An Exact Connection between two Solvable SDEs and a Non Linear - PowerPoint PPT Presentation

An Exact Connection between two Solvable SDEs and a Non Linear Utility Stochastic PDEs Mohamed MRAD Joint work with Nicole El Karoui e Paris VI, ´ Universit´ EcolePolytechnique, elkaroui@cmap.polytechnique.fr, mrad@cmap.polytechnique.fr with the financial support of the ”Fondation du Risque” and the F´ ed´ eration des banques Fran¸ caises 26 OCT 2010

Investment Banking and Utility Theory Some remarks on martingale theory and utility functions in Investment Banking from M. Musiela, T. Zariphopoulo, C. Rogers +alii (2002-2009) ◦ No clear idea how to specify the utility function. ◦ Classical or recursive utilities are defined in isolation to the investment opportunities given to an agent. ◦ Explicit solutions to optimal investment problems can only be derived under very restrictive model and utility assumptions, as Markovian assumption which yields to HJB PDEs. ◦ In non-Markovian framework, theory is concentrated on the problem of existence and uniqueness of an optimal solution, often via the dual representation of utility. ◦ The investor may want to use intertemporal diversification, i.e., implement short, medium and long term strategies ◦ Can the same utility function be used for all time horizons?

Consistent Dynamic Utility Let X be a convex family of positive portfolios, called Test porfolios Definition : An X -Consistent progressive utility U ( t , x ) process is a positive adapted random field s.t. ∗ Concavity assumption : for t ≥ 0, x > 0 �→ U ( t , x ) is an increasing concave function, (in short utility function) . ⋆ Consistency with the class 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 wealth X ∗ .

The General Market Model ◮ The security market consists of one riskless asset S 0 , dS 0 t = S 0 t r t dt , and d continuous risky assets S i , i = 1 .. d defined on a filtred Brownian space (Ω , F t ≥ 0 , P ) dS i t = b i t dt + σ i t . dW t , 1 ≤ i ≤ d S i t ◮ Risk premium vector, η t with b ( t ) − r ( t ) 1 = σ t η t Def A positive wealth process is defined as a pair ( x , π ), x > 0 is the initial value of the portfolio and π = ( π i ) 1 ≤ i ≤ d is the (predictable) proportion of each asset held in the portfolio, assumed to be S -integrable process. ◮ Thanks to AOA in the market, wealth process with π -strategy is driven by dX π t = r t dt + σ t π t . ( dW t + η t dt ) , X π t For simplicity we denote by R σ the range of the matrix σ := ( σ i ) i =1 ... d , κ := σπ, π ∈ R d . The class of Test portfolio in what follows is dX κ X := { ( X κ ) : = r t dt + κ t . ( dW t + η σ κ t ∈ R σ t t dt ) , t } X κ t

Consistent Utility of Itˆ o’s Type Let U be a dynamic utility (concave, increasing) , dU ( t , x ) = β ( t , x ) dt + γ ( t , x ) . dW t t ) is a supermartingale for X π ∈ X ( K ) and a martingale for the such that U ( t , X π optimal one. Open questions ◮ What about the drift β of the utility? ◮ What about the volatility γ of the utility? ◮ Under which assumptions on ( β, γ ) can one be sure that solutions are concave, increasing and consistent? Main difficulties come from the forward definition.

Stochastic calculus depending of a parameter From Kunita Book, Carmona-Nualart ◮ Let φ be a semimartingale random field satisfying d φ ( t , x ) = µ ( t , x ) dt + γ ( t , x ) . dW t , (1) ◮ The pair ( µ, γ ) is called the local characteristic of φ , and γ is referred as the volatility random field. ◮ A semimartingale random field φ is said to be Itˆ o-Ventzel regular if φ is a continuous C 2+ ... -process in x local characteristic ( µ, γ ) are C 1 in x additional assumptions as more regularity, uniform integrability are need to guarantee smoothness of φ and its derivatives, and the existence of regular version of these random fields

Itˆ o-Ventzel’s Formula (Kunita) ◮ Let φ and ψ be Itˆ o-Ventzel’s regular one-dimensional stochastic flows d φ ( t , x ) = µ ( t , x ) dt + γ ( t , x ) . dW t , d ψ ( t , x ) = α ( t , x ) dt + ν ( t , x ) . dW t . ◮ The compound random field φ o ψ ( t , x ) = φ ( t , ψ ( t , x )) is a regular semimartingale Itˆ o-Ventzel’s Formula d ( φ o ψ )( t , x ) = µ ( t , ψ ( t , x )) dt + γ ( t , ψ ( t , x )) . dW t φ x ( t , ψ ( t , x )) d ψ ( t , x ) + 1 2 φ xx ( t , x )( t , ψ ( t , x )) || ν ( t , x ) || 2 dt + + � γ x ( t , ψ ( t , x )) , ν ( t , x ) � dt . The volatility of φ o ψ is given by ν φ o ψ ( t , x ) = γ ( t , ψ ( t , x )) + φ x ( t , ψ ( t , x )) ν ( t , x ) .

