Quadratic Hedging for Sequential Claims with Random Weights in Discrete Time
aa r X i v : . [ q -f i n . M F ] M a y Quadratic Hedging for Sequential Claimswith Random Weights in Discrete Time
Jun Deng ∗ Bin Zou † This Version: May 14, 2020
Abstract
We study a quadratic hedging problem for a sequence of contingent claims with random weights indiscrete time. We obtain the explicit optimal hedging strategy in a recursive representation, without imposing the nondegeneracy condition on the model and square integrability on hedging strategies. Werelate the results to hedging under random horizon and fair pricing in the quadratic sense.
Keywords : binomial model; quadratic hedging; pricing; sequential claims
AMS Subject Classifications : 91G20, 93E20
Rooted from the classical mean-variance criterion in portfolio selection, quadratic hedging (also calledvariance-optimal hedging) is an essential topic and a popular criterion in the literature. A pioneering workin this area is Schweizer (1995), in which the author seeks an optimal strategy to minimize the expectedquadratic hedging error of a contingent claim in a discrete time model. In this paper, motivated by practicalproblems in finance and insurance, we extend the work of Schweizer (1995) by considering the quadratichedging problem for a sequence of contingent claims with random weights.Let us describe a standard quadratic hedging problem briefly. An investor faces the risk exposure of acontingent claim H N , with maturity at time N , and wants to construct a portfolio strategy ξ from tradableassets to hedge the claim H N . The objective is to find an optimal strategy ξ ⋆ to the following problem: V ( c ) := min ξ ∈A E (cid:2) ( H N − c − G N ( ξ )) (cid:3) , (1)where constant c is the initial capital (or interpreted as the hedging cost), G ( ξ ) is the cumulative gainprocess under strategy ξ , and A is the set of admissible strategies. Note that ξ ⋆ = ξ ⋆ ( c ) depends on the ∗ School of Banking and Finance, University of International Business and Economics, Beijing, China. Email: [email protected] † Corresponding author. Department of Mathematics, University of Connecticut, 341 Mansfield Road U1009, Storrs 06269-1009, USA. Email: [email protected]. Phone: +1-860-486-3921. c . If the minimum hedging error V ( c ) = 0 for some c in (1), then the portfolio ( c, ξ ⋆ ( c ))replicates the claim H N , and hence c is the fair price of H N at time 0. However, in a general incompletemarket, it is most likely V ( c ) > c and one may further consider min c ∈ R V ( c ) to find the “optimal”hedging cost c ⋆ , along with the corresponding optimal strategy ξ ⋆ ( c ⋆ ). In other words, an investor, withobjective given by (1), always prefers the portfolio ( c ⋆ , ξ ⋆ ( c ⋆ )) over any admissible portfolio ( c, ξ ).Problem (1) is first studied in continuous time by Duffie and Richardson (1991), in which H N is treatedas a non-tradable asset and a hedger dynamically trades another correlated asset in a standard geometricBrownian model. Schweizer (1992) generalizes the work of Duffie and Richardson (1991) by consideringa general claim that may depend on both assets. Further extension is done in Schweizer (1996) andPham et al. (1998) under a general semimartingale framework. There is an extensive body of literature bynow on this topic, and, to save space, we refer readers to Schweizer (2001, 2010) and the references thereinfor a detailed overview of pricing and hedging under a quadratic criterion in continuous time. We mentionthat static hedging under the quadratic criterion is also a popular topic, see, e.g., Carr and Madan (1998)and Leung and Lorig (2016).In this article, we extend the classic quadratic hedging Problem (1) to a sequence of contingent claims H = ( H n ) n =0 , , ··· ,N with random weights ω = ( ω n ) n =0 , , ··· ,N , where both H and ω are adapted to a givenfiltration and ω n ∈ [0 ,
