Time consistent portfolio strategies for a general utility function

Time consisten t p ortfolio strategies for a general utilit y function Oumar Mb odji ∗ oumarsoule@gmail.com F ebruary 23, 2026 Abstract. W e study the Merton portfolio management problem within a complete market, non constant time discoun t rate and general utilit y framework. The non constan t discoun t rate in troduces time inconsistency whic h can b e solv ed b y in tro ducing sub game p erfect strategies. Under some asymptotic assumptions on the utility function, w e sho w that the subgame p erfect strategy is the same as the optimal strategy , pro vided the discount rate is replaced b y the utilit y w eighted discoun t rate ρ ( t, x ) that dep ends on the time t and w ealth lev el x . A fixed p oin t iteration is used to find ρ . The consumption to w ealth ratio and the in vestmen t to wealth ratio are given in feedbac k form as functions of the v alue function. Key w ords. Portfolio optimization, Merton problem, time consistency , subgame p erfect strategies, non constan t time discount rates, general utilit y . AMS sub ject classifications. 60G35, 60H20, 91B16, 91B70 1 In tro duction This pap er is a contribution to the analysis of time consistent sto c hastic optimization in a complete mark et. The stock price is modeled as a geometric Brownian motion with constan t coefficients. The in vestor trades b et ween the sto c k and a riskless security mo deled with a constant in terest rate. He aims to maximize his exp ected utility for consumption and terminal wealth. The utility function is discounted with a v ariable discoun t rate ρ yielding time inconsistency . This mo del is the simplest extension of the mo dels in [4], [5], [10] for a general utility function. W e are still in a Mark ovian complete mark et setting and can use the theory of PDEs to build the solution. As in [10], [2], we define a v alue function that will solve an extended Hamilton-Jacobi-Bellman (HJB) equation. This pap er will sho w that the extended HJB equation can b e solv ed by means of a fixed p oin t problem. The w ork of [16] already solv ed the time consistency problem via a fixed p oin t methodology . The no velt y in our pap er is that w e define a marginal utilit y weigh ted discount rate that can be computed efficien tly via Monte-Carlo sim ulation and that gives a b etter economic intuition of the solution. Our present pap er uses the metho dology of [11] and works directly with the marginal v alue function V x ( t, x ) instead of the original v alue function V ( t, x ). Moreo ver, time-consistent p olicies are constructed in a feedbac k form from the first order conditions in the extended HJB equation. In the rest of this section, w e present the mark et model, define the discoun t function and the maximal exp ected utilit y problem. In Section 2, w e state the main results and introduce the fixed p oin t problem. Section 3 describ es the algorithm for constructing the subgame p erfect strategies. ∗ Indep endent Researc her 1 1.1 The Financial Market Consider a financial market consisting of a savings accoun t and one sto c k (the risky asset). W e assume a benchmark deterministic time horizon T . The sto c k price p er share follo ws an exp onen tial Bro wnian motion dS t = S t [ µ dt + σ dW ( t )] , 0 ≤ t ≤ T (1.1) where { W ( t ) } t ≥ 0 is a 1 − dimensional Brownian motion on a filtered probability space, (Ω , {F t } 0 ≤ t ≤ T , P ) . The filtration {F t } is the completed filtration generated by { W ( t ) } . The savings account accrues interest at the riskless rate r > 0. Let us denote θ ≜ µ − r σ (1.2) the market pric e of risk . W e place ourselves in a Mark ovian setting. There is one agent who is contin uously inv esting in the sto c k, is using the money market, and consuming. A t every time t , the agent c ho oses π ( t ), the ratio of w ealth in v ested in the risky asset and c ( t ) the ratio of w ealth consumed. Given an adapted pro cess { π ( t ) , c ( t ) } 0 ≤ t ≤ T , the equation describing the dynamics of wealth X π ,c ( t ) is given by : dX π ,c ( t ) = [ r − c ( t ) + σ θ π ( t )] X π ,c ( t ) dt + σ π ( t ) X π ,c ( t ) dW t (1.3) X π ,c (0) = x 0 the initial wealth x 0 b eing exogenously sp ecified. 1.2 Time preferences and risk preferences As seen in the in troduction, the time preference reflects the economic agent’s preference for immediate utilit y ov er dela yed utility . W e now define discount functions and discoun t rates. Definition 1.1. A discoun t function h : D = { 0 ≤ t ≤ s ≤ T } → R is a contin uous, p ositiv e function satisfying h ( t, t ) = 1. h ( t, s ) is the discoun t factor b etw een time t and s with t ≤ s . R emark 1.2 . W e take a discount form to b e of the general form h ( t, s ) b ecause, as noted in [13], [10] and [5], we ha ve to account for sto c hastic time horizons T which could b e the time of death of the agent. In that case, the discount function has to b e transformed and will take the general form h ( t, s ). W e can normalize b y dividing h ( t, s ) by h ( t, t ). Assumption 1.3 . The discount function h satisfies • 0 < inf ( t,s ) ∈ D h ( t, s ) ≤ sup ( t,s ) ∈ D h ( t, s ) < ∞ (1.4) • The functions ∂ h ( t,s ) ∂ s , ∂ h ( t,s ) ∂ t , ∂ 2 h ( t,s ) ∂ 2 t are con tinuous and b ounded. The discoun t rate is defined as ρ h ( t, s ) = ∂ h ( t, s ) ∂ t × 1 h ( t, s ) (1.5) In the case h ( t, s ) = H ( s − t ) for a certain C 1 function H on [0 , T ], we get: ρ h ( t, s ) = − H ′ ( s − t ) H ( s − t ) . Next, we define the agen t’s risk pr efer enc es . An economic agen t will hav e satisfaction U ( C ) from consuming an amount C . 2 Assumption 1.4 . The utility function U is twice contin uously differentiable. W e assume: 1. U is C 2 on (0 , ∞ ), strictly increas ing, strictly concav e and satisfies the Inada condition: U ′ (0) = ∞ , U ′ ( ∞ ) = 0. 2. Let R 1 ( x ) := − xU ′′ ( x ) U ′ ( x ) b e its relative risk av ersion co efficient. R 1 ( x ) is p ositiv e and b ounded aw a y from 0. That is, there exists r 1 , r 2 p ositiv e n umbers such that: r 1 ≤ R 1 ( x ) ≤ r 2 ∀ x > 0 . (1.6) R emark 1.5 . Note that condition 2 of Assumption 1.4 implies that the asymptotic elasticit y defined in [12], [15] AE ( U ) = lim inf x →∞ xU ′ ( x ) U ( x ) < 1. Utilit y functions of the form α x γ 1 γ 1 + (1 − α ) x γ 2 γ 2 , γ 1  = 0 , γ 2  = 0 satisfy assumption 1.4. The exp onen tial utility function do es not satisfy the assumption. 1.3 The in tertemp oral utility Let us now define the admissible strategies. Definition 1.6. An R 2 -v alued sto c hastic pro cess { π ( t ) , c ( t ) } 0 ≤ t ≤ T is called an admissible strategy process if: 1. it is progressively measurable with resp ect to the sigma algebra σ ( { W ( t ) } 0 ≤ t ≤ T ), 2. it satisfies c ( t ) ≥ 0 for all t almost surely and X π ,c ( T ) ≥ 0 , almost surely (1.7) 3. moreo v er, we require that E sup 0 ≤ s ≤ T | U ( c ( s ) X π ,c s ) | < ∞ , E sup 0 ≤ s ≤ T | U ( X π ,c s ) | < ∞ , a.s. (1.8) The last set of inequalities is purely technical and is satisfied for e.g. b ounded strategies. In order to ev aluate the p erformance of an inv estment-consumption strategy the in vestor uses an exp ected utilit y criterion. F or an admissible strategy pro cess { π ( s ) , c ( s ) } s ≥ 0 and its corresp onding w ealth pro cess { X π ,c ( s ) } s ≥ 0 , w e denote the in ter-temp oral utilit y by J ( t, x, π , c ) = E t  Z T t h ( t, s ) U ( c ( s ) X π ,c ( s )) ds + h ( t, T ) U ( X π ,c ( T ))  A natural ob jective for the decision maker is to maximize the ab ov e exp ected utility criterion. As shown in [5], the optimal strategy is time inconsistent when the discoun t function is not of the exp onen tial form. The decision-maker could implemen t tw o t yp es of strategies. He could pr e c ommit at time t 0 = 0 to follo w the optimal strategy and stay with it un til time T . Or he could implemen t a time consistent strategy that takes into account the fact that the decision maker’s preferences will change in the future. Time consisten t problems hav e b een studied extensiv ely in the literature. 