Drift Constraint Let U be a progressive utility of class C (2) in the sense of Kunita with local t ( t , x ) = − U x ( t , x ) characteristics ( β, γ ) and risk tolerance coefficient α U U xx ( t , x ) . We introduce the utility risk premium η U ( t , x ) = γ x ( t , x ) U x ( t , x ) . Then, for any admissible portfolio X κ , � � dU ( t , X κ U x ( t , X κ t ) X κ t κ t + γ ( t , X κ t ) = t ) . dW t t + 1 � � β ( t , X κ t ) + U x ( t , X κ t ) r t X κ 2 U xx ( t , X κ t ) Q ( t , X κ + t , κ t ) dt , � x κ t � 2 − 2 α U ( t , x )( x κ t ) . where x 2 Q ( t , x , κ ) η σ t + η U ,σ ( t , x ) � � := . Let γ σ x be the orthogonal projection of γ x on R σ . Let Q ∗ ( t , x ) = inf κ ∈R σ Q ( t , x , κ ); the minimum of this quadratic form is achieved at the optimal policy κ ∗ given by x κ ∗ U xx ( t , x ) ( U x ( t , x ) η σ 1 t + γ σ x ( t , x )) = α U ( t , x ) � η σ t + η U ,σ ( t , x ) � � t ( x ) = − x ( t , x )) || 2 = −|| x κ ∗ x 2 Q ∗ ( t , x ) 1 t ( x ) || 2 . U xx ( t , x ) 2 || U x ( t , x ) η σ t + γ σ = −

Verification Theorem: I Let U be a progressive utility of class C (2) in the sense of Kunita with local characteristics ( β, γ ). Hyp Assume the drift constraint to be Hamilton-Jacobi-Bellman nonlinear type β ( t , x ) = − U x ( t , x ) r t x + 1 2 U xx ( t , x ) � x κ ∗ t ( t , x ) � 2 (2) where κ ∗ is the optimal policy given by 1 x κ ∗ U xx ( t , x )( U x ( t , x ) η σ t + γ σ t ( x ) = − x ( t , x )) Then the progressive utility is solution of the following forward HJB-SPDE ( U x ( t , x )) 2 γ σ x ( t , x ) − U x ( t , x ) r t x + 1 U xx ( t , x ) || η σ U x ( t , x ) || 2 ) dt + γ ( t , x ) . dW t , � dU ( t , x ) = t + 2 and for any admissible wealth X κ t , the process U ( t , X κ t ) is a supermartingale.

Verification Theorem: II Theorem Under previous hypothesis, ◮ Assume that κ ∗ ( t , x ) is sufficiently smooth so that the equation dX ∗ t = X ∗ t ( r t dt + κ ∗ ( t , X ∗ t ) . ( dW t + η σ t dt ) has a (unique? strong ?) positive solution for any initial wealth x > 0. ⇒ Then, the progressive increasing utility U is a consistent utility, with optimal wealth X ∗ .

Inverse flows Let φ be a strictly monotone Itˆ o-Ventzel regular flow with inverse process ξ ( t , y ) = φ ( t , . ) − 1 ( y ). Assume d φ ( t , x ) = µ ( t , x ) dt + γ ( t , x ) . dW t , i) The inverse flow ξ ( t , y ) has as dynamics in old variables 2 ∂ y � γ ( t , ξ ) � 2 d ξ ( t , y ) = − ξ y ( t , y )( µ ( t , ξ ) dt + γ ( t , ξ ) . dW t ) + 1 φ x ( t , ξ ) dt ii) In terms of new variable, with ν ξ ( t , y ) = − ξ y γ ( t , ξ ) � � ν ξ ( t , y ) � 2 � 1 � d ξ ( t , y ) = ν ξ ( t , y ) . dW t + � 2 ∂ y − µ ( t , ξ ) ξ y ( t , y ) dt ξ y iii ) If φ = Φ x ( t , x ) with d Φ( t , x ) = M ( t , x ) dt + C ( t , x ) . dW t , then ξ = Ξ y ( t , y ) � C x ( t , ξ ) � 2 d Ξ( t , y ) = − C ( t , ξ ) . dW t − M ( t , ξ ) dt + 1 Φ xx ( t , ξ ) dt 2

Duality: Convex conjugate SPDE I Let U be a consistent progressive utility of class C (3) , in the sense of Kunita, satisfying the β constraint (2), then the convex conjugate def ˜ � � U ( t , y ) = inf x ∈ Q ∗ U ( t , x ) − x y satisfies + 1 γ y ( t , y ) � 2 − � ˜ � � d ˜ γ σ y ( t , y ) + y ˜ U yy ( t , y ) η σ t � 2 � + y ˜ � U ( t , y ) = � ˜ U y ( t , y ) r t dt 2˜ U yy ( t , y ) γ ( t , y ) = γ ( t , − ˜ + ˜ γ ( t , y ) . dW t with ˜ U y ( t , y )) . ◮ The drift ˜ β ( t , y ) is the value of an optimization program achieved on the optimal policy ν ∗ ( t , y ) = θ ∗ ( t , − ˜ y ( t , y ) / y ˜ γ ⊥ U ( t , y )) = − ˜ U yy ( t , y ). ◮ ˜ β can be written us the solution of the following optimization program � ˜ U y ( t , y ) r t − 1 γ y ( t , y ) 2 y 2 ˜ β ( t , y ) = y ˜ ˜ ν t ∈R σ, ⊥ {|| ν t − η σ t || 2 +2 ν t − η σ � � � U yy ( t , y ) inf . } t y ˜ U yy ( t , y ) γ y ( t , y ) / y ˜ U yy ( t , y ) = η U ( t , − ˜ U ( t , y )) = γ x ( t , − ˜ with − ˜ U ( t , y )) / y .