1] for all n . Precisely, we solve the following problem: V ( c ) := min ξ ∈A J ( ξ ; c ) := min ξ ∈A N X n =0 E h ω n (cid:0) H n − c − G n ( ξ ) (cid:1) i , (2)where c ∈ R is the initial capital (hedging cost) and A is the set of admissible strategies. It is clear thatthe standard Problem (1) is a special case of our Problem (2), by simply taking ω = · · · = ω N − = 0and ω N = 1. The extended problem arises naturally from practical finance and insurance concerns,such as hedging under random horizon and pricing path-dependent contingent claims. More economicinterpretations are discussed in Subsection 2.1.Solving Problem (1) in a discrete time model dates back to Sch¨al (1994) and Schweizer (1995). In Sch¨al(1994), the optimal hedging strategy is obtained under the assumption that E t [∆ S ] / p V t [∆ S ] is bounded,where E t [∆ S ] (resp. V t [∆ S ]) denotes the conditional mean (resp. variance) of the price changes of therisky asset S , which is equivalent to the so-called nondegeneracy (ND) condition in Schweizer (1995). Notethat the trading strategies in Sch¨al (1994) are not necessarily self-financing, but only mean-self-financing(see Eq.(3.6) there). Schweizer (1995) presents a more complete and general analysis of Problem (1) underthe ND condition, and obtains the optimal strategy in a recursive form. In both Sch¨al (1994) and Schweizer(1995) (and many works in continuous time), the existence of a solution to Problem (1) is obtained using theHilbert projection theorem, which requires the subspace of { G N ( ξ ) : ξ ∈ A} to be closed and in turn needsthe ND condition; while on the other hand, finding the optimal strategy is based on the KunitaWatanabedecomposition. To overcome the restriction of the ND condition, Melnikov and Nechaev (1999) study theconditional version of Problem (1), replacing E [ · ] by E [ ·|F ], where F is the sigma field at time 0 and2ay be non-trivial. ˇCern`y (2004) applies dynamic programming to study Problem (1). ˇCern`y and Kallsen(2009) utilizes the sequential regression method to derive the optimal strategy.Our paper contributes to the literature in three ways. First, Problem (2) is general enough to includingthe standard Problem (1), and, to the best of our knowledge, has not been studied before. Second, we obtainthe optimal hedging strategy ξ ⋆ , the value function V ( c ), and the minimum hedging cost c ⋆ = arg min V ( c )in closed forms. We do not impose the ND condition on the price process S or the square integrablecondition on the hedging strategies ξ . Third, we also consider a special quadratic hedging problem underrandom horizon τ , even the stopped market S τ may admit arbitrage opportunities.In the remaining of the paper, we formulate the problem in Section 2, and present the main results inSection 3. Two examples are given in Section 5. Technical proofs are placed in Section 6 and Appendix. We consider a discrete time market model with N periods. To simplify notations, we introduce two indexsets of time by T = { , , , · · · , N } and T + = { , , · · · , N } . Let us fix a filtered probability space(Ω , F , F = ( F n ) n ∈T , P ), which supports all the random objects considered in the paper. We consider arepresentative investor who can trade a risk-free asset (bond) and a risky asset (stock) in this market. Forconvenience, we normalize the risk-free asset and set the interest rate to be zero. The price process of therisky asset is given by an F -adapted and square-integrable process S := ( S n ) n ∈T . Introduce L ( P ) as the setof all square-integrable random variables under P . We denote the price increment ∆ S by ∆ S n = S n − S n − ,for all n ∈ T + . Denote P ( F ) the set of all F -predictable processes, i.e., if ψ = ( ψ n ) n ∈T + ∈ P ( F ), we have ψ n ∈ F n − . We set ψ = 0 for any predictable process ψ unless stated otherwise.A hedging strategy is a predictable process ξ = ( ξ n ) n ∈T + ∈ P ( F ), where ξ n is the number of shares inthe risky asset held by the investor from time ( n −
1) to time n . The cumulative gain process of strategy ξ is denoted by G ( ξ ) = ( G n ( ξ )) n ∈T . A strategy ξ is called self-financing if G n ( ξ ) = n X i =1 ξ i ∆ S i , n ∈ T + , and G ( ξ ) = 0 . (3)A strategy ξ is called admissible if it is predictable and self-financing. Denote the admissible set by A .In our setup, the risk exposure the investor faces is modeled by a sequence of contingent claims H withrandom weights ω . Denote such a sequential risk by F -adapted processes H = ( H n ) n ∈T and ω = ( ω n ) n ∈T ,where claim H n ∈ L ( P ) and weight ω n ∈ [0 ,
1] for all n . We also call ( H, ω ) a (contingent) claim.
Problem 2.1.
The investor, with initial capital c , aims to solve the quadratic hedging problem for asequence of randomly weighted claim ( H, ω ) formulated in Problem (2) , i.e., V ( c ) := min ξ ∈A J ( ξ ; c ) := min ξ ∈A N X n =0 E h ω n (cid:0) H n − c − G n ( ξ ) (cid:1) i , e call a solution ξ ⋆ = ξ ⋆ ( c ) to Problem (2) an optimal hedging strategy, and V ( c ) the value function orthe minimum hedging error. Remark 2.2.
