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Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes Roscoff, March 11-12, 2010 Consumption-investment problem with transaction costs Yuri Kabanov Laboratoire de Math´ ematiques, Universit´ e de Franche-Comt´ e March, 11-12, 2010 Yuri Kabanov HJB equations 1 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes History Merton. JET, 1971. Magill, Constantinides. JET, 1976. Davis, Norman. Math. Oper. Res., 1990. Shreve, Soner. AAP, 1994. K., Kl¨ uppelberg. FS, 2004. Benth, Karlsen, Reikvam. 2000. ... K., de Vali` ere. 200‘9. Yuri Kabanov HJB equations 2 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes Outline Consumption–investment without transaction costs 1 Models with transaction costs 2 Consumption–investment with L´ evy processes 3 Yuri Kabanov HJB equations 3 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes Classical Merton Problem We are given a stochastic basis with an m -dimensional standard Wiener process w . The market contains a non-risky security which is the num´ eraire , i.e. its price is identically equal to unit, and m risky securities with the price evolution dS i t = S i t ( µ i dt + dM i t ) , i = 1 , ..., m , (1) where M = Σ w is a (deterministic) linear transform of w . Thus, M is a Gaussian martingale with � M � t = At ; the covariance matrix A = ΣΣ ∗ is assumed to be non-degenerated. The dynamics of the value process : dV t = H t dS t − c t dt , (2) where the m -dimensional predictable process H defines the number of shares in the portfolio, c ≥ 0 is the consumption process. Yuri Kabanov HJB equations 4 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes Merton Problem : dynamics, constraints, and goal It is convenient to choose as the control the process π = ( α, c ) with α i t := H i t S i t / V t (the proportion of the wealth invested in the i th asset). Then the dynamics of the value process is : dV t = V t α t ( µ dt + dM t ) − c t dt , V 0 = x > 0 , (3) Constraints : α is bounded c is integrable, V = V x ,π ≥ 0 ; π = 0 after the bankruptcy. Infinite horizon. The investor’s goal : EJ π ∞ → max , (4) where � t J π e − β s u ( c s ) ds . t := (5) 0 where u is increasing and concave. For simplicity : u ≥ 0, u (0) = 0. A typical example : u ( c ) = c γ /γ , γ ∈ ]0 , 1[. The parameter β > 0 shows that the agent prefers to consume sooner than later. Yuri Kabanov HJB equations 5 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes Merton Problem : the Bellman function Define the Bellman function EJ π W ( x ) := sup ∞ , x > 0 . (6) π ∈A ( x ) By convention, A (0) := { 0 } and W (0) := 0. The Bellman function W inherits the properties of u . It is increasing (as A (˜ x ) ⊇ A ( x ) when ˜ x ≥ x ) and concave (almost obvious in H -parametrization). The process H = λ H 1 + (1 − λ ) H 2 admits the representation via α with (1 − λ ) V 2 λ V 1 α i = H i S i / V = α i α i 1 + 2 ; λ V 1 + (1 − λ ) V 2 λ V 1 + (1 − λ ) V 2 α is bounded when α j are bounded. Thus, π = ( α, λ c 1 + (1 − λ ) c 2 ) ∈ A ( x ) with x = λ x 1 + (1 − λ ) x 2 and W ( λ x 1 + (1 − λ ) x 2 ) ≥ EJ π ∞ ≥ λ EJ π 1 ∞ + (1 − λ ) EJ π 2 ∞ due to concavity of u . We obtain the concavity of W by taking supremum over π i . Yuri Kabanov HJB equations 6 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes Merton Problem : the result Theorem Let u ( c ) = c γ /γ , γ ∈ ]0 , 1[ . Assume that � � 1 β − 1 γ 1 − γ | A − 1 / 2 µ | 2 κ M := > 0 . (7) 1 − γ 2 Then the optimal strategy π o = ( α o , c o ) is given by the formulae 1 α o = θ := 1 − γ A − 1 µ, c o t = κ M V o t , (8) where V o is the solution of the linear stochastic equation dV o = V o t θ ( µ dt + dM t ) − κ M V o V o t dt , 0 = x . (9) The process V o is optimal and the Bellman function is � � x γ = m x γ . κ γ − 1 W ( x ) = /γ (10) M Yuri Kabanov HJB equations 7 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes Merton Problem - comments For the two-asset model � � µ 2 1 β − 1 γ κ M := > 0 . σ 2 1 − γ 2 1 − γ Notice that we cannot guarantee without additional assumptions that