3 1.4 Subgame p erfect strategies and v alue function W e now introduce a sp ecial class of time consistent strategies, which can also b e called sub game p erfe ct str ate gies . That is, w e consider that the decision-mak er at time t can commit his successors up to time t + ϵ , with ϵ → 0, and w e seek strategies whic h are optimal to implement right now conditioned on them b eing implemented in the future. This is made precise in the following formal definition. Definition 1.7. An admissible trading strategy { ¯ π ( s ) , ¯ c ( s ) } 0 ≤ s ≤ T is a sub game p erfe ct str ate gy if there exists a map G = ( G π , G c ) : [0 , T ] × R + → R × [0 , ∞ ) suc h that for any t ∈ [0 , T ] and w ealth lev el x > 0 at time t lim inf ϵ ↓ 0 J ( t, x, ¯ π , ¯ c ) − J ( t, x, π ϵ , c ϵ ) ϵ ≥ 0 , (1.9) where: ¯ π ( s ) = G π ( s, ¯ X ( s )) , ¯ c ( s ) = G c ( s, ¯ X ( s )) (1.10) and the wealth pro cess ¯ X ( s ) := X ¯ π , ¯ c ( s ) is a solution of the sto chastic differential equation (SDE): d ¯ X ( s ) = ¯ X ( s )[ r + σ θG π ( s, ¯ X ( s )) − G c ( s, ¯ X ( s ))] ds + σ G π ( s, ¯ X ( s )) ¯ X ( s ) dW ( s ) (1.11) The process { π ϵ ( s ) , c ϵ ( s ) } s ∈ [ t,T ] men tioned abov e is another inv estment-consumption strategy defined b y π ϵ ( s ) = ( G π ( s, X ϵ ( s )) , s ∈ [ t, T ] \ E ϵ,t π ( s ) , s ∈ E ϵ,t , (1.12) c ϵ ( s ) = ( G c ( s, X ϵ ( s )) , s ∈ [ t, T ] \ E ϵ,t c ( s ) , s ∈ E ϵ,t , (1.13) with E ϵ,t = [ t, t + ϵ ] , and { π ( s ) , c ( s ) } s ∈ E ϵ,t is any strategy for which { π ϵ ( s ) , c ϵ ( s ) } s ∈ [ t,T ] is an admissible p olicy . X ϵ is defined on [ t + ϵ, T ] b y the SDE: dX ϵ ( s ) = [ r − c ϵ ( s ) + σ θπ ϵ ( s )] X ϵ ( s ) ds + π ϵ ( s ) σ X ϵ ( s ) dW ( s ) X ϵ ( t + ϵ ) = X π ,c ( t + ϵ ) . (1.14) Dynamic programming is a v ery conv enien t wa y of formulating a large set of dynamic problems in financial economics. Most prop erties of this to ol are well established and understo o d. In dynamic programming, we introduce an ob ject called the value function that is obtained b y ev aluating a certain functional at our candidate solutions. The solutions of the dynamic programming problem are then the solutions of a certain equation called HJB and can be expressed en tirely in terms of the v alue function and its deriv atives. In optimization problems, the v alue function is the optimal v alue an agen t can derive from his maximization process. The pap er [5] uses the v alue function metho dology to characterize subgame p erfect strategies. W e will see that the v alue function can b e written as a function V ( t, x ) of time t and w ealth x and this allows us to find subgame p erfect strategies in a feedback form. F or fixed t, x , the strategy ( ¯ π , ¯ c ) can b e expressed as deterministic functions of V and its deriv atives with resp ect to x . W e start with a definition. Definition 1.8. Let V : [0 , T ] × (0 , ∞ ) → R , ( t, x ) 7→ V ( t, x ) b e a C 1 , 2 function. Supp ose V is strictly increasing, conca ve in the x v ariable. Let I ( x ) := ( U ′ ) − 1 ( x ) the inv erse marginal utilit y . Supp ose that { ¯ π ( s ) , ¯ c ( s ) , ¯ X ( s ) } s ∈ [0 ,T ] are subgame p erfect strategies with the corresp onding map ¯ π ( s ) = G π ( s, ¯ X ( s )) , ¯ c ( s ) = G c ( s, ¯ X ( s )) , (1.15) where G π ( t, x ) = − θ ∂ V ∂ x ( t, x ) σ x ∂ 2 V ∂ x 2 ( t, x ) , G c ( t, x ) = 1 x × I  ∂ V ∂ x ( t, x )  (1.16) 4 and ¯ X ( s ) is the wealth pro cess given by: d ¯ X ( s ) = [ r + σ θG π ( s, ¯ X ( s )) − G c ( s, ¯ X ( s ))] ¯ X ( s ) ds + σ G π ( s, ¯ X ( s )) ¯ X ( s ) dW ( s ) . (1.17) W e shall say that V is a v alue function if for all ( t, x ) ∈ [0 , T ] × (0 , ∞ ), we hav e: V ( t, x ) = J ( t, x, ¯ π , ¯ c ) (1.18) The economic in terpretation is v ery natural: if one applies the Mark ov strategy asso ciated with V b y the relations [(1.15), (1.16), (1.17), (1.18)] and computes the corresp onding v alue of the inv estor’s criterion starting from X t = x at time t , one gets precisely V ( t, x ). In other w ords, this is fundamen tally a fixed-p oin t characterization. In the next section, we give the main results of this pap er. 2 Main Results 2.1 The extended HJB As in [4], [5], [10], the time consisten t p olicy can b e determined by solving an HJB equation with a non lo cal term. [2] pro vides a more general setting and shows that the v alue function solves an extended HJB system along with a verification theorem. The authors [8], [9] ha ve shown the well p osedness of the extended HJB system. Our result is new in that it deals with a general utility function and pro vides a w ay to construct the solution. The follo wing prop osition gives the subgame p erfect strategies in terms of the v alue function. Prop osition 2.1. If the extende d HJB (2.3) has a C 1 , 2 solution V , then the sub game p erfe ct str ate gies ar e given by: ¯ c ( t, x ) = I ( V x ( t, x )) x (2.1) ¯ π ( t, x ) = − θ V x ( t, x ) σ xV xx ( t, x ) (2.2) The pro of comes from a simple calculation of the first order conditions for the quan tity ( ¯ π , ¯ c ) = arg max ( π ,c ) admissible {A π ,c V + U ( xc ) } . Theorem 2.2 (Extended HJB) . L et V : [0 , T ] × (0 , ∞ ) → R b e a C 1 , 2 function. Supp ose ( ¯ π , ¯ c ) is an admissible Markovian p olicy and that • V solves the extende d Hamilton Jac obi Bel lman e quation : ∂ V ∂ t ( t, x ) + sup ( π ,c ) admissible  A π ,c V ( t, x ) + U ( xc )  (2.3) = E t  Z T t ∂ h ( t, s ) ∂ t U ( ¯ c ( s ) X ¯ π , ¯ c ( s )) ds + ∂ h ( t, T ) ∂ t U ( X ¯ π , ¯ c ( T ))  wher e A π ,c V ( t, x ) = ( r + σ θ π − c ) x ∂ V ∂ x ( t, x ) + 1 2 σ 2 x 2 π 2 ∂ 2 V ∂ x 2 ( t, x ) (2.4) along with the b oundary c ondition V ( T , x ) = U ( x ) . 