We do not impose the ND condition on S or square-integrability on ξ , which are requiredin almost all existing works (see, e.g., Sch¨al (1994); Schweizer (1995); ˇCern`y and Kallsen (2009)). Oneexception is Melnikov and Nechaev (1999), where the problem is instead assumed to be well posed. Weshow, without these conditions, that V ( c ) < ∞ for all c ∈ R , and hence Problem (2) is well posed. First, Problem 2.1 can be linked to quadratic hedging problems under random horizon. To wit, let τ denote a random time, a positive F -measurable random variable taking values in T . Consider a quadratichedging problem with random horizon τ as follows:min ξ ∈A E ( H τ − c − G τ ( ξ )) , H n ∈ L ( P ) for all n ∈ T . (4)It is easy to see that Problem (4) is equivalent to Problem (2) given ω n = P ( τ = n |F n ) for all n ∈ T . Butsuch an equivalence fails in general. Especially, if we interpret τ as an ( F n )-stopping time, it is related toAmerican option pricing. At first glance, Problem (4) may not have a solution since the stopped market S τ may admit arbitrage opportunities (see, e.g., Aksamit et al. (2017, 2018); Choulli and Deng (2017)).In Example 5.2, we solve Problem 4 when the stopped market S τ does have arbitrage.Second, many practical problems in insurance can be formulated in the form of Problem (4). Forexample, we may interpret H τ as the payment of a life insurance contract, liquidated at the random deathtime τ , and consider an insurer who trades longevity bond to hedge such a risk in discrete time.Third, the problem also arises from tracking a benchmark index ( H n ) by trading available assets andevaluate the tracking performance using the quadratic criterion on a regular basis (say weekly) over afixed period (say one year). It is also related to optimal execution under the market-on-close benchmark(see, e.g., Frei and Westray (2018)). Finally, Problem 2.1 can serve as an upper bound or estimate of thepricing of many exotic options, such as Bermuda and Asian options. For instance, if we treat ( H n ) as theunderling asset ( S n ) and consider an average Asian option, we have E (cid:18) S + S + · · · + S N N − c − G N ( ξ ) (cid:19) ≤ N N X n =1 E h(cid:0) S n − c − b G n ( ξ ) (cid:1) i , with b G n ( ξ ) := N ∆ G n ( ξ ) . In this section, we present the main results of this paper, a closed-form solution to Problem (2), in Theorem3.4. We first derive a sufficient optimality condition of Problem (2) in the following.4 roposition 3.1.
An admissible hedging strategy ξ ⋆ is optimal to Problem (2) if it satisfies E " N X n = i ω n ( H n − c − G n ( ξ ⋆ )) ! ∆ S i (cid:12)(cid:12)(cid:12) F i − = 0 , for all i ∈ T + . (5) Proof.
Let ξ ∈ A be an arbitrary admissible strategy, and ξ ⋆ ∈ A satisfying condition (5). Noting ξ + ξ ⋆ ∈ A ,and the linearity of G by (3), we obtain J ( ξ ⋆ + ξ ; c ) = J ( ξ ⋆ ; c ) + N X n =0 E (cid:2) ω n G n ( ξ ) (cid:3) − N X i =1 E " ξ i E " N X n = i ω n ( H n − c − G n ( ξ ⋆ )) ! ∆ S i (cid:12)(cid:12)(cid:12) F i − , which leads to the desired result. Remark 3.2.
The optimality condition (5) is equivalent to the one obtained under the Hilbert projectiontheorem, see Eq.(2.15) in Schweizer (1995). But here we do not need the closedness condition of thesubspace { G N ( ξ ) } . To facilitate the presentation of the main results, we define the following predictable processes by β n := α n δ n and ρ n := η n δ n , for all n ∈ T + , (6)where α = ( α n ) n ∈T + , η = ( η n ) n ∈T + , and δ = ( δ n ) n ∈T + are given by α n := E (cid:20) ∆ S n (cid:20) N X i = n ω i i Y j = n +1 (1 − β j ∆ S j ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) F n − (cid:21) , (7) η n := E (cid:20) ∆ S n (cid:20) N X i = n H i ω i i Y j = n +1 (1 − β j ∆ S j ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) F n − (cid:21) , (8) δ n := E (cid:20) ∆ S n (cid:20) N X i = n ω i i Y j = n +1 (1 − β j ∆ S j ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) F n − (cid:21) . (9)As convention, we set a sum over an empty set to zero, a product over an empty set to one, and 0 / n = N , that α N = E [ ω N ∆ S N |F N − ] , η N = E [ ω N H N ∆ S N |F N − ] , δ N = E [ ω N ∆ S N |F N − ] , β N = α N δ N , ρ N = η N δ N , and use induction to complete the definitions for all n = N − , N − , · · · , Remark 3.3.
We show these processes are well defined by later using Proposition 6.1 and Corollary 6.2.
Theorem 3.4.
