The relations of Choquet Integral and G-Expectation
aa r X i v : . [ q -f i n . M F ] F e b The relations of Choquet Integral and G-Expectation
Ju Hong Kim
Department of Mathematics, Sungshin Women’s University, Seoul 02844, Republic of Korea
Abstract
In incomplete financial markets, there exists a set of equivalent martingale mea-sures (or risk-neutral probabilities) in an arbitrage-free pricing of the contingentclaims. Minimax expectation is closely related to the g -expectation which is thesolution of a certain stochastic differential equation. We show that Choquet ex-pectation and minimax expectation are equal in pricing European type options,whose payoff is a monotone function of the terminal stock price S T . Keywords:
Choquet expectation, G -expectation, Minimax expectation,Submodular Capacity, Comonotonicity
1. Introduction
Nonlinear expectations such as Choquet expectation, minimax expectationand g -expectation are applied to many areas like statistics, economics and fi-nance. Choquet expectation [3] has a difficulty in defining a conditional expec-tation. Wang [11] introduces the concept of conditional Choquet expectationwhich is the conditional expectation with respect to a submodular capacity.Choquet expectation [1, 5] is equivalent to the convex(or coherent) risk mea-sure if given capacity is submodular. G -expectation (see papers[4, 6, 7, 8, 9, 10,13] for the related topics) is the solution of the following nonlinear backwardstochastic differential equation(BSDE), y t = ξ + Z Tt g ( s, y s , z s ) ds − Z Tt z s dB s , ≤ t ≤ T. (1.1) G -expectation very much depends on the generator g in the BSDE (1.1). If g is sublinear with respect to z , then g -expectation is represented as y = sup Q ∈P E Q [ ξ ] ∀ ξ ∈ L (Ω , F T , P )where y t is the solution of the BSDE (1.1), E Q represents the expectation withrespect to Q and P is a set of risk-neutral probability measures. Minimax ex-pectation [12] is the expectation taken supremum or infimum over a set of prob-ability measures. Minimax expectation is very much related to g -expectation. Preprint submitted to Journal of mathematical analysis and applications February 23, 2021 n this paper, we will show that Choquet and minimax expectations areequal in pricing European type options, whose payoff is a monotone function ofthe terminal stock price S T . First, it is shown that the Choquet and minimaxexpectations are equal on the space of real-valued, bounded, F T -measurablefunctions, B (Ω , F T , P ). Second, the function space of B (Ω , F T , P ) is extendedto a monotone subset of L (Ω , F T , P ). G -expectation and Choquet expectation In this section, we define the upper and the lower Choquet expectations, andalso find the specific solution of the BSDE (1.1) when the generator g is sublinearwith respect to z . Minimax pricing rules are closely related to g -expectation,the solution of the BSDE (1.1).Let (Ω , F , P ) be a given completed probability space. Let (Ω , ( F t ) t ∈ [0 ,T ] , P )be the given filtered probability space. The filtration F t = σ { B s : s ≤ t } isgenerated by ( B t ) t ∈ [0 ,T ] , a one-dimensional standard Brownian motion. Let B (Ω , F T , P ) be the space of real-valued, bounded, F T -measurable functions,and let V : B (Ω , F T , P ) → R be a functional. Definition 2.1.
A set function c : F T → [0 ,
1] is called monotone if c ( A ) ≤ c ( B ) for A ⊂ B, and A, B ∈ F T and normalized if c ( ∅ ) = 0 and c (Ω) = 1 . The monotone and normalized set function is called a capacity . A monotone setfunction is called submodular or 2 -alternating if c ( A ∪ B ) + c ( A ∩ B ) ≤ c ( A ) + c ( B ) A, B ∈ F T . The risk of an asset position X + Y will be lower than the sum of each risk,because of the diversification effects. The property of comonotonicity is that ifthere is no way for X to serve as a hedge for Y , then it is simply adding up therisks.Two real functions X, Y ∈ B (Ω , F T , P ) are called comonotonic if[ X ( ω ) − X ( ω )][ Y ( ω ) − Y ( ω )] ≥ , ω , ω ∈ Ω . The functional V is said to be comonotonic additive if X, Y are comonotonic = ⇒ V ( X + Y ) = V ( X ) + V ( Y ) . Definition 2.2.