5 • ( ¯ π , ¯ c ) satisfies: ( ¯ π , ¯ c ) = arg max {A π ,c V ( t, x ) + U ( xc ); ( π , c ) admissible } (2.5) Then V is a value function. Mor e over { ¯ π ( s ) , ¯ c ( s ) , ¯ X ( s ) } is a sub game p erfe ct str ate gy (cf. Definition 1.7). Theorem 2.2 is pro ven in [2]. It is called a v erification theorem b ecause it allo ws us to c heck if a giv en v alue function is actually a subgame p erfect strategy . Next, w e define a strategy dep enden t discoun t rate that w e call utility w eighted discount rate. Substituting ¯ c, ¯ π by the expressions in prop osition 2.1, the extended HJB b ecomes: V t +  r x − θ 2 V x 2 V xx − I ( V x )  V x + U ( I ( V x )) = E t  Z T t ∂ h ( t, s ) ∂ t U ( I ( V x ( s, ¯ X s )) ds + ∂ h ( t, T ) ∂ t U ( ¯ X T )  In the spirit of [11], the x deriv ative of the extended HJB gives: V tx + ( r − θ 2 ) V x + ( r x − I ( V x )) V xx + ( θ V x ) 2 V xxx 2 V 2 xx ( t, x ) = ∂ ∂ x E t  Z T t ∂ h ( t, s ) ∂ t U ( ¯ c s ¯ X s ) ds + ∂ h ( t, T ) ∂ t U ( ¯ X T )  (2.6) = ρ ( t, x ) V x ( t, x ) (2.7) and equation (2.6) is known as the marginal extended HJB. The quan tity ρ ( t, x ) is defined as: ρ ( t, x ) = R T t ∂ h ( t,s ) ∂ t ∂ ∂ x E t [ U ( ¯ c s ¯ X s )] ds + ∂ h ( t,T ) ∂ t ∂ ∂ x E t [ U ( ¯ X T ))] R T t h ( t, s ) ∂ ∂ x E t [ U ( ¯ c s ¯ X s )] ds + h ( t, T ) ∂ ∂ x E t [ U ( ¯ X T ))] (2.8) Note that the denominator in the expression ab o ve is equal to V x ( t, x ) > 0. The quan tity ρ is an av erage discoun t rate weigh ted by the marginal utility . In particular, ρ ( t, x ) ∈  inf ∂ h ∂ t h , sup ∂ h ∂ t h  . In what follows, we write v ( t, x ) := V x ( t, x ) (2.9) Since all the terms in (2.6) inv olv e v and its deriv atives, we get a PDE for v : v t ( t, x ) + ( r − θ 2 ) v + ( r x − I ( v )) v x + θ 2 2 v 2 v xx v 2 x = ρ ( t, x ) v ( t, x ) (2.10) v ( T , x ) = U ′ ( x ) (2.11) v satisfies a non linear parab olic PDE that is p ossibly degenerate. A w ay around it, is to work with the x in verse of v ( t, x ) and get a linear PDE. Assuming that V is strictly concav e and C 2 as a function of x yields a unique p ( t, x ) such that v ( t, p ( t, x )) = x (2.12) p is the x -inv erse of the marginal v alue function v = V x . W e calculate partial deriv atives from equation (2.12): v x = 1 p x , v xx = − p xx p 3 x , v t = − p t p x (2.13) 6 where v is ev aluated at ( t, p ( t, x )) and p is ev aluated at ( t, x ). In terms of p , the marginal HJB (2.6) b ecomes ρ ( t, p ( t, x )) x = − p t ( t, x ) p x ( t, x ) + ( r p − I ( x )) 1 p x + ( r − θ 2 ) x − p xx 2 p 3 x ( − θ xp x ) 2 p t + θ 2 2 x 2 p xx − r p + I ( x ) + ( ρ ( t, p ( t, x )) + θ 2 − r ) xp x ( t, x ) = 0 (2.14) p ( T , x ) = I ( x ) (2.15) p solv es a linear parab olic PDE which can b e solved using standard tec hniques. W e can change the v ariable to y = log x Definition 2.3. F or fixed ( t, y ) ∈ [0 , T ] × R , define I 0 ( y ) = I ( e y ) and U 0 ( y ) := U ( I ( e y )), ¯ p ( t, y ) = p ( t, e y ) and ¯ ρ ( t, y ) = ρ ( t, p ( t, e y )). The functions ¯ ρ and ρ satisfy the relation ρ ( t, x ) = ¯ ρ ( t, log v ( t, x )) (2.16) Since ¯ p y ( t, y ) = e y p x ( t, e y ) = xp x ( t, x ) and ¯ p y y ( t, y ) = e 2 y p xx ( t, e y ) + e y p x ( t, e y ) = x 2 p xx ( t, x ) + xp x ( t, x ). W e get ¯ p t + θ 2 2 ¯ p y y +  θ 2 2 + ¯ ρ ( t, y ) − r  ¯ p y − r ¯ p + I 0 ( y ) = 0 (2.17) ¯ p ( T , y ) = I 0 ( y ) (2.18) W e make the following assumptions ab out the function ¯ ρ . Assumption 2.4 . The functions ¯ ρ ( t, y ) and ¯ ρ y ( t, y ) are contin uous and b ounded. ∀ t ∈ [0 , T ] , y ∈ R : | ¯ ρ ( t, y ) | ≤ || ρ || and | ¯ ρ y ( t, y ) | ≤ κ (2.19) W e will construct such a function at the end of this pap er. W e can still solve the equation ab o ve using the F eynman-Kac formula if we assume ¯ ρ is kno wn and is C 1 in the v ariables t, y . W e define the pro cess ¯ Y s b y the SDE: d ¯ Y s = ( − θ 2 2 − r + ¯ ρ ( s, ¯ Y s )) ds − θdW s , ¯ Y t = y (2.20) By F eynman Kac’s formula: ¯ p ( t, y ) = E  e − r ( T − t ) I 0 ( ¯ Y T + θ 2 ( T − t )) + Z T t e − r ( s − t ) I 0 ( ¯ Y s + θ 2 ( s − t )) ds     ¯ Y t = y  W e wan t to get b ounds for p , ¯ p and v . In the spirit of [11], we start b y in tro ducing some function spaces. Definition 2.5. F or each 0 < ν 1 ≤ ν 2 , denote by D 0 ( ν 1 , ν 2 ) the space of functions F : (0 , ∞ ) → (0 , ∞ ) of class C 1 suc h that ν 1 ≤ − xF ′ ( x ) F ( x ) ≤ ν 2 ∀ x ∈ (0 , ∞ ) (2.21) W e will need the following lemma: 7 Lemma 2.6. Supp ose F ∈ D 0 ( ν 1 , ν 2 ) and let f : (0 , ∞ ) → (0 , ∞ ) b e its inverse i.e. F ( f ( x )) = x . Then f ∈ D 0  1 ν 2 , 1 ν 1  . Pro of Replace x by f ( x ) in (2.21) to get: ν 1 ≤ − f ( x ) F ′ ( f ( x )) F ( f ( x )) ≤ ν 2 . Using the relations F ( f ( x )) = x and f ′ ( x ) = 1 F ′ ( f ( x )) , w e get: ν 1 ≤ − f ( x ) xf ′ ( x ) ≤ ν 2 i.e. 1 ν 2 ≤ − xf ′ ( x ) f ( x ) ≤ 1 ν 1 . □ Prop osition 2.7. F or t ∈ [0 , T ] , let r 1 ( t ) := r 1 e − κ ( T − t ) and r 2 ( t ) := r 2 e κ ( T − t ) . (2.22) Then v ( t, . ) ∈ D 0 ( r 1 ( t ) , r 2 ( t )) and p ( t, . ) ∈ D 0  1 r 2 ( t ) , 1 r 1 ( t )  . The pro of is in the app endix. Prop osition 2.8. The function 1 I 0 ( y ) × ∂ ¯ p ( t,y ) ∂ t is b ounde d and for k = 0 , 1 , 2 , the functions 1 I 0 ( y ) × ∂ k ¯ p ( t,y ) ∂ y k ar e b ounde d functions of ( t, y ) ∈ [0 , T ] × R . F urthermor e ther e ar e p ositive c ontinuous functions r 3 ( t ) , r 4 ( t ) such that r 3 ( t ) ≤ ¯ p ( t, y ) I 0 ( y ) ≤ r 4 ( t ) (2.23) The pro of will b e in the app endix. The function y 7→ ¯ p ( t, y ) is strictly decreasing. Since lim y →∞ I 0 ( y ) = 0 and lim y →−∞ I 0 ( y ) = ∞ , w e get the following result: Prop osition 2.9. F or every t ∈ [0 , T ] , the function y 7→ ¯ p ( t, y ) defines a bije ction fr om R to (0 , ∞ ) . And we have: lim y →∞ ¯ p ( t, y ) = 0 and lim y →−∞ ¯ p ( t, y ) = ∞ . Prop osition 2.10. The quantities ¯ π ( t, x ) = − θ v ( t, x ) σ xv x ( t, x ) ¯ c ( t, x ) = I ( v ( t, x )) x ar e b ounde d by θ r 1 ( t ) σ ≤ ¯ π ( t, x ) ≤ θ σ r 1 ( t ) and 1 r 4 ( t ) ≤ ¯ c ( t, x ) ≤ 1 r 3 ( t ) (2.24) Pro of Since v ( t, . ) ∈ D 0 ( r 1 ( t ) , r 2 ( t )), r 1 ( t ) ≤ − xv x ( t, x ) v ( t, x ) ≤ r 2 ( t ) . So θ σ r 1 ( t ) ≤ ¯ π ( t, x ) ≤ θ σ 1 r 1 ( t ) By making the change of v ariables x = ¯ p ( t, y ), we get: ¯ c ( t, x ) = I ( v ( t, x )) x = I ( v ( t, ¯ p ( t, y ))) ¯ p ( t, y ) = I 0 ( y ) ¯ p ( t, y ) Since b y Prop osition 2.8, r 3 ( t ) ≤ ¯ p ( t, y ) I 0 ( y ) ≤ r 4 ( t ) 8 w e conclude that 1 r 4 ( t ) ≤ ¯ c ( t, x ) ≤ 1 r 3 ( t ) □ Next, w e give a c haracterization of the wealth process. Theorem 2.11. L et t ∈ [0 , T ] and x > 0 . The we alth pr o c ess ¯ X s := X ¯ π , ¯ c ( s ) is given by ¯ X ( s ) = X ¯ π , ¯ c ( s ) = ¯ p ( s, ¯ Y ( s )) wher e ¯ X ( t ) = x = ¯ p ( t, y ) , y = log v ( t, x ) and ¯ Y satisfies the SDE (2.20) . The pro of is in the app endix. 2.2 Calculation of the quan tit y ¯ ρ W e use Theorem 2.11 to get ¯ c s ¯ X s = I ( v ( s, ¯ X ( s )) = I ( e ¯ Y s ) = I 0 ( ¯ Y s ) . ¯ X T = ¯ p ( T , ¯ Y T ) = I 0 ( ¯ Y T ) and the following: ∂ ∂ y = ∂ x ∂ y . ∂ ∂ x = ¯ p y ( t, y ) . ∂ ∂ x ρ ( t, ¯ p ( t, y )) = ∂ ∂ y E t  R T t ∂ h ( t,s ) ∂ t U 0 ( ¯ Y s ) ds + ∂ h ( t,T )) ∂ t U 0 ( ¯ Y T )  ∂ ∂ y E t  R T t h ( t, s ) U 0 ( ¯ Y s ) ds + h ( t, T ) U 0 ( ¯ Y T )  = ¯ ρ ( t, y ) (2.25) Th us, ¯ ρ is solution of a fixed p oin t op erator F . W e first start b y a definition. Definition 2.12. W e define the space B the space of functions ϕ : [0 , T ] × R 7→ R such that : • | ϕ ( t, y ) | ≤ || ρ || , ∀ t, y . • ( t, y ) 7→ ϕ ( t, y ) , ( t, y ) 7→ ϕ y ( t, y ) are b ounded con tinuous functions. Let κ > 0, we define B κ = { ϕ ∈ B | ∀ t ∈ [0 , T ] , y ∈ R : | ϕ y ( t, y ) | ≤ κ } F or ϕ ∈ B , Y ϕ is the solution of the SDE: d Y ϕ s = ( − θ 2 2 − r + ϕ ( s, Y ϕ s )) ds − θdW s , t ≤ s ≤ T , Y ϕ t = y (2.26) F or t ∈ [0 , T ] , y ∈ R , define F [ ϕ ]( t, y ) = ∂ ∂ y E t  R T t ∂ h ( t,s ) ∂ t U 0 ( Y ϕ s ) ds + ∂ h ( t,T )) ∂ t U 0 ( Y ϕ T )  ∂ ∂ y E t  R T t h ( t, s ) U 0 ( Y ϕ s ) ds + h ( t, T ) U 0 ( Y ϕ T )  (2.27) 9 Theorem 2.13. ¯ ρ is the unique fixe d p oint for the op er ator F in the sp ac e of functions