Dynamic Utilities. Nicole El Karoui, Mohamed M Rad. UPMC/Ecole Polytechnique, Paris

Transcription

1 Dynamic Utilities and Long Term Decision Making Nicole El Karoui, Mohamed M Rad UPMC/Ecole Polytechnique, Paris Ecole CEA EDF Inria, Rocquencourt, 15 Octobre 2012 Systemic Risk and Quantitative Risk

2 Plan 1 Utility function 2 Progressive and Consistent Dynamic Utilities Framework and Definition 3 Concavity and SDE 4 Dynamics of the conjugate dynamic Utility 5 Consistent Dynamic Utility 6 X -consistent utilities with given optimal portfolio 7 Examples based on Mixture of stochastic utilities Dynamic utility with consumption 8 Applications to Interest Rates

3 Plan 1 Utility function 2 Progressive and Consistent Dynamic Utilities 3 Concavity and SDE 4 Dynamics of the conjugate dynamic Utility 5 Consistent Dynamic Utility 6 X -consistent utilities with given optimal portfolio 7 Examples based on Mixture of stochastic utilities

4 Some History Generalities Since Bernoulli : Paradox of St Petersbourg Dou you prefer 1 dollar today or play to a lottery with gain 100 dollars Basically, Bernoulli assumed that the value given to a particular wealth amount depends on its relative importance to total wealth. The utility function is a function of capital that can associate a certainty equivalent to a given bet, as it is indifferent to take the bet or its certainty equivalent. Logarithmic utility u(x) = log x

5 Regular Utility Function A regular utility function u is a (positive) function defined on [0, ) Concave : E(u(X)) u(e(x)) Increasing Inada condition : u(x) is a C 2 -function with marginal utility u x(.), decreasing from + to 0. Convex Conjugate Utility ũ ũ is the Fenchel transform of u(. x) Under Inada condition, ũ(y) = sup x>0 ( u(t, x) x y ) The optimum is achieved at u x(x ) = y, and ũ y = (u x(.)) 1 ũ(y) = u( ũ y(y)) + yũ y(y) Certainty equivalent : E(u(X)) = u(c(x)) concavity = c(x) E(X)

6 Risk Aversion coefficient Quantities of interest Risk Aversion coefficient α(x) = u xx(x) u x (x), relative ˆα(x) = xu xx(x) u x (x), Risk tolerance coefficient τ(x) = (α(x)) 1 Typical example : power utility For α (0, 1), u(x) = x 1 α 1 α with conjugate ũ y(y) = y 1/α

7 Basic optimization problem Define the horizon T of the problem for given X convex family of random variables X T, and some state price density Y T with E(Y T ) 1, max{e(u(x)) X X, with budget constraint E(Y T X T ) x} Solution via duality : Lagrange multiplier technics The problem is equivalent to : max{e(u(x) + y(x Y T X T ) X X } If ũ y(yy T ) X, then optimum is XT = ũ y(yy T ) y is selected by achieved the budget constraint, if it is possible E[ ũ y (yy T )Y T ] = x

8 Performance measurement in debate Performance and risk measurement are fundamental in mathematical finance, risk-management and portfolio optimization An old new question For a long time, expected utility has been the standard for dynamic risk Extended into a robust formulation by taking into account ambiguity on the reference probability measure by min-max point of view max XT X min Q Q E Q [u(x T )] In relation with risk measures (Foellmer, Schied) defined as ρ(x T ) = max Q Q [E Q (X T ) α(q)]

9 Prospect theory Motivated by the difficulty to the agents to describe their own utility, experimental finance (Zhou) shows that prospect theory or distorsion point of view are more flexible than the robust point of view (Fritelli) max F ( E Q [u(ξ X T )] ) µ(dq) X T X M + 1

10 Dynamic Value Function of Classical Utility Problem Time consistency Given a utility function u(t, x) at given time horizon T, the problem at time r is to maximize over all admissible portfolios starting from (r, x), the conditional expected utility of the terminal wealth, V (r, x, (u, T )) = ess sup X X (r,x) E(u(T, X T )lf r ) Dynamic programming principle V (t, X t, (u, T )) = V (t, X t, (V (t + h,., (u, T )), t + h)), a.s. Maximum principle = Comparison theorem = concavity of V (r, x). Conc : V (t, x, (u, T )) = ess sup X X (t,x) E[V (t + h, X t+h, (u, T )) F t ]