W is finite. If the latter property holds, then, due to the concavity, W ( x ) is continuous for x > 0, but the question whether it is continuous at zero should be investigated specially. At last, when the utility u is a power function, the Bellman function W , if finite, is proportional to u . Indeed, the linear dynamics of the control system implies that W ( ν x ) = ν γ W ( x ) whatever is ν > 0, i.e. the Bellman function is positive homogeneous of the same order as the utility function. In a scalar case this homotheticity property defines, up to a multiplicative constant, a unique finite function, namely x γ . Yuri Kabanov HJB equations 8 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes HJB equation and verification theorem,1 For our infinite horizon problem the HJB is : � 1 � 2 | A 1 / 2 α | 2 x 2 f ′′ ( x ) + αµ xf ′ ( x ) − β f ( x ) − f ′ ( x ) c + u ( c ) sup = 0 ( α, c ) where x > 0 and sup is taken over α ∈ R d and c ∈ R + . Simple observation : Let f : R + → R + and π ∈ A ( x ). Put X f , x ,π = X f t = e − β t f ( V t ) + J π t where V = V x ,π . If f is smooth, by the Ito formula X f t = f ( x ) + D t + N s where (with L ( x , α, c ) = [ ... ] of the HJB equation) � t � t e − β s L ( V s , α s , c s ) ds , e − β s f ′ ( V s ) V s α s dM s . D t := N t := 0 0 The process N is a local martingale up to the bankruptcy time σ . That is, there are σ n ↑ σ such that the stopped processes N σ n are uniformly integrable martingales. If σ = ∞ and N is a martingale we take σ n = n . Yuri Kabanov HJB equations 9 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes HJB equation and verification theorem,2 If sup ( α, c ) [ ... ] ≤ 0, then N and X f t are supermartingales. Hence, t − Ee − β t f ( V t ) ≤ EX f EJ t = EX f t ≤ f ( x ) . Proposition If f is a supersolution of the HJB, then W ≤ f and, hence, W ∈ C ( R + \ { 0 } ) . If, moreover, f (0+) = 0 , then W ∈ C ( R + ) . Theorem Let f ∈ C ( R + ) ∩ C 2 ( R + \ { 0 } ) be a positive concave function solving the HJB equation, f (0) = 0 . Suppose that sup is attained on α ( x ) and c ( x ) where that α is bounded measurable, c ≥ 0 and the equation dV o t = V o t α ( V o t )( µ dt + dM t ) − c ( V o V o t ) dt , 0 = x , admits a strong solution V o t . If lim Ee − βσ n f ( V o σ n ) = 0 , then W = f and the optimal control π o = ( α ( V o ) , c ( V o )) . Yuri Kabanov HJB equations 10 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes Proof of the Merton Theorem, 1 The verification theorem is very efficient if we have a guess about the solution. It is the case when the utility is a power function : the problem is to find the constant! Put u ∗ ( p ) := sup c ≥ 0 [ u ( c ) − cp ]. For u ( c ) = c γ /γ we have u ∗ ( p ) = 1 − γ p γ/ ( γ − 1) . γ Expecting that f ′′ < 0, we find that the maximum of the quadratic form over α is attained at α o ( x ) = − A − 1 µ f ′ ( x ) xf ′′ ( x ) == A − 1 µ/ (1 − γ ) . Thus, the HJB equation is : 2 | A − 1 / 2 µ | 2 ( f ′ ( x )) 2 − 1 − β f ( x ) + 1 − γ ( f ′ ( x )) γ − 1 = 0 . γ f ′′ ( x ) γ Its solution f ( x ) = m x γ should have m = κ γ − 1 /γ . M The function α o ( x ) is constant, c o ( x ) = κ M x , and the equation pretending to describe the optimal dynamics is linear : Yuri Kabanov HJB equations 11 / 83

Consumption–investment without transaction costs Models with transaction costs Consumption–investment with L´ evy processes Proof of the Merton Theorem, 2 � � dV o dt + A − 1 µ 1 1 − γ | A − 1 / 2 µ | 2 − κ M t V o = 1 − γ dM , 0 = x . V o t Its solution is the geometric Brownian motion which never hits zero. Noticing that � A − 1 µ M � t = | A − 1 / 2 µ | 2 t , we have that �� � � | A − 1 / 2 µ | 2 t − κ M t + A − 1 µ 1 − γ − 1 1 1 V o t = x exp 1 − γ M t . (1 − γ ) 2 2 t ) p = x p e κ p t where κ p is a constant, the process N for this Since E ( V o control is a true martingale; we σ n = n . For p = γ the corresponding constant κ γ = 1 γ 1 − γ − γκ M = β − κ M . 2 Thus, t ) γ = x γ e − κ M t → 0 , e − β t E ( V o t → ∞ . The Merton theorem is proven. Yuri Kabanov HJB equations 12 / 83