B i.e. F [ ¯ ρ ] = ¯ ρ (2.28) Definition 2.14. F or 0 ≤ t ≤ s ≤ T and x > 0, let y = log v ( t, x ), define: ¯ α ( t, s, x ) = E t [ U ( ¯ c s ¯ X s ) | ¯ X t = x ] ; ¯ δ ( t, s, y ) = E t [ U 0 ( ¯ Y s ) | ¯ Y t = y ] (2.29) and G ( t, x ) := Z T t h ( t, s ) ¯ α ( t, s, x ) ds + h ( t, T ) ¯ α ( t, T , x ) (2.30) The goal is to show that G is a v alue function. W e start by getting the PDEs for ¯ δ , ¯ α . Prop osition 2.15. ¯ δ ( t, s, y ) satisfies the PDE: ¯ δ t + θ 2 2 ¯ δ y y + ( ¯ ρ ( t, y ) − r − θ 2 2 ) ¯ δ y ( t, s, y ) = 0 ; ¯ δ ( s, s, y ) = U 0 ( y ) (2.31) ¯ α ( t, s, x ) satisfies the PDE: ( ¯ α t + θ 2 v 2 2 v 2 x ¯ α xx + ( r x − I ( v ( t, x )) − θ 2 v v x ( t, x )) ¯ α x ( t, s, x ) = 0 ¯ α ( s, s, x ) = U 0 (log v ( s, x )) (2.32) F or the pro of, see the app endix. In the next prop osition, w e show that G x = v . Prop osition 2.16. F or t ∈ [0 , T ] , x ∈ (0 , ∞ ) : G x ( t, x ) = v ( t, x ) (2.33) The pro of is in the app endix. The next result shows that G ( t, x ) is a v alue function. Theorem 2.17. The function G ( t, x ) = E t  Z T t h ( t, s ) U (¯ c ( s ) ¯ X ( s )) ds + h ( t, T ) U ( ¯ X ( T ))  is a value function. It is the C 1 , 2 ([0 , T ] × (0 , ∞ )) solution of the extende d HJB G t ( t, x ) + sup ( π ,c ) admissible { σ 2 π 2 x 2 2 G xx ( t, x ) + ( r − c + θ π ) xG x ( t, x ) + U ( xc ) } = E t  Z T t ∂ h ( t, s ) ∂ t U ( ¯ c ( s ) ¯ X ( s )) ds + ∂ h ( t, T ) ∂ t U ( ¯ X ( T ))  (2.34) The sub game p erfe ct we alth pr o c ess is given by ¯ X ( s ) = X ¯ π , ¯ c ( s ) . The pro of is in the app endix. 10 3 Algorithm for constructing the subgame p erfect strategies W e are given an inv estor with initial wealth x 0 at time t = 0. W e w ant to construct the subgame p erfect strategies { ¯ c s , ¯ π s , ¯ X s , 0 ≤ s ≤ T } on the time interv al [0 , T ]. Recall I = U ′− 1 is the inv erse of the marginal utilit y , I 0 ( y ) = I ( e y ) and U 0 ( y ) = U ( I 0 ( y )). Step 1: Find the fixed p oin t for the op erator F . F or ϕ : [0 , T ] × R 7→ R b e a b ounded function with b ounded con tinuous first deriv atives in t and y , define Y t,y ; ϕ s as the solution of the SDE d Y t,y ; ϕ s = ( − θ 2 2 − r + ϕ ( s, Y ϕ s )) ds − θdW s , Y t,y ; ϕ t = y Define the op erator F [ ϕ ]( t, y ) = ∂ ∂ y E t  R T t ∂ h ( t,s ) ∂ t U 0 ( Y t,y ; ϕ s ) ds + ∂ h ( t,T )) ∂ t U 0 ( Y t,y ; ϕ T )  ∂ ∂ y E t  R T t h ( t, s ) U 0 ( Y t,y ; ϕ s ) ds + h ( t, T ) U 0 ( Y t,y ; ϕ T )  Find a fixed p oin t for the op erator F i.e. ¯ ρ such that: F [ ¯ ρ ]( t, y ) = ¯ ρ ( t, y ) ∀ t ∈ [0 , T ] , y ∈ R ¯ ρ can b e found using succe ssiv e appro ximations: ϕ 0 = 0 , ϕ n +1 = F [ ϕ n ] for all n . W e define the pro cess ¯ Y s = Y ¯ ρ s . Step 2: Define the function ¯ p ( t, y ) by: ¯ p ( t, y ) = E  e − r ( T − t ) I 0 ( ¯ Y T + θ 2 ( T − t )) + Z T t e − r ( s − t ) I 0 ( ¯ Y s + θ 2 ( s − t )) ds     ¯ Y t = y  Giv en t, y , define c ∗ ( t, y ) = I 0 ( y ) ¯ p ( t, y ) , π ∗ ( t, y ) = − θ ¯ p y ( t, y ) σ ¯ p ( t, y ) Step 3: Given an initial w ealth x 0 > 0 at time t = 0, we need to find y 0 suc h that x 0 = ¯ p (0 , y 0 ). F or s ∈ [0 , T ], with initial w ealth x 0 at t = 0, we can construct a subgame p erfect strategy: { π ∗ ( s, ¯ Y 0 ,y 0 s ) , c ∗ ( s, ¯ Y 0 ,y 0 s ) , ¯ p ( s, ¯ Y 0 ,y 0 s ) , 0 ≤ s ≤ T } References [1] T. Bj¨ ork, A. Murgo ci, and X. Y. Zhou (2014), Me an-varianc e p ortfolio optimization with state- dep endent risk aversion, Mathematical Finance, 24(1), 1-24. [2] T. Bj¨ ork, M. Khapk o, and A. Murgo ci (2017), A the ory of markovian time inc onsistent sto chastic c ontr ol in c ontinuous time, Finance and Sto c hastics, 21, 331-360. [3] S. Chang, P . Cosman, L. Milstein (Nov. 2011) Chernoff-T yp e Bounds for the Gaussian Err or F unc- tion, IEEE T ransactions on communications. [4] I. Ek eland, and T. A. Pirvu (2008), Investment and c onsumption without c ommitment, Mathematics and Financial Economics, 2(1), 57-86. [5] I. Ekeland, O. Mbo dji, and T. A. Pirvu (2012), Time c onsistent p ortfolio management, SIAM Journal on Financial Mathematics, 3(1), 1-32. 11 [6] L.C. Ev ans (1998), Partial differ ential e quations, Graduate studies in Mathematics, AMS. [7] M. G. Garroni, and J.L. Menaldi (2012), Gr e en functions for se c ond or der p ar ab olic inte gr o-differ ential pr oblems, Longman Scientific and T echnical. [8] Q. Lei and C.S. Pun (2024) On the Wel l-Pose dness of Hamilton-Jac obi-Bel lman Equations of the Equilibrium T yp e , SSRN [9] Q. Lei and C.S. Pun (2023) Nonlo c ality, nonline arity, and time inc onsistency in sto chastic differ ential games , Mathematical Finance [10] O. Mb odji, and T. A. Pirvu (2025), Portfolio time c onsistency and utility weighte d disc ount r ates, Mathematics and Financial Economics. [11] S. Nadto c hiy , and T. Zariphop oulou (2010) An appr oximation scheme for solution to the optimal investment pr oblem in inc omplete markets, Siam Journal of Financial Mathematics. [12] D. Kramko v and W. Sc hacherma yer (1999) The asymptotic elasticity of utility functions and optimal investment in inc omplete markets. The annals of Applied Probabilit y . [13] T. A. Pirvu, and H. Zhang (2014), Investment-c onsumption with r e gime-switching disc ount r ates, Mathematical So cial Sciences, 71, 142-150. [14] P . E. Protter (Second Edition), Sto chastic Inte gr ation and Differ ential Equations, Springer. [15] Sc hacherma y er, W. (2002). Optimal Investment in Inc omplete Financial Markets. In: Geman, H., Madan, D., Plisk a, S.R., V orst, T. (eds) Mathematical Finance — Bachelier Congress 2000. Springer Finance. Springer, Berlin, Heidelb erg. [16] J. Y ong (2012), Time inc onsistent optimal c ontr ol pr oblems and the e quilibrium HJB e quation, Math- ematical Con trol and Related Fields, 2(3), 271-329. 4 App endix Preliminary inequalities and gro wth b ounds Lemma 4.1. R e c al l the r elations I 0 ( y ) := I ( e y ) and U 0 ( y ) := U ( I 0 ( y )) . Sinc e U ′ ∈ D 0 ( r 1 , r 2 ) , I ∈ D 0  1 r 2 , 1 r 1  1 r 2 ≤ − I ′ 0 ( y ) I 0 ( y ) ≤ 1 r 1 (4.1) F urthermor e, for y , z ∈ R : I 0 satisfies: | I 0 ( z ) − I 0 ( y ) | I 0 ( y ) ≤ 1 r 1 .e | z − y | r 1 | z − y | ; I 0 ( z ) I 0 ( y ) ≤ e | z − y | r 1 (4.2) Ther e is β 1 , β 2 > 0 such that: β 1 e − β 2 | z − y | ≤ U ′ 0 ( z ) U ′ 0 ( y ) ≤ β 2 e β 2 | z − y | (4.3) | U 0 ( z ) − U 0 ( y ) | | U ′ 0 ( y ) | ≤ β 2 | z − y | .e β 2 | z − y | (4.4) 12 Pro of The first inequality comes from | I ′ 0 ( y ) | I 0 ( y ) = | e y I 0 ( y ) .U ′′ ( I 0 ( y )) | = 1 R 1 ( I 0 ( y )) 1 r 2 ≤ | I ′ 0 ( y ) I 0 ( y ) | ≤ 1 r 1 If w e integrate betw een y and z , w e get: e − | y − z | r 1 ≤ I 0 ( z ) I 0 ( y ) ≤ e | y − z | r 1 By the mean v alue theorem, there is z 0 ∈ [ y , z ] such that: | I 0 ( z ) − I 0 ( y ) | I 0 ( y ) ≤ | z − y | | I ′ 0 ( z 0 ) | I 0 ( y ) ≤ 1 r 1 .e | z − y | r 1 | z − y | W e also hav e r 2 1 e − (1+ 1 r 1 ) | z − y | ≤ U ′ 0 ( z ) U ′ 0 ( y ) = I ′ 0 ( z ) e z I ′ 0 ( y ) e y ≤ e (1+ 1 r 1 ) | z − y | r 2 1 (4.5) By the mean v alue theorem, there exists z 0 ∈ [ y , z ] such that: | U 0 ( z ) − U 0 ( y ) | | U ′ 0 ( y ) | ≤ | z − y | | U ′ 0 ( z 0 ) | | U ′ 0 ( y ) | ≤ | z − y | r 2 1 · e (1+ 1 r 1 ) | z − y | W e take β 1 := r 2 1 ; β 2 := 1 + 1 r 2 1 . □ Lemma 4.2. Supp ose Z is distribute d as N (0 , 1) an d κ > 0 . Then E [ e − κ | Z | ] = 2 e κ 2 2 N ( − κ ) ; E [ e κ | Z | ] = 2 e κ 2 2 N ( κ ) (4.6) wher e N is the distribution function of the normal distribution. F urthermor e, for x ≥ 0 , ac c or ding to [3], we have the lower b ound: N ( − x ) ≥ 3 4 exp( − 4 x 2 ) (4.7) Pro of W e prov e (4.6) by direct computation of the exp ectation. □ F undamen tal solution and regularity The follo wing lemma will allo w us to get estimates for the op erator F . Lemma 4.3. L et n 1 , n 2 , q 1 , q 2 ∈ R and supp ose q 1 q 2 > 0 . The quotient n 1 + n 2 q 1 + q 2 is b etwe en n 1 q 1 and n 2 q 2 i.e. n 1 + n 2 q 1 + q 2 ∈  n 1 q 1 , n 2 q 2  . 