11 Dynamic point of view Dynamic recursive utility Expected utility is overly restrictive in expressing reasonable risk aversion in temporal setting Intertemporal substitution and risk aversion are inflexibly linked Stochastic Differential Utility (Duffie, Epstein, Skiadas...) : the local variation is depending on the expected future utility ; BSDE s point of view du t (ξ T ) = g(t, U t, Z t )dt Z t dw t, U T (ξ T ) = u(ξ T ) for given u For a given terminal utility function u, A solution consists into two processes The progessive utility U t(ξ T ) The progressive diffusion coefficient Z t

12 Investment Banking and Utility Theory Remarks and Comments from M.Musiela, T.Zariphopoulo ( ) Classical or recursive utilities are defined in isolation to the investment opportunities given to agent The investor may want to use intertemporal diversification, I.e. implement short, medium, long term strategies Need of intertemporal consistency of optimal strategies. Can the same utility function be used for all time horizon? At the optimum the investor should become indifferent to the investment horizon. which utility for long term decision making? + C.Rogers +Berier+Tehranchi, Henderson-Obson, Zitkovic ( )

13 7 Examples based on Mixture of stochastic utilities Plan 1 Utility function 2 Progressive and Consistent Dynamic Utilities Framework and Definition 3 Concavity and SDE 4 Dynamics of the conjugate dynamic Utility 5 Consistent Dynamic Utility 6 X -consistent utilities with given optimal portfolio

14 Progressive Utility Definition of Progressive Utility A progressive utility is a positive family U = {U(t, x) : t 0, x > 0} Progressivity : for any x > 0, t U(t, x) is a progressive random field Concavity : for t 0, x > 0 U(t, x) is an increasing concave function. Inada condition : U(., x) is a C 2 -function with marginal utility U x(.,.), decreasing from + to 0. Initial condtion : u a deterministic positive C 2 -utility function with Inada condition Convex Conjugate Dynamic Utility Ũ Ũ is the Fenchel transform of U(. x) Under Inada condition, Ũ(t, y) = sup x>0,x Q + ( U(t, x) x y ) The optimum is achieved at U x(t, x ) = y, and Ũ y(t, ) = (U x(t,.)) 1 (y) Ũ(t, y) = U(t, Ũ y(t, y)) + yũ y(t, y)

15 Consistent Dynamic Utility Let X be a convex family of non negative portfolios, called Test porfolios An X -consistent dynamic utility U(t, x) is a progressive utility s.t Consistency with the family of test portfolios For any admissible wealth process X X, E(U(t, X t )) < + and E(U(t, X t )/F s ) U(s, X s ), s t. Existence of optimal For any initial wealth x > 0, there exists an optimal wealth process (benchmark) X X, X 0 = x, U(s, X s ) = E(U(t, X t ) F s) s t. In short for any admissible wealth X X, U(t, X t ) is a supermartingale, and a martingale for the optimal benchmark X.

16 A General Market Model I Incomplete Market : Let W be a n-brownian motion, a short rate process r t and a risk premium vector η t, and X the class of (positif) wealth processes X κ driven by the self-financing equation [ dxt κ = Xt κ rt dt + κ t.(dw t + ηt σdt)], ηt σ, κ t R σ t σ t is the dxn volatility matrix, and σ t.σ t is invertible. Let π t be the wealth proportions invested in the different assets, and κ t = σ t π t, Constraints : R σ t is a family of adapted subvector spaces in R n, typically R σ t = σ t (R d ), d n.