Let c : F T → [0 ,
1] be a capacity. The
Choquet expectation with respect to c is defined as Z Ω X dc := Z −∞ ( c ( X > x ) − dx + Z ∞ c ( X > x ) dx, X ∈ L (Ω , F T , P ) . B (Ω , F T , P ). Theorem 2.3 ([15]) . Let V be a functional from B (Ω , F T , P ) to R . The fol-lowing statements are equivalent. V is normalized, monotone, and comonotonic additive. There exists a unique capacity c : F T → [0 , such that V ( X ) = Z −∞ ( c ( X > x ) − dx + Z ∞ c ( X > x ) dx ∀ X ∈ B (Ω , F T , P ) . (2.1)Let g : Ω × [0 , T ] × R × R n → R be a function that ( y, z ) g ( t, y, z ) ismeasurable for each ( y, z ) ∈ R × R n and satisfy the following conditions | g ( t, y, z ) − g ( t, ¯ y, ¯ z ) | ≤ K ( | y − ¯ y | + | z − ¯ z | ) (2.2a) ∀ t ∈ [0 , T ] , ∀ ( y, z ) , (¯ y, ¯ z ) ∈ R × R n , for some K > , Z T | g ( t, , | dt < ∞ , (2.2b) g ( t, y,
0) = 0 for each ( t, y ) ∈ [0 , T ] × R . (2.2c)The space L (Ω , F T , P ) is defined as L (Ω , F T , P ) := { ξ | ξ is F T -measurable random variable and E [ | ξ | ] < ∞} . Theorem 2.4 ([14]) . For every terminal condition ξ ∈ L ( F T ) := L (Ω , F T , P ) the following backward stochastic differential equation − dy t = g ( t, y t , z t ) dt − z t dB t , ≤ t ≤ T, (2.3a) y T = ξ (2.3b) has a unique solution ( y t , z t ) t ∈ [0 ,T ] ∈ L F ([0 , T ]; R ) × L F ([0 , T ]; R n ) . Definition 2.5.
For each ξ ∈ L ( F T ) and for each t ∈ [0 , T ] g − expectation of ξ and the conditional g − expectation of ξ under F t is respectively defined by E g [ ξ ] := y , E g [ ξ |F t ] := y t , where y t is the solution of the BSDE (2.3).Let { S t } be the stock price evolving as a stochastic differential equation dS t S t = µ t dt + σ t dB t where { µ t } is a market return rate, and { σ t } is a market volatility.3n a Black-Scholes world, there exists a unique risk-neutral probability mea-sure Q defined as dQdP = e − R T ( µs − rσs ) ds + R T ( µs − rσs ) dB s , where r is a riskless interest rate. In a real world, the parameters µ t and σ t are not known exactly. We assume that µ t belong to some interval, i.e. µ t ∈ [ r − kσ t , r + kσ t ] for a constant k >
0. Then the risk-neutral probabilitymeasures belong to P = ( Q ν : dQ ν dP = e − R T | ν s | ds + R T ν s dB s , sup t ∈ [0 ,T ] | ν t | ≤ k ) where ν t := ( µ t − r ) /σ t . There are two pricing methods of a contingent claim ξ , i.e. minimax pricing rules which are E [ ξ ] := inf Q ∈P E Q [ ξ ] , ¯ E [ ξ ] := sup Q ∈P E Q [ ξ ] . Let ξ ∈ L (Ω , F T , P ). The conditional g -expectations ¯ E [ ξ |F t ] and E [ ξ |F t ]are given as¯ E [ ξ |F t ] = ess sup Q ∈P E Q [ ξ |F t ] , E [ ξ |F t ] = ess inf Q ∈P E Q [ ξ |F t ] , (2.4)which are the solutions of BSDE (2.3) when the generators are g ( t, y t , z t ) = k | z t | and g ( t, y t , z t ) = − k | z t | respectively. The equations (2.4) will be proved inLemma 2.1.It is clear that¯ E [ ξ |F ] = ¯ E [ ξ ] := sup Q ∈P E Q [ ξ ] , E [ ξ |F ] = E [ ξ ] := inf Q ∈P E Q [ ξ ] . The upper and the lower Choquet integrals(or expectations) are respectivelydefined as ¯ V ( ξ ) := Z −∞ (¯ c ( ξ > x ) − dx + Z ∞ ¯ c ( ξ > x ) dx,V ( ξ ) := Z −∞ ( c ( ξ > x ) − dx + Z ∞ c ( ξ > x ) dx, where ¯ c and c are defined as¯ c ( A ) = sup Q ∈P Q ( A ) and c ( A ) = inf Q ∈P Q ( A ) for A ∈ F T . We will use the notation of ¯ V ( ξ ) := R ξ d ¯ c and V ( ξ ) := R ξ dc , or sometimesintegration notation just for the convenience of proof.4t can be easily seen that V ( ξ ) ≤ E [ ξ ] ≤ ¯ E [ ξ ] ≤ ¯ V ( ξ ) . In the complete market where P has a single element, we can see that V ( ξ ) = E [ ξ ] = ¯ E [ ξ ] = ¯ V ( ξ ) . Theorem 2.6 ([1]) . Suppose that g satisfies the condition (2.2a)-(2.2c). Thenthere exists a Choquet integral whose restriction to L (Ω , F T , P ) is equal to a g -expectation if and only if g does not depend on y and is linear in z , that is,there exists a continuous function ν t such that g ( y, z, t ) = ν t z. The Theorem 2.6 implies that the generator g in (2.3) should be linearfunction for both Choquet integral and g -expectation to be equal. We will showthat ¯ E [ ξ |F t ] and E [ ξ |F t ] are the solutions of the BSDEs (2.5) in the followingLemma 2.1. Lemma 2.1.