13 Pro of Let x := n 1 + n 2 q 1 + q 2 . W e ha ve: x − n 1 q 1 = n 2 q 1 − n 1 q 2 q 1 ( q 1 + q 2 ) ; x − n 2 q 2 = n 1 q 2 − n 2 q 1 q 2 ( q 1 + q 2 ) x − n 1 q 1 and x − n 2 q 2 ha ve opp osite signs, therefore x ∈ [ n 1 q 1 , n 2 q 2 ]. □ Definition 4.4. F or ϕ : [0 , T ] × R 7→ R in B and 0 ≤ t 1 ≤ t 2 ≤ T , we define the follo wing quantities || ϕ || t 1 ,t 2 = sup ( t,y ) ∈ [ t 1 ,t 2 ] × R | ϕ ( t, y ) | (4.8) || ϕ y || t 1 ,t 2 = sup ( t,y ) ∈ [ t 1 ,t 2 ] × R | ϕ y ( t, y ) | (4.9) F or ϕ ∈ B , recall that Y ϕ ; t,y u , t ≤ u ≤ T by the SDE: ( d Y ϕ ; t,y u = ( ϕ ( u, Y ϕ ; t,y u ) − r − θ 2 2 ) du − θdW u , Y ϕ ; t,y t = y (4.10) Define δ ϕ ( t, s, y ) := E P t  U 0 ( Y ϕ ; t,y s )  and F [ ϕ ]( t, y ) = R T t ∂ h ( t,s ) ∂ t δ ϕ y ( t, s, y ) ds + ∂ h ( t,T ) ∂ t δ ϕ y ( t, T , y ) R T t h ( t, s ) δ ϕ y ( t, s, y ) ds + h ( t, T ) δ ϕ y ( t, T , y ) (4.11) Note that δ ϕ y ( t, s, y ) := E P t,y  U ′ 0 ( Y ϕ s ) e R s t ϕ y ( u,Y t,y u ) du  < 0 (4.12) So the denominator in the expression of F [ ϕ ] is negative. W e wan t to find ¯ ρ such that F [ ¯ ρ ]( t, y ) = ¯ ρ ( t, y ). T o pro ve it, we ha ve to find a suitable space of functions ϕ ( t, y ) such that F is a con traction and use a fixed p oin t theorem. The function δ ϕ satisfies the following PDE: ( δ ϕ t ( t, s, y ) + θ 2 2 δ ϕ y y ( t, s, y ) + ( ϕ ( t, y ) − r − θ 2 2 ) δ ϕ y ( t, s, y ) = 0 δ ϕ ( s, s, y ) = U 0 ( y ) (4.13) for all t ∈ [0 , s ] and y ∈ R . W rite F = F 1 F 0 (4.14) where F 1 resp. F 0 are the numerator and denominator in the expression of F . Define F y [ ϕ ]( t, y ) := ∂ F [ ϕ ]( t, y ) ∂ y W e define F 0 y , F 1 y in a similar manner. W e hav e: F y = F 1 y − F F 0 y F 0 i.e. ∂ F [ ϕ ]( t, y ) ∂ y = R T t ( ∂ h ( t,s ) ∂ t − h ( t, s ) F [ ϕ ]( t, y )) δ ϕ y y ( t, s, y ) ds + ( ∂ h ( t,T ) ∂ t − h ( t, T ) F [ ϕ ]( t, y )) δ ϕ y y ( t, T , y ) R T t h ( t, s ) δ ϕ y ( t, s, y ) ds + h ( t, T ) δ ϕ y ( t, T , y ) (4.15) 14 Definition 4.5. Let ϕ 1 , ϕ 2 ∈ B κ . F or 0 ≤ t ≤ s ≤ T and y ∈ R : ϵ ϕ 1 ,ϕ 2 ( t, s, y ) = δ ϕ 2 ( t, s, y ) − δ ϕ 1 ( t, s, y ) (4.16) ϵ satisfies the PDE: ϵ ϕ 1 ,ϕ 2 t + θ 2 2 ϵ ϕ 1 ,ϕ 2 y y + ( ϕ 2 ( t, y ) − r − θ 2 2 ) ϵ ϕ 1 ,ϕ 2 y ( t, s, y ) = − ( ϕ 2 ( t, y ) − ϕ 1 ( t, y )) δ ϕ 1 y ( t, s, y ) (4.17) ϵ ϕ 1 ,ϕ 2 ( s, s, y ) = 0 (4.18) W e wan t to get b ounds for the quan tities | F [ ϕ 2 ]( t, y ) − F [ ϕ 1 ]( t, y ) | , | F [ ϕ 2 ] y ( t, y ) − F [ ϕ 1 ] y ( t, y ) | in terms of ϵ ϕ 1 ,ϕ 2 and its deriv atives. In tegral estimates Prop osition 4.6. L et b b e a p ositive c onstant. L et a ( t, y ) b e a function define d on [0 , T ] × R . Supp ose a is c ontinuous, b ounde d and uniformly Lipschitz in the variable y uniformly in t i.e. | a ( t, y 2 ) − a ( t, y 1 ) | ≤ || a || L | y 2 − y 1 | ∀ t ∈ [0 , T ] , y 1 , y 2 ∈ R . Consider the Cauchy pr oblem − u t ( t, y ) + bu y y ( t, y ) + a ( t, y ) u y ( t, y ) = 0 (4.19) Then for l = 0 , 1 , 2 , the fundamental solution Γ( t, y , τ , z ) , 0 ≤ τ < t ≤ T , y , z ∈ R satisfies the b ound : | ∂ l ∂ y l Γ( t, y ; τ , z ) | ≤ C (1 + || a || L ) × ( t − τ ) − 1+ l 2 exp( − c | y − ζ | 2 t − τ ) (4.20) wher e c , C ar e c onstants indep endent of t, y , || a || L . Pro of This result is a consequence of the construction of Γ using the parametrix metho d as in [7]. Let Γ b ( t, y ) = 1 √ 4 π bt e − y 2 4 bt and F ( t, y , τ , ζ ) = − a ( t, y ) ∂ ∂ y Γ b ( y − ζ , t − τ ). Let Q ( t, y , τ , ζ ) satisfy the V olterra equation: Q ( t, y , τ , ζ ) = F ( t, y , τ , ζ ) + Z t τ ds Z R F ( t, y , s, z ) Q ( s, z , τ , ζ ) dz Γ 1 ( t, y , τ , ζ ) = Z t τ ds Z R Γ b ( t − s, y − z ) Q ( s, z , τ , ζ ) dz Finally , the fundamental solution of the PDE (4.19) is Γ( t, y , τ , z ) = Γ b ( t − τ , y − z ) + Γ 1 ( t, y , τ , z ) The function F satisfies a Lipschitz prop ert y in y , the function Q satisfies a similar estimate. The estimate (4.20) for | ∂ l ∂ y l Γ( t, y , τ , z ) | by following the pro ofs in Chapter 5 of [7] [ Lemma 3.1, Lemma 3.3 and Theorem 3.5]. The constan t c dep ends on b and C dep ends on b and T . □ W e will need the following calculations that will help us estimate the deriv ativ es of parab olic PDE solutions. 15 Lemma 4.7. F or c > 0 , τ > t ≥ 0 . n 0 = 0 , 1 and α ∈ R . We c an get explicit expr essions for the inte gr al I l n 0 ,α = Z R ( τ − t ) − 1+ l 2 | ζ − y | n 0 e − c | y − ζ | 2 τ − t e α | ζ − y | dζ (4.21) I l 0 ,α = ( τ − t ) − l 2 √ 2 π e α 2 ( τ − t ) 4 c √ 2 c N ( α r τ − t 2 c ) (4.22) I l 1 ,α = ( τ − t ) 1 − l 2 c ×  1 + α √ π r τ − t c e α 2 ( τ − t ) 4 c N ( α r τ − t 2 c )  (4.23) Estimates for ¯ δ , ϵ and their deriv ativ es. Prop osition 4.8. L et t ∈ [0 , T ] , y ∈ R , ϕ 1 , ϕ 2 ∈ B : Ther e exists s 0 , s 1 , s 2 , s 3 ∈ [ t, T ] such that: | F [ ϕ 2 ]( t, y ) − F [ ϕ 1 ]( t, y ) | ≤ 2 || ρ || . | ϵ ϕ 1 ,ϕ 2 y ( t, s 0 , y ) | | δ ϕ 1 y ( t, s 0 , y ) | (4.24) | ∂ F [ ϕ 2 ]( t, y ) ∂ y − ∂ F [ ϕ 1 ]( t, y ) ∂ y | ≤ 2 || ρ || . | ϵ ϕ 1 ,ϕ 2 y y δ ϕ 1 y ( t, s 1 , y ) | + 4 || ρ || . | δ ϕ 1 y y δ ϕ 1 y ( t, s 2 , y ) | × | ϵ ϕ 1 ,ϕ 2 y δ ϕ 1 y ( t, s 3 , y ) | (4.25) Pro of F [ ϕ 2 ]( t, y ) − F [ ϕ 1 ]( t, y ) = F 1 [ ϕ 2 ] F 0 [ ϕ 2 ] ( t, y ) − F 1 [ ϕ 1 ] F 0 [ ϕ 1 ] ( t, y ) = F 1 [ ϕ 2 ] − F 1 [ ϕ 1 ]( t, y ) F 0 [ ϕ 1 ]( t, y ) − F [ ϕ 2 ]( t, y ) × ( F 0 [ ϕ 2 ] − F 0 [ ϕ 1 ]( t, y )) F 0 [ ϕ 1 ]( t, y ) = R T t h ( t, s )( ρ h ( t, s ) − F [ ϕ 2 ]( t, y )) ϵ ϕ 1 ,ϕ 2 y ( t, s, y ) ds + h ( t, T )( ρ h ( t, T ) − F [ ϕ 2 ]( t, y )) ϵ ϕ 1 ,ϕ 2 y ( t, T , y ) R T t h ( t, s ) δ ϕ 1 y ( t, s, y ) ds + h ( t, T ) δ ϕ 1 y ( t, T , y ) By lemma 4.3 F [ ϕ 2 ]( t, y ) − F [ ϕ 1 ]( t, y ) ∈  R T t h ( t, s )( ρ h ( t, s ) − F [ ϕ 2 ]( t, y )) ϵ ϕ 1 ,ϕ 2 y ( t, s, y ) ds R T t h ( t, s ) δ ϕ 1 y ( t, s, y ) ds , ( ρ h ( t, T ) − F [ ϕ 2 ]( t, y )) ϵ ϕ 1 ,ϕ 2 y ( t, T , y ) δ ϕ 1 y ( t, T , y )  and the extended mean v alue theorem yields the existence of s 0 ∈ [ t, T ] suc h that: | F [ ϕ 2 ]( t, y ) − F [ ϕ 1 ]( t, y ) | ≤ | ( ρ h ( t, s 0 ) − F [ ϕ 2 ]( t, y )) .