For any fixed initial capital c , we define ξ ⋆ ( c ) := ( ξ ⋆n ( c )) n ∈T + by ξ ⋆n ( c ) := ρ n − β n (cid:0) c + G n − ( ξ ⋆ ( c )) (cid:1) , (10) where ρ = ( ρ n ) n ∈T + and β = ( β n ) n ∈T + are given by (6) . The following results hold true: V ( c ) = J ( ξ ⋆ ; c ) < ∞ for all c ∈ R , and the strategy ξ ⋆ ( c ) defined in (10) solves Problem (2) . (b) The value function of Problem (2) is given by V ( c ) = c N X n =0 E ( Z n ) − c N X n =0 E ( Z n H n ) + N X n =0 E ω n H n − n X i =1 ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) , (11) where Z = ( Z n ) n ∈T is defined by Z n := ω n n Y i =1 (1 − β i ∆ S i ) . (12)(c) Fix an arbitrary but fixed non-negative integer n in T , i.e., n = 0 , , · · · , N . We have for all k = 0 , , · · · , n + 1 that (setting G − = 0 ) H n − c − G n ( ξ ⋆ ( c )) = H n − n X i = k ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) − (cid:0) c + G k − ( ξ ⋆ ( c )) (cid:1) n Y i = k (1 − β i ∆ S i ) . (13) Corollary 3.5.
If the random weights process ω = ( ω n ) n ∈T degenerates into a sequence of constants and S is an F -martingale, then we have ξ ⋆n ( c ) = ρ n = P Ni = n ω i E h H i ∆ S n (cid:12)(cid:12)(cid:12) F n − iP Ni = n ω i E h ∆ S n (cid:12)(cid:12)(cid:12) F n − i , n ∈ T + , which is independent of the initial capital c . In this section, we explore the connection between hedging and pricing of a sequence of contingent claimsin the quadratic sense. Key results are presented in Theorem 4.1. Following the setup in Problem (2), weformulate the quadratic pricing of the claim (
H, ω ) as follows: V ⋆ = min c ∈ R V ( c ) = min c ∈ R + min ξ ∈A N X n =0 E (cid:2) ω n ( H n − c − G n ( ξ )) (cid:3) . (14)The financial interpretation of Problem (14) is that one chooses an initial capital c , along with a self-financing strategy ξ , to minimize the quadratic hedging error of the contingent claim ( H, ω ). Theorem 4.1.
Problem (14) has an optimal solution ( c ⋆ , ξ ⋆ ( c ⋆ )) given by c ⋆ = P Nn =0 E ( Z n H n ) P Nn =0 E ( Z n ) and ξ ⋆ ( c ⋆ ) by (10) , (15) where Z = ( Z n ) n ∈T is defined in (12) . roof. We have P Nn =0 E ( Z n ) ≥ V ⋆ = J ( ξ ⋆ ( c ⋆ ); c ⋆ ) ≤ J ( ξ ⋆ ( c ); c ) ≤ J ( ξ ; c ) for all c ∈ R and ξ ∈ A .This ends the proof.Similar to Problem (4), we can reformulate the quadratic pricing problem in (14) under a randomhorizon τ , and apply Theorem 4.1 to obtain the solution to such a problem. Corollary 4.2.
Let τ be a random time and ( H, ω ) be a contingent claim, with ω n = P ( τ = n |F n ) . Theminimum capital c ⋆ , given in (15) , solves the following pricing problem under random horizon: min c ∈ R min ξ ∈A E h(cid:0) H τ − c − G τ ( ξ ) (cid:1) i . Furthermore, if τ is independent of S and S is an F -martingale, the minimum capital c ⋆ is equal to c ⋆ = P Nn =1 E ( H n ) · P ( τ = n ) . Remark 4.3.
Denote e Z = ( e Z n ) n ∈T , where e Z n = Z n / P Ni =0 E [ Z i ] . Then c ⋆ given in (15) can be rewritten as c ⋆ = P Nn =0 E [ e Z n H n ] . Therefore, e Z can be seen as a “fair” pricing measure for the contingent claim ( H, ω ) .Furthermore, if τ degenerates to a constant N , Corollary 4.2 is reduced to Corollary 3.2 in Schweizer (1995)and e Z is reduced to a signed probability measure absolutely continuous with respect to P . Throughout this section, we consider a two-period binomial model (Ω , F , ( F n ) n =0 , , , P ) specified as follows: • Ω = { x , x , x , x } , F = {∅ , Ω } , F = σ ( { x , x } , { x , x } ), and F = F = 2 Ω . • P ( x ) = p , P ( x ) = P ( x ) = pq , and P ( x ) = q , where 0 < p < q = 1 − p . • The stock price S evolves by: at time 0, S = 1; at time 1, S ( { x , x } ) = u and S ( { x , x } ) = d ; attime 2, S ( { x } ) = u , S ( { x } ) = S ( { x } ) = ud , and S ( { x } ) = d , where 0 < d < < u . In this example, we consider a contingent claim (
H, ω ) given by H = 0 , H = a { x ,x } , H = b { x } + b { x } ; ω = 0 , ω = 1 , ω = { x } , (16)where a , b and b are constants and denotes an indicator function.Using (6)-(9), we first compute their values at time 2: α = E [ ω ∆ S |F ] = u ( d − q { x ,x } , η = E [ ω H ∆ S |F ] = b u ( d − q { x ,x } , and δ = E [ ω ∆ S |F ] = u ( d − q { x ,x } , which imply β = α δ = { x ,x } u ( d −
1) and ρ = η δ = b { x ,x } u ( d − . (17)7t time 1, notice that ω (1 − β ∆ S ) ≡
0. We then obtain α = E [ ω ∆ S ] = ( u − p + ( d − q , η = E [ ω H ∆ S ] = ( d − a q , and δ = ( u − p + ( d − q , leading to β = α δ = ( u − p + ( d − q ( u − p + ( d − q and ρ = η δ = a ( d − q ( u − p + ( d − q . (18)By (12), we get Z = Z = 0 and Z = 1 − β ∆ S . We then have: Corollary 5.1.