17 A General Market Model II η σ t R σ t defined as the projection of η t on R σ t is the minimal risk premium, All processes are adapted with good integrability properties Def A process Y is said to be a state price density or (adjoint process) if for any κ R σ, Y. X κ. is a local martingale there exists ν R σ, : dy ν t Y ν t = r t dt + (ν t η σ t ).dw t, ν t R σ, t

18 Progressive Utility of Itô Type Assume the progressive utility U to be a family of Itô semimartingales with local characteristics (β, γ) (β=drift, γ= diffusion) du(t, x) = β(t, x)dt + γ(t, x)dw t Assume the conjugate progressive utility Ũ to be also of Itô type. Open questions at this stage dũ(t, x) = β(t, x)dt + γ(t, x)dw t Under which assumptions on (β, γ) the solution is concave and increasing, What kind of relationship between (β, γ) and ( β, γ)? Under which assumptions on (β, γ) only, Ũ is also of Itô type Main difficulties come from the forward definition : Absence of maximum principle or comparison theorem.

19 Plan 1 Utility function 2 Progressive and Consistent Dynamic Utilities 3 Concavity and SDE 4 Dynamics of the conjugate dynamic Utility 5 Consistent Dynamic Utility 6 X -consistent utilities with given optimal portfolio 7 Examples based on Mixture of stochastic utilities

20 Concavity and Stochastic DIfferential Equation Concavity and SDE Let us consider a progressive differentiable random field U, such that U and U x are Itô random fields with local characteristics (β, γ) and (β x, γ x ). (i) NECESSARY CONDITION If U is a progressive utility with conjugate Ũ. Then U x (t,.) is decreasing in x from to 0, with inverse Ũy(t,.). du x (t,.x) = β x (t, x)dt + γ x (t, x).dw t (ii) Intrinsic SDE Then U x (., x) = Z. (u x (x)), where Z. (z) is a strong solution of the following intrinsic SDE, dz t = µ(t, Z t )dt + σ(t, Z t ) dw t, Z 0 = z with coefficients µ(t, z) = β x (t, Ũy(t, z)), σ(t, z) := γ x ( t, Ũ y (t, z) with µ(t, 0) = 0, σ(t, 0) = 0 which is increasing and differentiable on z with range (0, ).

21 Utility and Primitive of Intrinsic SDE dz t = µ(t, Z t )dt + σ(t, Z t )dw t, Z 0 = z Characterization as primitive of monotone SDE If the SDE has a unique strong solution Z. (z), increasing and differentiable in z from 0 to, For any utility u, Z t (u x (x)) is positive, decreasing progressive random field, with range (, 0). If Z. (u x (x)) is integrable in a neighborhood of x = 0, the primitive {U(t, x) = x 0 Z t(u x (z))dz, t 0, x > 0} is a progressive utility.

22 SDE with random coefficients Protter, Kunita books Lipschitz condition Let the one-dimensional SDE, dz t = µ(t, Z t )dt + σ(t, Z t )dw t, Assume there exists C t and K t with T 0 (C t + Kt 2 )dt < +. Assume that µ(t, 0) 0, σ(t, 0) 0. and µ(t, x, ω) µ(t, y, ω) C t (ω) x y, σ(t, x, ω) σ(t, y, ω) K t (ω) x y Then, for any z R + there exists a unique strong solution Z z of the SDE increasing with respect to its initial condition Z 0 = z. The range of the map z Z (., z) is ]0, + [ and Z (., z) is integrable near to 0 and to infinity.

23 Applications to Progressive Utility SUFFICIENT CONDITIONS If there exist random Lipschitz bounds C t and K 2 t integrable in time such that a.s, β x (t, x) C t U x (t, x), γ x (t, x) K t U x (t, x) β xx (t, x) C t U xx (t, x), γ xx (t, x) K t U xx (t, x) Then the derivatives of the coefficients µ x and σ x are spatially bounded, then the SDE has unique strong solution and U is a progressive utility.

24 Plan 1 Utility function 2 Progressive and Consistent Dynamic Utilities 3 Concavity and SDE 4 Dynamics of the conjugate dynamic Utility 5 Consistent Dynamic Utility 6 X -consistent utilities with given optimal portfolio 7 Examples based on Mixture of stochastic utilities

25 Itô s Ventcell Formula For technical regularity problems see books of Kunita, or Carmona-Nualart. The identity Ũ(t, y) = U(t, Ũy(t, y)) + yũy(t, y) is based on the C 2 random field U along the random process Ũy(t, y). Need to extension of the Itô s formula. Itô s Ventcell Formula Let F (t, x) be a C 2 Itô random field (β, γ), such that F x (t, x) is associated with (β x, γ x ). For any Itô semimartingale X, df (t, X t ) = β(t, X t )dt + γ(t, X t ).dw t + F x (t, X t )dx t F xx(t, X t ) dx t + df x (t, x), dx t x=xt