For ξ ∈ L (Ω , F T , P ) , let ( Y t , z t ) and ( y t , z t ) be the unique solu-tion of the following BSDEs Y t = ξ + Z Tt k | z s | ds − Z T z s dB s , t ∈ [0 , T ] , (2.5a) y t = ξ − Z Tt k | z s | ds − Z T z s dB s , t ∈ [0 , T ] (2.5b) respectively. Then Y t and y t are respectively represented as Y t = ess sup Q ∈P E Q [ ξ |F t ] = ¯ E [ ξ |F t ] , (2.6a) y t = ess inf Q ∈P E Q [ ξ |F t ] = E [ ξ |F t ] . (2.6b) Proof.
First, we show (2.6a). Let ν t = k sgn( z t ). Then sup t ∈ [0 ,T ] | ν t | ≤ k . If wedefine z νt as z νt = exp (cid:18) − Z t | ν s | ds + Z t ν s dB s (cid:19) , ≤ t ≤ T, then ( z νt ) ≤ t ≤ T is a P -martingale since dz νt /z νt = ν t · dB t . Also z νT is a P -densityon F T since 1 = z ν = E [ z νT ].Define an equivalent martingale probability measure Q ν and a Brownianmotion ¯ B t as dQ ν dP = e − R T | ν s | ds + R T ν s dB s , ¯ B t = B t − Z t ν s ds. Q ν ∈ P , and Girsanov’s theorem implies that { ¯ B t } is a Q ν -Brownianmotion.The BSDE (2.5a) is expressed as Y t = ξ − Z Tt z θs d ¯ B s . So we get Y t = E Q ν [ ξ | F t ] ≤ ess sup Q ∈P E Q [ ξ |F t ] . (2.7)Let { θ t } be a adapted process satisfyingsup t ∈ [0 ,T ] | θ t | ≤ k. Consider the following BSDE Y θt = ξ + Z Tt θ s z θs ds − Z Tt z θs dB s , t ∈ [0 , T ] . (2.8)Define an equivalent martingale probability measure Q θ and a Brownian motion¯ B θt as dQ θ dP = e − R T | θ s | ds + R T θ s dB s , ¯ B θt = B t − Z t θ s ds. Then Q θ ∈ P , and Girsanov’s theorem implies that { ¯ B θt } is a Q θ -Brownianmotion. The BSDE (2.8) is expressed as Y θt = ξ − Z Tt z s d ¯ B θs . So we get Y θt = E Q θ [ ξ | F t ] . Since θ t z t ≤ k | z t | for all ( z t , t ) ∈ R × [0 , T ], the Comparison Theorem appliedto (2.5a) and (2.8), implies that E Q θ [ ξ | F t ] = Y θt ≤ Y t ∀ t ∈ [0 , T ]Hence we obtain ess sup Q ∈P E Q [ ξ |F t ] ≤ Y t . (2.9)The inequalities (2.7) and (2.9) implies that¯ E [ ξ |F t ] := ess sup Q ∈P E Q [ ξ |F t ]6s the solution of (2.5a).In the same fashion, we can show that E [ ξ |F t ] := ess inf Q ∈P E Q [ ξ |F t ]is the solution of (2.5b) by setting ν t = − k sgn( z t ).