Let ( H, ω ) be given by (16) . For any initial capital c , the optimal quadratic hedging strategy ξ ⋆ = ξ ⋆ ( c ) to Problem (2) is given by ξ ⋆ = ρ − cβ and ξ ⋆ = ρ − β ( c + ξ ( u − , where β i and ρ i , i = 1 , , are defined in (17) and (18) . The minimum capital c ⋆ to Problem (14) is givenby c ⋆ = a ( u − d − u + d − . In the second example, we study an quadratic hedging problem under random horizon as formulated in(4). We set up the contingent claim H by H = a , H = a { x ,x } + a { x ,x } , H = b { x } + b { x } + b { x } + b { x } , (19)where all the a i and b i ’s are constants. The random time τ is defined by τ = 0 · { x ,x } + 1 · { x } + 2 · { x } . (20)To use the results from Theorem 3.4, we require ω n = P ( τ = n |F n ) for all n = 0 , ,
2, which yields ω = p , ω = p { x ,x } , and ω = { x } .Similar to the previous example, we carry out calculations by (6)-(9) and obtain α = d ( d − q { x ,x } , η = d ( d − b q { x ,x } , and δ = d ( d − q { x ,x } , implying β = { x ,x } d ( d − and ρ = b { x ,x } d ( d − . At time1, we compute α = ( d − pq , η = ( d − a pq , δ = ( d − pq , leading to β = d − and ρ = a d − .The sequence ( Z n ) n =0 , , , defined in (12), reads in this example as Z = p and Z = Z = 0. In turn,we get P n =0 E [ Z n ] = p and P Nn =0 E [ Z n H n ] = a p . An application of Theorems 3.4 and 4.1 yields: Corollary 5.2.
Let H and τ be defined by (19) and (20) , and ω n = P ( τ = n |F n ) for n = 0 , , . For anyinitial capital c , the optimal hedging strategy ξ ⋆ = ξ ⋆ ( c ) to Problem (2) is given by ξ ⋆ ( c ) = a − cd − and ξ ⋆ ( c ) = b − a d ( d − { x ,x } . (21) The minimum capital c ⋆ to Problem (14) is c ⋆ = a . Proposition 5.3.
Let the assumptions in Corollary 5.2 hold. We have:
The optimal strategy ξ ⋆ ( c ⋆ ) with initial capital c ⋆ = a , where ξ ⋆ is given by (21) , replicates thecontingent claim H = ( H , H , H ) on Ω , { x , x } , and { x } , respectively. (b) The stopped market S τ admits arbitrage.Proof. Assertion (a) can be verified by using (21) from Corollary 5.2. To show Assertion (b), take anadmissible strategy φ = ( φ , φ ) with φ = φ = −
1. Then, by (3), we obtain G = φ ∆ S τ = (1 − d ) { x ,x } ≥ ,G = φ ∆ S τ + φ ∆ S τ = (1 − d ) { x ,x } + d (1 − d ) { x } ≥ , and P ( G >
0) = q >
0. Hence, φ is an arbitrage strategy, which proves Assertion (b). In this section, we provide the proof to Theorem 3.4. To that end, we first present several preliminaryresults. We define processes A = ( A n ) n ∈T , B = ( B n ) n ∈T , C = ( C n ) n ∈T , and D = ( D n ) n ∈T by A n := N X i = n ω i i Y j = n +1 (1 − β j ∆ S j ) , B n := A n ∆ S n , C n := β n B n , D n := N X i = n ω i i Y j = n +1 (1 − β j ∆ S j ) , (22)where β is defined in (6). By (22) and the definition of α = ( α n ) n ∈T + in (7), we easily deduce that A N = ω N and E [ A n |F n ] = E [ A n +1 |F n ] + ω n − α n +1 β n +1 , n = 0 , , · · · , N − . (23) Lemma 6.1.
Let processes A , B , C , and D be defined by (22) . We have: (1) A , B , and C are squareintegrable; and (2) E [ A n |F n ] = E [ D n |F n ] ≤ N − n + 1 , ∀ n ∈ T . (24)Assertion (2) in Lemma 6.1 can be shown by backward induction, while Assertion (1) is proved byusing the Cauchy-Schwarz inequality, ω n ∈ [0 ,
1] and (24). Please see Appendix A for the complete proof.The following is an immediate application of Lemma 6.1.