26 Conjugate utility dynamics Recall the identity Ũ(t, y) = U(t, Ũy(t, y)) + yũy(t, y) Apply this result to F (t, x) = U(t, x) + xy with X t = Ũy(t, y) (assumed to be Itô), by observing that F x (t, x) = 0 when x = Ũy(t, y). Dynamics of the conjuguate utility Assume (U, Ũ) with characteristics (β, γ) and ( β, γ) and (U x, Ũy) associated with the derivatives. dũ(t, y) =γ(t, Ũy(t, y)).dw t + β(t, Ũy(t, y))dt + where γ x ( t, Ũ y (t, z)) = σ(t, z) 1 2Ũyy(t, y) γ x ( t, Ũ y (t, y) ) 2 dt

27 Marginal Conjuguate Utility Dynamics of the marginal conjuguate utility Let (µ, σ) be the random coefficients of the SDE associated with U x ( µ(t, z) = β x (t, Ũy(t, z)), σ(t, z) := γ x t, Ũ y (t, z) Define L σ,µ to be the adjoint operator, L σ,µ = 1 2 y( σ(t, y) 2 y ) µ(t, y) y. Then the inverse of U x, Ũy is a monotonic solution of the SPE, with initial condition Ũy(0, y) = ũ y (y), and conversely by verification. dg(t, y) = G y (t, y)σ(t, y).dw t + L σ,µ (G)(t, y)dt Other application : dynamic copula

28 Consistent Dynamic Utilities Assume U to be X -consistent. How express on (β, γ) the supermartingale property of U(t, X κ. ) Is the convex conjugate utility associated with the same kind of optimization problem? Existence of optimal solutions? In the classical backward framework, Open questions By maximum principle, U x(t, X t (x)) = Y t (u x(x)). Y. (y) is the optimal solution of the dual problem Is these properties still hold true Regularity of Xt (x) and Yt (y) with respect of their initial condition? If X t (x) is monotone, U x(t, x)) = Y t (u x((x t (x)) 1 ))?

29 Plan 1 Utility function 2 Progressive and Consistent Dynamic Utilities 3 Concavity and SDE 4 Dynamics of the conjugate dynamic Utility 5 Consistent Dynamic Utility 6 X -consistent utilities with given optimal portfolio 7 Examples based on Mixture of stochastic utilities

30 Drift Constraint Let U be a Itô-Ventzel regular utility and X. κ an admissible wealth du(t, x) = β(t, x)dt + γ(t, x)dw t, Itô-Ventcel Formula du(t, X κ t ) = β(t, X κ +U x (t, X κ t )X κ t κ t dx t + HJB type constraints dx κ t = X κ t [r t dt + κ t.(dw t + η σ t dt)], t )dt + γ(t, Xt κ ).dw t + γ x (t, Xt κ ), Xt κ κ t dt. ( U x (t, Xt κ )r t Xt κ U xx(t, Xt κ )(Xt κ ) 2 κ t 2) dt du(t, Xt κ ) = ( U x (t, Xt κ )Xt κ ( + β(t, Xt κ ) + U x (t, Xt κ )r t Xt κ κ t + γ(t, X κ t ) ).dw t U xx(t, X κ t )Q(t, X κ t, κ t ) ) dt, where Q(t, x, κ) := xκ 2 + 2xκ. ( U x (t, x)ηt σ + γ x (t, x) ). U xx (t, x)

31 Verification Theorem Verification Theorem Let γ σ x be the orthogonal projection of γ x on R σ ; and Q (t, x) = inf κ R σ Q(t, x, κ) ; The minimum of this quadratic form is achieved at the optimal policy κ ( xκ t (x) = 1 U xx (t,x) Ux(t, x)ηt σ + γx σ (t, x) ) x 2 Q (t, x) = 1 U xx (t,x) 2 U x(t, x)η σ t + γ σ x (t, x)) 2 = xκ t (x) 2 Drift constraint β(t, x) = U x (t, x)r t x U xx(t, x) xκ t (t, x) 2 Volatility The volatility γ(t, x) verifies U x(t, x)ηt σ + γ x(t, x) = xu xx(t, x)κ t (x) ν (t, x) : ν (t, x) R σ, t Decreasing utility When γ(t, x) 0, classical optimal strategy U x(t, x)η σ t = U xx(t, x)xκ t (x).