ϵ ϕ 1 ,ϕ 2 y ( t, s 0 , y ) δ ϕ 1 y ( t, s 0 , y ) | Using the inequalities | F [ ϕ ] | ≤ || ρ || and | ρ h | ≤ || ρ || , w e get inequalit y (4.8). ∂ F [ ϕ 2 ]( t, y ) ∂ y − ∂ F [ ϕ 1 ]( t, y ) ∂ y = R T t ( ∂ h ( t,s ) ∂ t − h ( t, s ) F [ ϕ 2 ]( t, y )) δ ϕ 2 y y ( t, s, y ) ds R T t h ( t, s ) δ ϕ 2 y ( t, s, y ) ds − R T t ( ∂ h ( t,s ) ∂ t − h ( t, s ) F [ ϕ 1 ]( t, y )) δ ϕ 1 y y ( t, s, y ) ds R T t h ( t, s ) δ ϕ 1 y ( t, s, y ) ds = R T t h ( t, s ) ×  ( F [ ϕ 1 ] − F [ ϕ 2 ]( t, y )) δ ϕ 1 y y ( t, s, y ) + ( ρ h ( t, s ) − F [ ϕ 2 ]( t, y )) ϵ ϕ 1 ,ϕ 2 y y ( t, s, y )  ds R T t h ( t, u ) δ ϕ 1 y ( t, u, y ) du + R T t h ( t, s )( F [ ϕ 1 ]( t, y ) − ρ h ( t, s )) δ ϕ 1 y y ( t, s, y ) ds R T t h ( t, u ) δ ϕ 1 y ( t, s, y ) ds × R s t h ( t, s ) ϵ ϕ 1 ,ϕ 2 y ( t, s, y ) ds R T t h ( t, s ) δ ϕ 2 y ( t, s, y ) ds  16 T o simplify the calculations ab ov e, we ha ve omitted the term in ( t, T , y ). W e can apply the mean v alue theorem to the quotien t of integrals and lemma 4.3: there exists s 0 , s 1 , s 2 , s 3 ∈ [ t, T ] suc h that | ∂ F [ ϕ 2 ]( t, y ) ∂ y − ∂ F [ ϕ 1 ]( t, y ) ∂ y | ≤ | ( ρ h ( t, s 1 ) − F [ ϕ 2 ]( t, y )) ϵ ϕ 1 ,ϕ 2 y y δ ϕ 1 y ( t, s 1 , y ) | + | ( − ρ h ( t, s 0 ) + F [ ϕ 2 ]( t, y )) ϵ ϕ 1 ,ϕ 2 y δ ϕ 1 y ( t, s 0 , y ) . δ ϕ 1 y y δ ϕ 1 y ( t, s 1 , y ) | + | ( F [ ϕ 1 ]( t, y ) − ρ h ( t, s 2 )) δ ϕ 1 y y δ ϕ 1 y ( t, s 2 , y ) × ϵ ϕ 1 ,ϕ 2 y δ ϕ 1 y ( t, s 3 , y ) | Using the inequalities | F [ ϕ ] | ≤ || ρ || and | ρ h | ≤ || ρ || , w e get the result. □ W e need to find upp er b ounds for the following quantities | δ ϕ i yy δ ϕ i y ( t, s, y ) | , for i = 1 , 2 and for | ϵ ϕ 1 ,ϕ 2 y δ ϕ 1 y ( t, s, y ) | , | ϵ ϕ 1 ,ϕ 2 yy δ ϕ 1 y ( t, s, y ) | . The follo wing lemma establishes b ounds for δ ϕ y ( t, s, y ). Lemma 4.9. L et t, s ∈ [0 , T ] , t ≤ s, y ∈ R , ϕ ∈ B κ : ther e ar e p ositive c onstants k 10 , k 20 indep endent of κ such that k 10 e − κ ( s − t ) ≤ | δ ϕ y ( t, s, y ) U ′ 0 ( y ) | ≤ k 20 e κ ( s − t ) (4.26) | δ 0 y y ( t, s, y ) U 0 ( y ) | ≤ β 2 k 20 (4.27) Pro of Note that if ϕ ∈ B , D ( s ) := ∂ Y ϕ s ∂ y = exp  Z s t ∂ ϕ ∂ y ( u, Y ϕ u ) du  (4.28) Recall that δ ϕ ( t, s, y ) = E P t,y  U 0 ( Y ϕ s )  and δ ϕ y ( t, s, y ) = E P t,y  U ′ 0 ( Y ϕ s ) e R s t ϕ y ( u,Y ϕ u ) du  th us, | δ ϕ y ( t, s, y ) U ′ 0 ( y ) | ≤ E P t,y  | U ′ 0 ( Y ϕ s ) U ′ 0 ( y ) | e R s t ϕ y ( u,Y ϕ u ) du  ≤ β 2 E t [ e β 2 | Y ϕ s − y | e R s t ϕ y ( u,Y ϕ u ) du ] ≤ 2 β 2 e ( β 2 ( r + θ 2 2 +2 || ρ || )+ β 2 2 θ 2 2 )( s − t )+ κ ( s − t ) N ( β 2 θ √ s − t ) ≤ 2 β 2 e ( β 2 ( r + θ 2 2 +2 || ρ || )+ β 2 2 θ 2 2 )( s − t )+ κ ( s − t ) Similarly , | δ ϕ y ( t, s, y ) U ′ 0 ( y ) | ≥ 2 β 1 e − ( β 2 ( r + θ 2 2 +2 || ρ || )+ β 2 2 θ 2 2 − κ ( s − t ) N ( − β 2 θ √ s − t ) ≥ 3 β 1 2 e − β 2 ( r + θ 2 2 +2 || ρ || )( s − t )+ β 2 2 θ 2 2 )( s − t ) − κ ( s − t ) − 4 β 2 2 θ 2 ( s − t ) where w e used the inequalit y N ( − x ) ≥ 3 4 e − 4 x 2 ∀ x > 0. 17 Define the p ositiv e num b ers k 10 := 3 β 1 2 e  − β 2 ( r + θ 2 2 +2 || ρ || ) − 7 β 2 2 θ 2 2  T ; k 20 := 2 β 2 e  β 2 ( r + θ 2 2 +2 || ρ || )+ β 2 2 θ 2 2  T (4.29) W e hav e the inequalities k 10 e − κ ( s − t ) ≤ | δ ϕ y ( t, s, y ) U ′ 0 ( y ) | ≤ k 20 e κ ( s − t ) (4.30) Let Y 0 s b e the solution of the SDE (4.13), where we take ϕ = 0. The functions ϕ and ∂ ϕ ∂ y are b ounded contin uous functions of ( t, y ). W e can use Chapter 5, theorem 7.6 of (Bro wnian Motion and Stochastic Calculus): let R ϕ ( t, y ; τ , ζ ) b e the fundamen tal solution asso ciated to δ ϕ giv en by the parab olic PDEs (4.13). δ ϕ ( t, s, y ) = Z R R ϕ ( y , s − t, ζ , 0) U 0 ( ζ ) dζ Then, b y prop osition 4.6, we hav e the following estimate for R ϕ | ∂ l R ϕ ∂ y l ( y , t, ζ , τ ) | ≤ C (1 + κ )( τ − t ) − 1+ l 2 exp  − c | y − ζ | 2 τ − t  (4.31) where l = 0 , 1 , 2. The p ositiv e constants c, C dep end on r , θ, T , || ϕ || , β 2 and do not dep end on κ . Lemma 4.10. Ther e is k 30 > 0 and C 0 > 0 dep ending only on r, θ , || ρ || , T , β 2 such that | δ 0 y y ( t, s, y ) U ′ 0 ( y ) | ≤ k 30 ; | F [0] y ( t, y ) | ≤ C 0 ∀ t, y . W e can estimate | δ 0 yy ( t,s,y ) U ′ 0 ( y ) | easier b ecause the fundamental solution will b e of the form R 0 = R 0 ( s − t, y − ζ ). Using an integration by parts, we get: δ 0 y y ( t, s, y ) = Z R R 0 y y ( y − ζ , s − t ) U 0 ( ζ ) dζ = Z R R 0 y ( y − ζ , s − t ) U ′ 0 ( ζ ) dζ W riting | R 0 y ( y − ζ , s − t ) | ≤ C ( s − t ) − 1 2 exp( − c | y − ζ | 2 s − t ) | δ 0 y y ( t, s, y ) U ′ 0 ( y ) | ≤ Z R β 2 e β 2 | ζ − y | C ( s − t ) − 1 2 exp( − c | y − ζ | 2 s − t ) dζ = C β 2 I 0 0 ,β 2 ≤ C β 2 r π c e β 2 2 T 4 c := k 30 □ If we study the expression of F y [ ϕ ]( t, y ), we notice that the n umerator and denominator are con tinuous functions of the v ariable t, y . W e can use the extended mean v alue theorem whic h is applied to a quotient of in tegrals. By the mean v alue theorem there exists s 0 ∈ [ t, T ] suc h that | ∂ F [ ϕ ]( t, y ) ∂ y | ≤ 2 || ρ || . | δ ϕ y y ( t, s 0 , y ) δ ϕ y ( t, s 0 , y ) | F or a fixed t ∈ [0 , T ] , s ∈ [ t, T ] , y ∈ R , w e need to find b ounds for δ ϕ yy ( t,s,y ) δ ϕ y ( t,s,y ) . 18 Prop osition 4.11. L et t, s ∈ [0 , T ] , t ≤ s, y ∈ R , ϕ ∈ B κ : Ther e is a p ositive c onstant C > 0 indep endent on κ such that for al l 0 ≤ t ≤ s ≤ T : | δ ϕ y y ( t, s, y ) U ′ 0 ( y ) | ≤ k 30 + C (1 + κ ) 2 √ s − t (4.32) Pro of W e write δ ϕ ( t, s, y ) = δ 0 ( t, s, y ) + ϵ 0 ,ϕ ( t, s, y ), and apply the triangular inequality: | δ ϕ y y ( t, s, y ) U ′ 0 ( y ) | ≤ | δ 0 y y ( t, s, y ) U ′ 0 ( y ) | + | ϵ 0 ,ϕ y y ( t, s, y ) U ′ 0 ( y ) | The first term is b ounded b y k 30 , the second term can b e rewritten ϵ 0 ,ϕ y y ( t, s, y ) = Z s t Z R | R ϕ y y ( s − t, y , s − τ , ζ ) ϕ ( τ , ζ ) δ 0 y ( τ , s, ζ ) | dζ dτ = Z s t Z R R ϕ y y ( s − t, y , s − τ , ζ )( ϕ ( τ , ζ ) δ 0 y ( τ , s, ζ ) − ϕ ( τ , y ) δ 0 y ( τ , s, y )) dζ dτ The term in parenthesis can b e b ounded using the mean v alue theorem. There is ζ 0 ∈ [ y , ζ ] such that: | ϕ ( τ , ζ ) δ 0 y ( τ , s, ζ ) − ϕ ( τ , y ) δ 0 y ( τ , s, y )) | ≤ ( | ϕ y ( τ , ζ 0 ) δ 0 y ( τ , s, ζ ) | + | ϕ ( τ , ζ 0 ) δ 0 y y ( τ , s, ζ 0 ) | ) . | ζ − y | ≤ ( κk 20 + || ρ || k 30 ) | U ′ 0 ( y ) | . | ζ − y | .e β 2 | ζ − y | W e use lemma 4.7 to get: | ϵ 0 ,ϕ y y ( t, s, y ) U ′ 0 ( y ) | ≤ Z s t Z R C (1 + κ )( τ − t ) − 3 / 2 e − c | y − ζ | 2 τ − t | ζ − y | e β 2 | ζ − y | ( κk 20 + || ρ || k 30 ) dζ dτ ≤ C c Z s t ( τ − t ) − 1 / 2 (1 + κ )( κk 20 + || ρ || k 30 )  1 + β 2 r π ( τ − t ) c e β 2 2 ( τ − t ) 4 c  dτ | ϵ 0 ,ϕ y y ( t, s, y ) U ′ 0 ( y ) | ≤ 2 C c ( k 20 + || ρ || k 30 )(1 + κ ) 2  1 + β 2 r π ( s − t ) c e β 2 2 ( s − t ) 4 c  × √ s − t So there is C > 0 dep ending on the co efficien ts of the PDE i.e. on || ρ || , r , θ , β 2 , T such that | δ ϕ y y ( t, s, y ) U ′ 0 ( y ) | ≤ k 30 + C (1 + κ ) 2 √ s − t □ No w, let ϕ 1 , ϕ 2 ∈ B κ . W e can estimate | F [ ϕ 2 ] − F [ ϕ 1 ]( t, y ) | in terms of | ϵ ϕ 1 ,ϕ 2 y | so we need an upp er b ound. ϵ ϕ 1 ,ϕ 2 ( t, s, y ) = − Z s t dτ Z R R ϕ 2 ( y , s − t, ζ , s − τ )( ϕ 2 ( τ , ζ ) − ϕ 1 ( τ , ζ )) δ ϕ 1 y ( τ , s, ζ ) dζ ϵ ϕ 1 ,ϕ 2 y ( t, s, y ) = − Z s t dτ Z R R ϕ 2 y ( y , s − t, ζ , s − τ )( ϕ 2 ( τ , ζ ) − ϕ 1 ( τ , ζ )) δ ϕ 1 y ( τ , s, ζ ) dζ W e write | ( ϕ 2 ( τ , ζ ) − ϕ 1 ( τ , ζ )) δ ϕ 1 y ( τ , s, ζ ) | ≤ | U ′ 0 ( ζ ) | .k 20 e κ ( s − τ ) | ϕ 2 ( τ , ζ ) − ϕ 1 ( τ , ζ ) | 19 and | R ϕ 2 y ( y , s − t, ζ , s − τ ) | ≤ C (1 + κ )( τ − t ) − 1 e − c | y − ζ | 2 τ − t to get | ϵ ϕ 1 ,ϕ 2 y ( t, s, y ) U ′ 0 ( y ) | ≤ Z s t Z R C (1 + κ )( τ − t ) − 1 e − c | y − ζ | 2 τ − t .e β | ζ − y | .k 20 e κ ( s − τ ) || ϕ 2 − ϕ 1 || t,s dζ dτ ≤ Z s t C (1 + κ )( τ − t ) − 1 2 √ π e β 2 ( τ − t ) 4 c 1 √ c N ( β r τ − t 2 c ) .k 20 e κ ( s − t ) || ϕ 2 − ϕ 1 || t,s dτ | ϵ ϕ 1 ,ϕ 2 y ( t, s, y ) U ′ 0 ( y ) | ≤ C. (1 + κ ) e κ ( s − t ) √ s − t. || ϕ 2 − ϕ 1 || t,s for a constant C > 0 dep ending only on r , θ , T , β 2 , || ρ || . Th us | F [ ϕ 2 ] − F [ ϕ 1 ]( t, y ) | ≤ 2 || ρ || C e 2 κ ( T − t ) . (1 + κ ) k 10 . √ T − t × || ϕ 2 − ϕ 1 || t,T W e hav e ϵ ϕ 1 ,ϕ 2 y y ( t, s, y ) = − Z s t dτ Z R R ϕ 2 y y ( y , s − t, ζ , s − τ )( ϕ 2 ( τ , ζ ) − ϕ 1 ( τ , ζ )) δ ϕ 1 y ( τ , s, ζ ) dζ = − Z s t dτ Z R R ϕ 2 y y ( y , s − t, ζ , s − τ )  ( ϕ 2 ( τ , ζ ) − ϕ 1 ( τ , ζ )) δ ϕ 1 y ( τ , s, ζ ) − ( ϕ 2 ( τ , y ) − ϕ 1 ( τ , y )) δ ϕ 1 y ( τ , s, y )  dζ By the mean v alue theorem, there is ζ ∈ [ y , ζ ] such that the term in paren thesis is equal to ( ζ − y )  ( ϕ 2 y − ϕ 1 y ) δ ϕ 1 y ( τ , s, ζ 0 ) + ( ϕ 2 − ϕ 1 ) δ ϕ 1 y y ( τ , s, ζ 0 )  and it can b e b ounded by | ζ − y | . | U ′ 0 ( ζ 0 ) | .  || ϕ 2 y − ϕ 1 y || t,s k 20 e κ ( s − t ) + || ϕ 2 − ϕ 1 || t,s ( k 30 + C (1 + κ ) 2 √ s − t )  W e now use the inequalities | R ϕ 2 y y ( y , s − t, ζ , s − τ ) | ≤ C (1 + κ )( τ − t ) − 3 / 2 e − c | y − ζ | 2 τ − t and Lemma 4.7 to get | ϵ ϕ 1 ,ϕ 2 y y ( t, s, y ) U ′ 0 ( y ) | ≤ Z s t dτ Z R C (1 + κ )( τ − t ) − 3 / 2 e − c | y − ζ | 2 τ − t | ζ − y | .e β | ζ − y | ×  || ϕ 2 y − ϕ 1 y || t,s k 20 e κ ( s − t ) + || ϕ 2 − ϕ 1 || t,s ( k 30 + C (1 + κ ) 2 √ s − t )  dζ ≤ 2 C ( s − t ) 1 2 c ×  1 + β r π ( s − t ) c e β 2 ( s − t ) 4 c  × (1 + κ ) ×  || ϕ 2 y − ϕ 1 y || t,s k 20 e κ ( s − t ) + || ϕ 2 − ϕ 1 || t,s ( k 30 + C (1 + κ ) 2 √ s − t )  | ϵ ϕ 1 ,ϕ 2 y y ( t, s, y ) U ′ 0 ( y ) | ≤ C ′ (1 + κ ) 3 e κ ( s − t ) √ s − t ×  || ϕ 2 y − ϕ 1 y || t,s + || ϕ 2 − ϕ 1 || t,s  for a p ositiv e constant C ′ dep ending on the co efficien ts of the PDE i.e. on || ρ || , r, θ , β 2 , T . 20 Estimates for the op erator F W e combine the estimates for | ϵ ϕ 1 ,ϕ 2 y ( t,s,y ) U ′ 0 ( y ) | , | ϵ ϕ 1 ,ϕ 2 yy ( t,s,y ) U ′ 0 ( y ) | with prop osition 4.8 to get the following prop osi- tion. Prop osition 4.12. L et κ > 0 , ϕ 1 , ϕ 2 ∈ B κ . Ther e ar e universal c onstants K 0 , K 1 > 0 indep endent of κ , t, y such that for al l t ∈ [0 , T ] , y ∈ R : | F [ ϕ 2 ] − F [ ϕ 1 ]( t, y ) | ≤ K 0 (1 + κ ) e 2 κ ( T − t ) √ T − t × || ϕ 2 − ϕ 1 || t,T (4.33) | F [ ϕ 2 ] y − F [ ϕ 1 ] y ( t, y ) | ≤ K 1 (1 + κ ) 3 e 2 κ ( T − t ) √ T − t ×  || ϕ 2 − ϕ 1 || t,T + || ϕ 2 y − ϕ 1 y || t,T  (4.34) Recall that by Lemma 4.10, there is C 0 > 0 indep enden t of κ, t, y suc h that | F [0] y ( t, y ) | ≤ C 0 ∀ ( t, y ) ∈ [0 , T ] × R . If we c ho ose κ = max( || ρ || , 2 C 0 ) (4.35) and ν ∈ (0 , T ) small enough such that: K 0 (1 + κ ) e 2 κT √ ν ≤ 1 4 ; K 1 (1 + κ ) 3 e 2 κT √ ν ≤ 1 4 (4.36) W e define the following space of functions whic h is a restriction of B κ to [ T − ν , T ] × R . B (1) κ = { ϕ : [ T − ν , T ] × R → R , con tinuous in t, y ; C 1 in y ; | ϕ ( t, y ) | ≤ || ρ || ; | ϕ y ( t, y ) | ≤ κ } (4.37) F or ϕ, ϕ 1 , ϕ 2 ∈ B (1) κ , w e hav e: || F [ ϕ 2 ] y − F [ ϕ 1 ] y || T − ν,T + || F [ ϕ 2 ] − F [ ϕ 1 ] || T − ν,T ≤ 1 2 ( || ϕ 2 y − ϕ 1 y || T − ν,T + || ϕ 2 − ϕ 1 || T − ν,T ) F urthermore, F [ ϕ ]( t, y ) and F [ ϕ ] y ( t, y ) are quotients of integrals and therefore are contin uous func- tions of t, y . W e also hav e: || F [ ϕ ] y || T − ν,T ≤ || F [ ϕ ] y − F [0] y || T − ν,T + || F [0] y || T − ν,T ≤ κ 4 + 1 4  || ρ || + κ  ≤ κ Th us, F is a con traction for the norm ϕ 7→ || ϕ || T − ν,T + || ϕ y || T − ν,T in the complete con vex subset B (1) κ , therefore it admits a unique fixed p oint Φ. F [Φ]( t, y ) = Φ( t, y ) ∀ ( t, y ) ∈ [ T − ν, T ] × R (4.38) W e can build Φ on [ T − 2 ν, T ] × R by defining the space B (2) κ of functions ϕ : [ T − 2 ν, T ] × R → R suc h that: • ϕ ( t, y ) , ϕ y ( t, y ) are contin uous in t, y . • | ϕ ( t, y ) | ≤ || ρ || ; | ϕ y ( t, y ) | ≤ κ ; ϕ ( t, y ) = Φ( t, y ) ∀ t ∈ [ T − ν, T ] , y ∈ R . The function f ( t, y ) = ( Φ( t, y ) for ( t, y ) ∈ [ T − ν, T ] × R Φ( T − ν , y ) for ( t, y ) ∈ [ T − 2 ν , T − ν ) × R W e can construct Φ in this w ay on interv als [ T − ( k + 1) ν, T − k ν ] for k = 1 , 2 , · · · . Φ ∈ B κ and F [Φ]( t, y ) = Φ( t, y ) for t ∈ [0 , T ] , y ∈ R . 21 W e take ¯ ρ := Φ. And this ends the pro of of the fixed p oin t result. Pro of Pro of proposition 2.7 Note that I ′ 0 ( y ) = e y I ′ ( e y ). Since U ′ ∈ D 0 ( r 1 , r 2 ), this implies that its inv erse I ∈ D 0  1 r 2 , 1 r 1  i.e. 1 r 2 ≤ − xI ′ ( x ) I ( x ) ≤ 1 r 1 . Thus, 1 r 2 ≤ − I ′ 0 ( y ) I 0 ( y ) ≤ 1 r 1 ∀ y ∈ R (4.39) W e also hav e: − ¯ p y ( t, y ) ¯ p ( t, y ) = − E t h R T t e R s t ( ¯ ρ y ( u, ¯ Y u ) − r ) du I ′ 0 ( ¯ Y s + θ 2 ( s − t )) ds + e R T t ( ¯ ρ y ( u, ¯ Y u ) − r ) du I ′ 0 ( ¯ Y T + θ 2 ( T − t )) i E t h R T t e − r ( s − t ) I 0 ( ¯ Y s + θ 2 ( s − t )) ds + e − r ( T − t ) I 0 ( ¯ Y T + θ 2 ( T − t )) i And since for all t, y , | ¯ ρ y ( t, y ) | ≤ κ , we get the estimate: 1 r 2 e − κ ( T − t ) ≤ − ¯ p y ( t, y ) ¯ p ( t, y ) ≤ 1 r 1 e κ ( T − t ) (4.40) And since ¯ p ( t, y ) = p ( t, e y ), w e get 1 r 2 ( t ) ≤ − xp x ( t, x ) p ( t, x ) ≤ 1 r 1 ( t ) (4.41) i.e. p ( t, . ) ∈ D 0  1 r 2 ( t ) , 1 r 1 ( t )  . This implies v ( t, . ) the x -inv erse of p ( t, . ) is well defined. By Lemma 2.6, v ( t, . ) ∈ D 0 ( r 1 ( t ) , r 2 ( t )). □ Pro of Prop osition 2.8 W e w an t to sho w that | 1 I 0 ( y ) × ∂ l ¯ p ( t,y ) ∂ y l | is b ounded. The pro of is similar to the pro of of the boundedness of | 1 U ′ 0 ( y ) × ∂ l ¯ δ ∂ y l | . W e just hav e to replace the boundary function U 0 b y I 0 and the constan t β 2 b y 1 r 1 as the estimates of Lemma 4.1 show. □ Pro of Prop osition 2.10 Let y 1 ≤ y 2 , b y integrating the expression (4.39) b et w een y 1 and y 2 : 1 r 2 . ( y 2 − y 1 ) ≤ − log ( I 0 ( y 2 ) /I 0 ( y 1 )) ≤ 1 r 1 . ( y 2 − y 1 ) e − y 2 − y 1 r 1 ≤ I 0 ( y 2 ) I 0 ( y 1 ) ≤ e − y 2 − y 1 r 2 Next, w e get a b ound for ¯ p ( t,y ) I 0 ( y ) . ¯ p ( t, y ) I 0 ( y ) = E t  Z T t e − r ( s − t ) I 0 ( ¯ Y s + θ 2 ( s − t )) I 0 ( y ) ds + e − r ( T − t ) I 0 ( ¯ Y T + θ 2 ( T − t )) I 0 ( y )  W e write E t e − 1 r 1 . | ¯ Y s + θ 2 ( s − t )) − y | ≤ E t  I 0 ( ¯ Y s + θ 2 ( s − t )) I 0 ( y )  ≤ E t e 1 r 1 . | ¯ Y s + θ 2 ( s − t )) − y | 2 e − ( r + θ 2 2 + || ρ || ) r 1 . + θ 2 2 r 2 1 )( s − t ) N ( − θ √ s − t/r 1 ) ≤ E t  I 0 ( ¯ Y s + θ 2 ( s − t )) I 0 ( y )  ≤ 2 e ( r + θ 2 2 + || ρ || r 1 . + θ 2 2 r 2 1 )( s − t ) N ( θ √ s − t/r 1 ) 1 2 e  − r + θ 2 2 + || ρ || r 1 − 3 θ 2 2 r 2 1  ( s − t ) ≤ E t  I 0 ( ¯ Y s + θ 2 ( s − t )) I 0 ( y )  ≤ 2 e  r + θ 2 2 + || ρ || r 1 + θ 2 2 r 2 1  ( s − t ) 22 T aking the integral b et ween t and T yields r 3 ( t ) and r 4 ( t ) suc h that 0 < r 3 ( t ) ≤ ¯ p ( t, y ) I 0 ( y ) ≤ r 4 ( t ) □ Pro of Prop osition 2.15 W e use the fact that | U ′ 0 ( y ) | ≤ C e c | y | for constants c, C > 0, By in tegrating, w e get that | U 0 ( y ) | is b ounded by a function of the form C e c | y | . The PDE (2.31) has a unique C 1 , 2 solution therefore, we can apply the F eynman Kac form ula to get the result. F or ¯ α , w e notice that a 2 ( t, x ) := θ 2 v 2 2 x 2 v 2 x is b ounded aw ay from zero and a 1 ( t, x ) := r − I ( v ( t,x )) x − θ 2 v xv x ( t, x ) is b ounded. W e get a PDE of the form: f t ( t, x ) + a 2 ( t, x ) x 2 f xx ( t, x ) + a 1 ( t, x ) xf x ( t, x ) = 0 W e integrate the inequality r 1 ( t ) ≤ − xv x ( t,x ) v ( t,x ) ≤ r 2 ( t ) b et w een 1 and x to get v ( t, 1) x − r 2 ( t ) ≤ v ( t, x ) ≤ v ( t, 1) x − r 1 ( t ) if x ≥ 1 v ( t, 1) x − r 1 ( t ) ≤ v ( t, x ) ≤ v ( t, 1) x − r 2 ( t ) if 0 < x < 1 The b oundary | U 0 (log v ( s, x )) | can b e b