Corollary 6.2.
The processes β , ρ , α , η , and δ given in (6) - (9) are well defined. Lemma 6.3.
We have, for all n ∈ T + , that: E ∆ S n N X i = n ω i i X j = n +1 ρ j ∆ S j i Y k = j +1 (1 − β k ∆ S k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n − = E ∆ S n N X i = n ω i H i i X j = n +1 β j ∆ S j i Y k = j +1 (1 − β k ∆ S k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n − . β n η n = α n ρ n from (6), see Appendix B for details. We are now readyto show the three assertions in Theorem 3.4 and complete this task in four parts. Proof to Theorem 3.4.
Part 1:
We first show (13) in Assertion (c) holds by backward induction. We fix aninteger n ∈ T . When k = n + 1, (13) is trivial. Next suppose (13) holds for all k = n + 1 , n, n − , · · · , l + 1,our goal is to verify that (13) also holds for k = l . To that end, we obtain (denoting ξ ⋆ = ξ ⋆ ( c )) H n − c − G n ( ξ ⋆ ) = H n − n X i = l +1 ρ i ∆ S i n Y j = i +1 (1 − β i ∆ S i ) − (cid:0) c + G l ( ξ ⋆ ) (cid:1) n Y i = l +1 (1 − β i ∆ S i ) (25)= H n − n X i = l +1 ρ i ∆ S i n Y j = i +1 (1 − β i ∆ S i ) − (cid:0) c + G l − ( ξ ⋆ ) (cid:1) n Y i = l +1 (1 − β i ∆ S i ) − (cid:0) ρ l − β l (cid:0) c + G l − ( ξ ⋆ ) (cid:1)(cid:1) ∆ S l n Y i = l +1 (1 − β i ∆ S i ) (26)= H n − n X i = l +1 ρ i ∆ S i n Y j = i +1 (1 − β i ∆ S i ) − ρ l ∆ S l n Y i = l +1 (1 − β i ∆ S i ) − (cid:0) c + G l − ( ξ ⋆ ) (cid:1) (1 − β l ∆ S l ) n Y i = l +1 (1 − β i ∆ S i )= H n − n X i = l ρ i ∆ S i n Y j = i +1 (1 − β i ∆ S i ) − (cid:0) c + G l − ( ξ ⋆ ) (cid:1) n Y i = l (1 − β i ∆ S i ) , which arrives at the wanted result for k = l . To derive (25), we use the assumption that (13) holds for k = l + 1. To derive (26), we use G l ( ξ ⋆ ) = G l − ( ξ ⋆ ) + ξ ⋆l ∆ S l from (3) and the expression of ξ ⋆ from (10).The last two equalities are due to straightforward calculations (e.g., distribute the last term in (26) andcollect like terms).Assertion (c) is now proved. Part 2:
We show Problem (2) is well posed. That is, we prove V ( c ) = J ( ξ ⋆ ; c ) < ∞ for any c ∈ R , whichis done by checking E [ ω n ( H n − c − G n ( ξ ⋆ )) ] < ∞ for all n ∈ T . To achieve this purpose, we obtain E (cid:16) ω n ( H n − c − G n ( ξ ⋆ )) (cid:17) = E (cid:16) ω n (cid:16) H n − n X i =0 ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) − c n Y i =0 (1 − β i ∆ S i ) (cid:17) (cid:17) (take k = 0 in (13)) ≤ E (cid:16) ω n H n + ω n (cid:16) n X i =0 ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) (cid:17) + ω n c n Y i =0 (1 − β i ∆ S i ) (cid:17) (Cauchy-Schwatz) ≤ E (cid:0) H n (cid:1) + 3 E (cid:16) ω n (cid:16) n X i =0 ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) (cid:17) (cid:17) + 3 c E ( D ) ( ω n ∈ [0 ,
1] and (22))10 E (cid:0) H n (cid:1) + 3( n + 1) n X i =0 E (cid:16) ρ i ∆ S i N X j = i ω j j Y l = i +1 (1 − β l ∆ S l ) (cid:17) + 3 c ( N + 1) (take E for D in (24))= 3 E (cid:0) H n (cid:1) + 3 c ( N + 1) + 3( n + 1) n X i =0 E (cid:18) η i δ i (cid:19) (use (6) and (9)) ≤ E (cid:0) H n (cid:1) + 3 c ( N + 1)+ 3( n + 1) n X i =0 E " δ i E " N X j = i H j (cid:12)(cid:12)(cid:12) F i − · E " ∆ S i N X j = i ω j j Y k = i +1 (1 − β k ∆ S k ) (cid:12)(cid:12)(cid:12) F i − = δ i ((8) and H¨older)= 3 E (cid:0) H n (cid:1) + 3 c ( N + 1) + 3( n + 1) n X i =0 N X j = i E [ H j ] < + ∞ . In particular, we obtain V ( c ) = J ( ξ ⋆ ; c ) < ∞ without imposing the ND condition and ξ ⋆ ∈ L ( P ). Part 3:
We show ξ ⋆ = ξ ⋆ ( c ) given by (10) satisfies the sufficient condition (5) in Proposition 3.1, andhence is optimal to Problem (2). The proof below is based on backward induction.When n = N , by using (3), we have E (cid:2) ω N ( H N − c − G N ( ξ ⋆ )) · ∆ S N (cid:12)(cid:12) F N − (cid:3) = E (cid:2) ω N H N ∆ S N (cid:12)(cid:12) F N − (cid:3) − ξ ⋆N E (cid:2) ω N ∆ S N (cid:12)(cid:12) F N − (cid:3) − ( c + G N − ( ξ ⋆ )) E (cid:2) ω N ∆ S N (cid:12)(cid:12) F N − (cid:3) = η N − δ N ξ ⋆N − α N ( c + G N − ( ξ ⋆ )) , which vanishes with ξ ⋆N = ρ N − β N ( c + G N − ( ξ ⋆ )), where ρ N = η N /δ N and β N = α N /δ N by (6).Next suppose the desired statement is true for n = N, N − , · · · , k + 1. We aim to prove the samestatement holds for n = k as well. We first recall a useful identity (which can be proven by induction) l Y i = k +1 (1 − a i ) = 1 − l X i = k +1 a i l Y j = i +1 (1 − a j ) , (27)where k and l are fixed integers, and a = ( a n ) is any sequence. We then obtain E " N X n = k ω n ( H n − c − G n ( ξ ⋆ )) ! · ∆ S k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F k − = E N X n = k ω n H n − n X i = k +1 ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) ∆ S k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F k − − E N X n = k ω n ( c + G k ( ξ ⋆ )) n Y j = k +1 (1 − β j ∆ S j ) ∆ S k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F k − (by (13))= E " ∆ S k N X n = k ω n H n ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F k − − E " ∆ S k N X n = k ω n (cid:16) n X i = k +1 ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F k − ( c + G k − ( ξ ⋆ )) E " N X n = k ω n n Y j = k +1 (1 − β j ∆ S j ) ! ∆ S k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F k − − ξ ⋆k E " N X n = k ω n n Y j = k +1 (1 − β j ∆ S j ) ! ∆ S k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F k − (by (3))= E " ∆ S k N X n = k ω n H n ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F k − − α k ( c + G k − ( ξ ⋆ )) − δ k ξ ⋆k (by (7) , (9) , (23)) − E " ∆ S k N X n = k ω n H n n X i = k +1 β i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) !!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F k − (by Lemma 6 . E " ∆ S k " N X n = k ω n H n n Y i = k +1 (1 − β i ∆ S i ) F k − − α k ( c + G k − ( ξ ⋆ )) − δ k ξ ⋆k (by (27))= η k − α k ( c + G k − ( ξ ⋆ )) − δ k ξ ⋆k = 0 , (by (8) and (10))which confirms the induction indeed holds for n = k .By definition (10), ξ ⋆ is predictable and self-financing, and hence solves Problem (2). Part 4:
We show that the value function V ( c ) is given by (11). Taking k = 1 in (13) for all n ∈ T , we get N X n =0 E (cid:0) ω n ( H n − c − G n ( ξ ⋆ )) (cid:1) = N X n =0 E ω n H n − n X i =1 ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) − c n Y i =1 (1 − β i ∆ S i ) ! ! = c E N X n =0 ω n n Y i =1 (1 − β i ∆ S i ) ! − c N X n =0 E ω n H n n Y i =1 (1 − β i ∆ S i ) ! + N X n =0 E ω n H n − n X i =1 ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) ! ! + 2 c · CT , (28)where the Cross-Term CT := P Nn =0 E (cid:16) ω n (cid:16)P ni =1 ρ i ∆ S i Q nj = i +1 (1 − β j ∆ S j ) (cid:17) Q nk =1 (1 − β k ∆ S k ) (cid:17) .Using (23), we obtain that E N X n =0 ω n n Y i =1 (1 − β i ∆ S i ) ! = E [ D ] = E [ A ] = N X n =0 E [ Z n ] ≥ , (29)where Z n is defined in (12). Also by (12), the second term in (28) becomes 2 c P Nn =0 E ( H n Z n ). By comparingwith (11), we see that Assertion (b) is proved if CT = 0, which is done in the sequel: CT = E " N X i =1 i Y k =1 (1 − β k ∆ S k ) E " N X n = i ω n ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) · (1 − β i ∆ S i ) n Y k = j +1 (1 − β k ∆ S k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F i − = E " N X i =1 i Y k =1 (1 − β k ∆ S k ) E " N X n = i ω n ρ i ∆ S i n Y j = i +1 (1 − β j ∆ S j ) (1 − β i ∆ S i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F i − E " N X i =1 i Y k =1 (1 − β k ∆ S k ) ρ i E " ∆ S i N X n = i ω n n Y j = i +1 (1 − β j ∆ S j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F i − = α i by (7) and (23) − E " N X i =1 i Y k =1 (1 − β k ∆ S k ) ρ i β i E " ∆ S i N X n = i ω n n Y j = i +1 (1 − β j ∆ S j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F i − = δ i by (9) = E " N X i =1 i Y k =1 (1 − β k ∆ S k ) · ρ i ( α i − β i δ i ) | {z } =0 by (6) = 0 , where in the first line we have used Q nk =1 = Q ik =1 Q ik = i Q nk = j +1 to simplify the computations on condi-tional expectation.The proof to the main theorem, Theorem 3.4, is now complete. Acknowledgments.