32 Optimal Wealth Utility Stochastic PDE If κ (t, x) is sufficiently smooth so that x > 0 the equation dx t = Xt [ rt dt + κ t (Xt ).(dw t + ηt σ dt) ] has at least one positive solution X, then U(t, Xt ) is a local martingale. if the local martingale ( U(t, Xt )) is a martingale, then the t 0 progressive utility U is a X -consistent stochastic utility with optimal wealth process X. The semimartingale U x (t, Xt ) is a state price density process, du x (t, Xt ) = U x(t, Xt )[ r t dt + ( η U, (t, Xt ) ησ t )) ].dw t where η U, t (t, x) = γ x U x (t, x) is the orthogonal utility risk premium t

33 Conjugate SPDE Convex conjugate SPDE Let Ũ be the conjugate of U, with Itô-Ventzel regularity, then dũ(t, y) = β(t, Ũ y(t, y))dt + γ(t, Ũ y(t, y))dw t where β(t, x) = β(t, x) 1 γ x(t, x) 2 2 U xx(t, x) β(t, x) is the solution of a minimization program achieved by the projection of η σ t ν (t, x) U x(t, x) γ x(t, x) on (R σ ), defined before as In new variable, γ(t, y) = γ(t, Ũ y(t, y)), β(t, y) = β(t, Ũ y(t, y)) β(t, y) = r t yũ y(t, y) + 1 2Ũ yy ( ( η σ t yũ yy + γ,σ y ) 2 γ y 2) (t, y)

34 Convex consistent dual utility Consistent Conjuguate Utility Under previous assumption, The conjugate utility Ũ(t, y) is a convex decreasing stochastic flow, consistent with the family Y of semimartingales Y ν, defined from dy ν t = Y ν t [ r t dt + (ν t η σ t )dw t, ν t (R σ t ) ] There exists a dual optimal choice ν (t, y) = ν (t, Ũ y(t, y)) From any y > 0, the optimal dual process Yt (y) = Yt ν (y) satisfies Yt (u x(x)) = U x(t, Xt (x)) If X t (x) is strictly monotone in x, by taking its inverse X (t, x), we obtain that U x(t, x) = Y t (u x ((X (t, x))). No trivial calculation via stochastic calculus method.

35 New parametrization of the SPDE Volatility versus optimal strategies There is a one to one correspondence between the derivative of the volatility γ x and the optimal strategies κ and ν, γ σ x (t, x) = U xx (t, x)xκ (t, x) U x (t, x)η σ t γ x (t, x) = U x (t, x)ν, (t, U x (t, x)) Using the notation f (t) = t f (s)ds for the primitive of f, we have 0 γ σ (t, x) = U xx (t, x)xκ (t, x) U(t, x)η σ t γ (t, x) = ν, (t, U x (t, x))

36 Plan 1 Utility function 2 Progressive and Consistent Dynamic Utilities 3 Concavity and SDE 4 Dynamics of the conjugate dynamic Utility 5 Consistent Dynamic Utility 6 X -consistent utilities with given optimal portfolio 7 Examples based on Mixture of stochastic utilities

37 Dynamic Utilities with given optimal portfolio Methodology : Let us start with a given optimal portfolio X. (x), In the classical utility optimization backward problem, He & Huang (1994) (in Markovian framework) try to characterize the terminal utility function, with a given optimal wealth X. Constraint on X also. Also interesting point of view of C.Rogers and co author. In the forward problem The problem is to diffuse the initial utility u using the information given by the path of X. Observe that : The diffusion is not on u but on the derivative u x We have also to give the optimal state price density The only constraints are monotonicity of the both diffusion X and Y with respect to their initial condition or equivalently some Lispchitz condition on their coefficients

38 Main Result Reverse Engineering Assumption Assume the two equations admit monotonic solutions dxt = Xt [ rt dt + κ t (Xt ).(dw t + ηt σ dt) ] dyt = Yt (x) [ r t dt + ( νt (Yt ) + ηt σ ] )dw t with inverse processes X and Y Assume that u xx (x)xt (x) has a limit when x goes to infinity Construction Define the processes U and Ũ by U(t, x) = x 0 Y t (u (X (t, z)))dz, Ũ(t, y) = + y X t ( ũ y(y(t, z)))dz. U is a X -consistent stochastic utility satisfying the HJB type SPDE, and Ũ its Y -consistent conjugate utility with optimal proc. X and Y.