ounded b y C (1 + x c 1 + x − c 2 ) for constants C, c 1 , c 2 > 0 so the PDE (2.32) has a unique C 1 , 2 solution. By the F eynman-Kac theorem, w e get that ¯ α satisfies the PDE (2.32). □ Prop osition 4.13. L et f , g b e two functions define d r esp e ctively on [0 , T ] × R and R . We supp ose f and g ar e c ontinuous and satisfy a gr owth c ondition | f ( t, y ) | + | g ( y ) | ≤ C e c | y | for al l t ∈ [0 , T ] , y ∈ R . L et u b e a classic al solution of the he at e quation − u t ( t, y ) + u y y ( t, y ) + f ( t, y ) = 0; u (0 , y ) = g ( y ) (4.42) Then u is smo oth on (0 , T ] × R . The pro of can b e found in [6], Section 2.3, Theorem 8. Next, w e prov e that G x = v . Pro of Prop osition 2.16 Define q ( t, y ) := e y ¯ p y ( t, y ) ; L ( t, y ) = G ( t, ¯ p ( t, y )) (4.43) Since y 7→ ¯ p ( t, y ) is a bijection from R onto (0 , ∞ ), it is enough to show that L y ( t, y ) = q ( t, y ) for all t, y . W e can see that by doing an affine change of v ariables, we can reduce the PDEs defining ¯ p and ¯ δ to the heat equation with a source term. W e can then use prop osition 4.13 to establish the smo othness of ¯ p and ¯ δ . ¯ p ty , ¯ p y yy , ¯ δ ty , ¯ δ y yy are w ell defined. W e also ha ve: q t = e y ¯ p ty ; q y = e y ¯ p y + e y ¯ p y y = q + e y ¯ p y y ; q y y = − q + 2 q y + e y ¯ p y yy q satisfies the PDE: 23 q t + θ 2 2 q y y + ( ¯ ρ ( t, y ) − r − θ 2 2 ) q y − ρ ( t, y ) q ( t, y ) + e y I ′ 0 ( y ) = 0 ; q ( T , y ) = e y I ′ 0 ( y ) (4.44) Since ¯ α ( t, s, ¯ p ( t, y )) = ¯ δ ( t, s, y ) w e get L ( t, y ) = Z T t h ( t, s ) ¯ δ ( t, s, y ) ds + h ( t, T ) ¯ δ ( t, T , y ) and L y satisfies the PDE L ty + θ 2 2 L y yy + ( ¯ ρ ( t, y ) − r − θ 2 2 ) L y y − ¯ ρ ( t, y ) L y + e y I ′ 0 ( y ) = − ¯ δ y ( t, t, y ) + h ( t, T )[ ¯ δ t + θ 2 2 ¯ δ y y + ( ¯ ρ ( t, y ) − r − θ 2 2 ) ¯ δ y ( t, T , y )] + Z T t ∂ h ∂ t ( t, s ) ¯ δ y ( t, s, y ) ds + ∂ h ∂ t ( t, T ) ¯ δ y ( t, T , y ) + Z T t h ( t, s ) × [ ¯ δ t + θ 2 2 ¯ δ y y + ( ¯ ρ ( t, y ) − r − θ 2 2 ) ¯ δ y ( t, s, y )] ds = − e y I ′ 0 ( y ) + e y I ′ 0 ( y ) − ¯ ρ ( t, y ) L y ( t, y ) + Z T t ∂ h ∂ t ( t, s ) ¯ δ y ( t, s, y ) ds + ∂ h ∂ t ( t, T ) ¯ δ y ( t, T , y ) = ( F [ ¯ ρ ]( t, y ) − ¯ ρ ( t, y )) × L y ( t, y ) = 0 And since ¯ ρ ( t, y ) = F [ ¯ ρ ]( t, y ), the last quan tity is equal to zero. W e also hav e L y ( T , y ) = ¯ δ y ( T , T , y ) = U ′ 0 ( y ) = e y I ′ 0 ( y ). The functions L y and q are solutions of the same parab olic PDE and they are b ounded by C e c | y | for p ositiv e constan ts c, C that are big enough, therefore by uniqueness of the solution of the PDE: L y ( t, y ) = q ( t, y ) Since L y ( t, y ) = ¯ p y ( t, y ) G x ( t, ¯ p ( t, y )), q ( t, y ) = e y ¯ p y ( t, y ) and ¯ p ( t, y ) = p ( t, e y ), w e get: G x ( t, ¯ p ( t, y )) = e y i.e. G x ( t, p ( t, x )) = x , ∀ x > 0. And since p ( t, x ) is the x -in verse of v ( t, x ), w e conclude that G x ( t, x ) = v ( t, x ) for all x > 0. □ Pro of Theorem 2.11 The SDE d ¯ X s = [ r − ¯ c ( s, ¯ X s ) + θσ ¯ π ( s, ¯ X s )] ¯ X s ds + σ ¯ π ( s, ¯ X s ) ¯ X s dW s (4.45) has contin uously differentiable co efficients with b ounded deriv atives. T o see that, we express the deriv a- tiv es ∂ ( x ¯ c ( t,x )) ∂ x , ∂ ( x ¯ π ( t,x )) ∂ x in terms of x = ¯ p ( t, y ). ∂ ( x ¯ c ( t, x )) ∂ x = ∂ ( I ( v ( t, x )) ∂ x = v x ( t, x ) .I ′ ( v ( t, x )) = v x ( t, x ) U ′′ ( I ( v ( t, x )) Since v ( t, ¯ p ( t, y )) = e y , taking the y -deriv ative yields: ¯ p y ( t, y ) v x ( t, ¯ p ( t, y )) = e y (4.46) ∂ ( x ¯ c ( t, x )) ∂ x = e y ¯ p y ( t, y ) U ′′ ( I 0 ( y )) = U ′ ( I 0 ( y )) I 0 ( y ) ¯ p y ( t, y ) I 0 ( y ) U ′′ ( I 0 ( y )) 24 Then, w e use the fact that both U ′ ( I 0 ( y )) I 0 ( y ) U ′′ ( I 0 ( y )) and I 0 ( y ) ¯ p y ( t,y ) are bounded indep enden tly of t, y to conclude that there is L 1 > 0 suc h that | ∂ ( x ¯ c ( t, x )) ∂ x | ≤ L 1 ∀ t, x Similarly , in terms of the v ariable y , ∂ ∂ y = ∂ x ∂ y . ∂ ∂ x = ¯ p y ( t, y ) . ∂ ∂ x ∂ ( x ¯ π ( t, x )) ∂ x = − θ σ ∂ ∂ x ( v ( t, x ) v x ( t, x ) = − θ σ ¯ p y ( t, y ) ∂ ∂ y  e y ¯ p y ( t, y ) e y  = − θ ¯ p y y ( t, y ) σ ¯ p y ( t, y ) By prop osition 2.8, − θ ¯ p yy ( t,y ) σ ¯ p y ( t,y ) is uniformly b ounded. There is L 2 > 0 indep enden t of t, x suc h that: | ∂ ( x ¯ π ( t, x )) ∂ x | ≤ L 2 ∀ t, x So, b y Chapter V, Theorem 39 of [14], the SDE (4.45) has a unique classical solution. F or t ≤ s ≤ T , w e hav e: ¯ X s = x exp  Z s t  r − ¯ c + θ σ ¯ π − σ 2 ( ¯ π ) 2 ( u, ¯ X u ) 2  du + Z s t σ ¯ π ( u, ¯ X u ) dW u  (4.47) By Prop osition 2.10, ¯ π ( t, x ) , ¯ c ( t, x ) are uniformly b ounded indep enden tly of x . So ∀ s ∈ [ t, T ] : ¯ X s < ∞ P a.s. Note that: ¯ π ( t, ¯ p ( t, y )) = − θ σ ¯ p y ( t, y ) ¯ p ( t, y ) (4.48) By Ito’s lemma: d ¯ p ( s, ¯ Y s ) = [ ¯ p t + θ 2 2 ¯ p y y + ( ¯ ρ ( s, ¯ Y s ) − θ 2 2 − r ) ¯ p y ] ds − θ ¯ p y ( s, ¯ Y s ) dW s = ( r ¯ p ( s, ¯ Y s ) − I 0 ( ¯ Y s ) − θ 2 ¯ p y ( s, ¯ Y s )) ds − θ ¯ p y ( s, ¯ Y s ) dW s In the last equality , we ha ve used PDE (2.17). Thus, d ¯ p ( s, ¯ Y ( s )) = [ r − ¯ c ( s, ¯ p ( s, ¯ Y s )) + θσ ¯ π ( s, ¯ p ( s, ¯ Y s ))] ¯ p ( s, ¯ Y s ) ds + σ ¯ π ( s, ¯ p ( s, ¯ Y s )) ¯ p ( s, ¯ Y s ) dW s If we choose y = ¯ Y t suc h that ¯ p ( t, y ) = x = ¯ X t , the tw o pro cesses ¯ X s and ¯ p ( s, ¯ Y s ) satisfy the same SDE. By uniqueness of the solution, we hav e the equality almost surely: ¯ X s = ¯ p ( s, ¯ Y s ) ∀ s ∈ [ t, T ] . The relation y = log v ( t, x ) comes from the fact that p is the x in verse of v . □ Pro of Theorem 2.17 W e calculate the partial deriv ativ es of G ( t, x ) = R T t h ( t, s ) ¯ α ( t, s, x ) ds + h ( t, T ) ¯ α ( t, T , x ). G t ( t, x ) = − h ( t, t ) ¯ α ( t, t, x ) + Z T t h ( t, s ) ¯ α t ( t, s, x ) ds + h ( t, T ) ¯ α t ( t, T , x ) (4.49) + Z T t ∂ h ( t, s ) ∂ t ¯ α ( t, s, x ) ds + ∂ h ( t, T ) ∂ t ¯ α ( t, T , x ) (4.50) 25 G x ( t, x ) = Z T t h ( t, s ) ¯ α x ( t, s, x ) ds + h ( t, T ) ¯ α x ( t, T , x ) (4.51) G xx ( t, x ) = Z T t h ( t, s ) ¯ α xx ( t, s, x ) ds + h ( t, T ) ¯ α xx ( t, T , x ) (4.52) Recall that ¯ α satisfies equation (2.32): ¯ α t + θ 2 v 2 2 v 2 x ¯ α xx + ( r x − I ( v ( t, x )) − θ 2 v v x ( t, x )) ¯ α x ( t, s, x ) = 0 ; ¯ α ( s, s, x ) = U ( I ( v ( s, x ))) therefore G t ( t, x ) + θ 2 v 2 2 v 2 x G xx ( t, x ) +  r x − I ( v ( t, x )) − θ 2 v v x ( t, x )  G x ( t, x ) + U ( I ( v ( t, x )) = Z T t ∂ h ( t, s ) ∂ t ¯ α ( t, s, x ) ds + ∂ h ( t, T ) ∂ t ¯ α ( t, T , x ) Reorganizing, and using the fact that G x = v , w e get equation (2.34). W e conclude by noting that G is strictly increasing ( G x = v > 0) and strictly concav e ( G xx ( t, x ) = v x ( t, x ) = 1 p x ( t,v ( t,x )) < 0) so the sup in the extended HJB is attained at ¯ π , ¯ c . G is a solution of the extended HJB. G t ( t, x ) + sup π ,c {A π ,c G ( t, x ) + U ( xc ) } = E t  Z T t ∂ h ( t, s ) ∂ t U ( ¯ c ( s ) ¯ X ( s )) ds + ∂ h ( t, T ) ∂ t U ( ¯ X ( T ))  W e can then use the v erification theorem for the extended HJB system. The function G ( t, x ), along with the controls ¯ π , ¯ c is a solution of the extended HJB system to conclude that { ¯ π ( s, ¯ X s ) , ¯ c ( s, ¯ X s ) , ¯ X s , 0 ≤ s ≤ T } defines a subgame p erfect strategy . W e hav e th us constructed the v alue function V ( t, x ) = G ( t, x ). □ 26