e T is uniformly bounded is under consideration.

27 Remarks Thank you for your attention!