39 Plan 1 Utility function 2 Progressive and Consistent Dynamic Utilities 3 Concavity and SDE 4 Dynamics of the conjugate dynamic Utility 5 Consistent Dynamic Utility 6 X -consistent utilities with given optimal portfolio 7 Examples based on Mixture of stochastic utilities

40 Linear optimal processes I Assume that Xt (x) = xxt (1) = x X t and Yt (y) = yxt (1) = y Y t, and Yt Xt a true martingale. The optimal policies X t and Y t The inverse processes are X (t, z) = z/x t do not depend on x and y and Y(t, z) = z/y t For any consistent stochastic utility U, with initial utility u and linear optimal portfolios U x(t, X t (x)) = Y t (u x(x)) = U(t, x) = Y t X t u(x/x t ) There exists an equivalent martingale measure Q, and a numeraire X t such that in the new market consistent utility martingale. Û(t, x) = Y t X t u( x) is a Q dynamic

41 Power Utilities Power utility In particular if u (α) (x) = x 1 α 1 α, α ]0, 1], then U(α) (t, x) = Z (α) t u (α) (x) is a power utility with stochastic adjustment factor Z (α) t = Y t (X t ) α There exists an optimal par (Xt, Y ) with (ακ t = ηt σ, ν = 0) such that Z (α) t is decreasing. Markovian case : Markovian setting for the market with factors ξ t, Z (α) t = φ (α) (t, ξ t ) Backward calibration (He and Huang (1994) in the classical framework) Find conditions to have U (α) (T, x) = u (α) (x) Z (α) T = 1 = X T = (Y T ) 1/α

42 Utility ambiguity Let us consider a agent with ambiguity on his utility ; he can made his choice in a family Ũα of consistent stochastic utilities. α is a parameter with values in I, equipped with a priori probability measure µ(dα) the investor decides to allocate the initial wealth x into different initial wealths x α (x) according to its anticipation, x = I x α(x)µ(dα) he is looking for an optimal strategy as mixture of the individual optimal strategies monotonic in x Xt (x) = I X,α t (x α (x))µ(dα) The main assumption is that the dual utility is a mixture of individual dual utilities Ũ µ (t, y) = Ũ α (t, y)µ(dα)

43 Dual Mixture Dual Mixture,and Sup-convolution Let Ũµ (t, y) = I Ũα (t, y)µ(dα) Ũ µ (t, y) is a convex consistent dual utilities with the same set of admissible state prices densities, if and only if there exists an admissible adjoint process Y t (y) optimal for all utilities Ũα Let ũ µ (y) = I ũα (y)µ(dα) the dual initial utility. Then the optimal wealth of the primal problem is Xt (x) = ( Ũµ y t, Y t (u x µ (x)). It is a mixture of optimal wealths Xt (x) = I X,α t (x α (x))µ(dα), x α (x) = ũy α (u x (x)) Sup-convolution result : Equilibrium, Pareto optimality, U µ (t, x) = ess sup{ U α (t, x α )µ(dα) x α µ(dα) = x} I I

44 Decreasing Utilities Applying this point of view to power decreasing utilities, we obtain one part of the beautiful result of Thaleia,and Rogers, that is the mixture is still decreasing consistent utility, but not the fact there not other decreasing utility. Interpretation in terms of sup convolution may be interesting

45 Dynamic utilities from consumption and wealth I Extend the previous results to the case of dynamic utility functions to take into account that the preferences of the agent may changes with time. To get rid of the dependency on the maturity T. References : Musiela & Zariphopoulou, El Karoui & Mrad (dynamic utility functions from terminal wealth), Berrier & Rogers & Tehranchi (dynamic utility functions from consumption and terminal wealth).