The research of Jun Deng is supported by the National Natural Science Foundationof China (11501105) and the Fundamental Research Funds for the Central Universities in UIBE (19YB10).
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Proof to Lemma 6.1
Proof.
We first prove Assertion (2) by induction. When n = N , we get A N = D N = ω N ∈ [0 , N, N − , · · · , n + 1, where n < N . We need to show that (24) is alsotrue for n . To such a purpose, we compute E [ D n |F n ] = E (cid:2) ω n + N X i = n +1 ω i i X j = n +1 (1 − β j ∆ S j ) (cid:12)(cid:12) F n (cid:3) (by (22))= E (cid:2) ω n + (1 − β n +1 ∆ S n +1 ) · E [ D n +1 |F n +1 ] (cid:12)(cid:12) F n (cid:3) (by tower rule)= ω n + E (cid:2) (1 − β n +1 ∆ S n +1 ) E [ D n +1 |F n +1 ] | {z } = E [ A n +1 |F n +1 ] (cid:12)(cid:12) F n (cid:3) (by assumption)+ β n +1 E (cid:2) ∆ S n +1 D n +1 |F n +1 (cid:3)| {z } = δ n +1 (by (9))= ω n + E [ A n +1 |F n ] − β n +1 α n +1 + β n +1 δ n +1 (by (7)-(8))= ω n + E [ A n +1 |F n ] − β n +1 α n +1 (by (6))= E [ A n |F n ] . (by (23))Recall α n +1 β n +1 = β n +1 δ n +1 and δ n +1 ≥
0, and E [ A n +1 |F n +1 ] ≤ N − n by assumption, we then have E [ A n |F n ] ≤ ω n + E (cid:2) E [ A n +1 |F n +1 ] (cid:12)(cid:12) F n (cid:3) ≤ N − n + 1 , which, together with the above results, confirms (24) holds for all n ∈ T .Our next objective is to show Assertion (1). To that end, we deduce E (cid:2) A n |F n (cid:3) = E N X i = n ω i i Y j = n +1 (1 − β j ∆ S j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n (by (22)) ≤ ( N − n + 1) E N X i = n ω i i Y j = n +1 (1 − β j ∆ S j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n (By Cauchy-Schwarz and ω i ∈ [0 , N − n + 1) E [ D n |F n ] (by (22)) ≤ ( N − n + 1) , (by (24))which readily shows A n ∈ L ( P ) for all n ∈ T . Using this result, we immediately obtain the squareintegrability of B by E [ B n ] = E (cid:2) A n ∆ S n (cid:3) = E (cid:2) ∆ S n E [ A n |F n ] (cid:3) ≤ ( N − n + 1) E (cid:2) ∆ S n (cid:3) < ∞ , ∀ n ∈ T , where we have used the fact that S ∈ L ( P ). Lastly, to see C is also square integrable, we obtain E (cid:2) C n (cid:3) = E (cid:2) β n ∆ S n A n (cid:3) = E (cid:2) β n ∆ S n E [ A n |F n ] E [ D n |F n ] (cid:3) (By equality in (24))15 ( N − n + 1) E (cid:2) β n δ n (cid:3) = ( N − n + 1) E (cid:20) α n δ n (cid:21) (By (24) and (9)) ≤ ( N − n + 1) . (By Cauchy-Schwarz and ω i ∈ [0 , B Proof to Lemma 6.3
Proof.
By definition (6), we readily see β n η n = α n ρ n , which reads as E β n ∆ S n N X i = n ω i H i i Y j = n +1 (1 − β j ∆ S j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n − = E ρ n ∆ S n N X i = n ω i i Y j = n +1 (1 − β j ∆ S j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n − , Using the above result, we derivel.h.s. = E ∆ S n N X j = n +1 ρ j ∆ S j N X i = j ω i i Y k = j +1 (1 − β k ∆ S k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n − = E ∆ S n N X j = n +1 β j ∆ S j N X i = j ω i H i i Y k = j +1 (1 − β k ∆ S k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n −