46 Dynamic utility with consumption Definition : Dynamic utility from consumption U 1 and U 2 two positive progressive utilities For all admissible wealth and consumption processes (X x,c,κ (.), c(.)) satisfying dx x,c,κ t = c t dt + dx x,κ t with X x,c,κ t 0 U 2 (t, X x,c,κ t ) + t 0 U1 (s, c s)ds is a supermartingale. There exists an optimal pair (X (.), c (.)) for which it is a martingale. Dual structure A pair of conjugate dynamic utility functions For all state price density Y ν (y), the following process is a submartingale : Ũ2 (t, Yt ν (y)) + t 0 Ũ1 (s, Ys ν (y))ds. Their exists an optimum ν with a martingale property Ũ 1 (t, Y ν t (y)) = c t (y), y = u1 c (c 0 )

47 Plan 1 Utility function 2 Progressive and Consistent Dynamic Utilities 3 Concavity and SDE 4 Dynamics of the conjugate dynamic Utility 5 Consistent Dynamic Utility 6 X -consistent utilities with given optimal portfolio 7 Examples based on Mixture of stochastic utilities

48 Motivations Thanks to Isabelle Camilier, PhD Embedded long term interest rate risk in longevity-linked securities (maturity up to years.) Because of the lack of liquidity for long horizon, the standard financial point of view cannot be easily extended. Abundant literature on the economic aspects of long-term policy-making (Ekeland, Gollier, Weitzman...), often motivated by ecological issues (Hourcade & Lecocq)

49 The Ramsey Rule in Economics I Computation of a long term discount factor R 0 (T ). A representative agent with : u utility function β pure time preference parameter c aggregate consumption. Often a priori hypothesis are made on the form of the consumption function. Ramsey rule (link between consumption and discounting) : R 0 (T ) = β 1 [ u ] T log E (c T ) u. (c 0 ) (deduced from the maximization of the agent s intertemporal utility from consumption with infinite horizon)

50 The Ramsey Rule in Economics II Very popular particular case (Ramsey, 1928) : R 0 (T ) = β + γg, β pure time preference parameter, γ risk aversion, g growth rate. Example : Stern review on climate change (2006), with γ = 1, g = 1.3%, β = 0.1% R 0 (T ) = 1.4%. Controversy between economists concerning parameters values. R 0 (T ) = 1.4% : $ 1 million in 100 years $ 250,000 today. R 0 (T ) = 3.5% : $ 1 million in 100 years $ 32,000 today.

51 US Yields Curves :

52 Consumption Optimization and Ramsey rule Pathwise Link between the state price density process and the marginal utility from consumption. Uc 1 (t, ct ) Uc 1 (0, c0 ) = Y t y t = exp( r s ds)l t (y) 0 where L t (y) is the optimal change of proba measure with volatility η σ t + ν t (c 0 ) Taking the expectation under the historical probability : [ ] [ E P U 1 c (t,c t ) = E P exp( ] t Uc 1 (0,c0 ) 0 r sds)l t (c 0 ). In the classical Backward case, c0 depends on the expectation of some function of the optimal path. In the forward case, c0 does not depend on the future

53 Yields curve and Ramsey rule The yield curve in incomplete market The Ramsey rule is similar to price all zero-coupons (in incomplete market) using the Davis Rule that is under the optimal dual probability L t (y) The optimal probability L t (y) depends only of the risk aversion and the initial consumption rate The yield curve is given as the optimal expectation of the discounted factor. Acceptable for small trade for large trade use a second order correction term (indifference pricing?) depending on the size of the trade

54 Conclusion All Consistent dynamic utilities with a large degree of regularity continuous strictly may be generated as above. Valid for (R σ t (x), t 0, x > 0) supposed only convex sets. Valid also for other classical optimization problem. Work in progress : Progressive utility and consumption. Model with jump. Application to the state dependent utilities. Ref Paper : El Karoui N. and M rad M. : An Exact Connection between two Solvable SDEs and a Non Linear Utility Stochastic PDEs ( ) Arxiv.

55 Thank You for your attention!