3. Choquet expectation and minimax expectation
In this section, we show that Choquet expectation and minimax expectationare equal in pricing European type options, whose payoff is a monotone functionof the terminal stock price S T . We also prove that the minimax expectationattains a maximum or a minimum on the set of equivalent martingale probabilitymeasures which is weakly compact.At the expiration date T , let the stock price S T ∈ L (Ω , F T , P ) be a uniquesolution of the following SDE dS t = µ t S t dt + σ t S t dB t , t ∈ [0 , T ] . Let Φ be a monotone function such that Φ( S T ) ∈ L (Ω , F T , P ). Let ( Y t , z t ) and( y t , z t ) be the unique solution of the following BSDE Y t = Φ( S T ) + Z Tt µ s | z s | ds − Z T z s dB s ,y t = Φ( S T ) − Z Tt µ s | z s | ds − Z T z s dB s , respectively.In Lemma 2.1, we have shown that Y t = ¯ E [Φ( S T ) |F t ] , y t = E [Φ( S T ) |F t ] . For example, in the option pricing, the monotone functions Φ( x ) = ( x − K ) + or Φ( x ) = ( K − x ) + is the payoff function of European call or put option,respectively. Here K is an exercise price of the option. We want to show that¯ E [Φ( S T )] = ¯ V [Φ( S T )] , E [Φ( S T )] = V [Φ( S T )] , where ¯ V and V are the upper and lower Choquet expectations, respectively.Since ¯ E [ ξ ] is defined as ¯ E [ ξ ] := sup Q ∈P E Q [ ξ ] , it is obvious that ¯ E is normalized and monotone.For each i = 1 ,
2, let the random variables ξ ′ i s be comonotonic functions.7 E is comonotonic additive since¯ E [ ξ + ξ ] = sup Q ∈P E Q [ ξ + ξ ] = sup Q ∈P E Q [ ξ ] + sup Q ∈P E Q [ ξ ] = ¯ E [ ξ ] + ¯ E [ ξ ] . So Theorem 2.3 says that there exists a unique capacity c : F T → [0 , E [ X ] = Z −∞ ( c ( X > x ) − dx + Z ∞ c ( X > x ) dx ∀ X ∈ B (Ω , F T , P ) . (3.1)If we take X = I A for A ∈ F T , then (3.1) becomes¯ E [ I A ] = Z −∞ ( c ( I A > x ) − dx + Z ∞ c ( I A > x ) dx. (3.2)Thus we have c ( A ) = sup Q ∈P Q [ A ] := ¯ c ( A ) . (3.3)So we have c = ¯ c .Therefore, the equation (3.1) becomes¯ E [ X ] = Z −∞ (¯ c ( X > x ) − dx + Z ∞ ¯ c ( X > x ) dx := ¯ V ( X ) ∀ X ∈ B (Ω , F T , P ) . From now on, we will show that the equation (3.1) can be extended from B (Ω , F T , P ) to a set of the monotone functions which is a subset of L (Ω , F T , P ). Lemma 3.1.
The capacity ¯ c in (3.3) is submodular.Proof. It’s easily shown that ¯ c is monotone and normalized. Since I A ∪ B and I A ∩ B are a pair of comonotone functions for all A, B ∈ F T , the comonotonicityof ¯ E implies¯ c ( A ∩ B ) + ¯ c ( A ∪ B ) = ¯ E [ I A ∩ B ] + ¯ E [ I A ∪ B ] = ¯ E [ I A ∩ B + I A ∪ B ]= ¯ E [ I A + I B ] ≤ ¯ E [ I A ] + ¯ E [ I B ] = ¯ c ( A ) + ¯ c ( B ) . So the proof is done.
Lemma 3.2. ¯ E [ ξ ] := sup Q ∈P E Q [ ξ ] is L -continuous for comonotonic functions ξ ∈ L (Ω , F T , P ) .Proof. Let ξ and ξ be comonotonic functions. Since ¯ E is comonotonic additive, | ¯ E [ ξ ] − ¯ E [ ξ ] | = | ¯ E [ ξ − ξ ] | = (cid:12)(cid:12)(cid:12) sup Q ∈P E Q [ ξ − ξ ] (cid:12)(cid:12)(cid:12) ≤ sup Q ∈P E Q [ | ξ − ξ | ] = ¯ E [ | ξ − ξ | ] . E is L -bounded. Let an adapted process { θ t } bounded by k be such that dQ θ dP = e − R T | θ s | ds + R T θ s dB s . By the H¨older’s inequality, we have E Q θ ( | ξ | ) = E (cid:18) | ξ | dQ θ dP (cid:19) ≤ ( E [ | ξ | ]) (cid:18) E (cid:20)(cid:12)(cid:12)(cid:12) dQ θ dP (cid:12)(cid:12)(cid:12) (cid:21)(cid:19) = ( E [ | ξ | ]) (cid:16) E h e − R T | θ s | ds + R T θ s dB s + R T | θ s | ds i(cid:17) = ( E [ | ξ | ]) (cid:16) e R T | θ s | ds E h e − R T | θ s | ds + R T θ s dB s i(cid:17) ≤ ( E [ | ξ | ]) e k T . So we get ¯ E [ ξ ] = sup Q ∈P E Q [ ξ ] ≤ ( E [ | ξ | ]) e k T . (3.4)Thus we have | ¯ E [ ξ ] − ¯ E [ ξ ] | ≤ ( E [ | ξ − ξ | ]) e k T . Therefore, ¯ E is L -continuous for the comonotonic random variables.On L (Ω , F T , P ), denote Choquet integral as Z Ω X dc := Z −∞ ( c ( X > x ) − dx + Z ∞ c ( X > x ) dx ∀ X ∈ L (Ω , F T , P ) , just for the convenience of proof. Theorem 3.1 ([2]) . Let
X, Y be real-valued measurable functions defined on Ω .If a capacity c is submodular and < p, q < ∞ with p + q = 1 , then Z Ω | XY | dc ≤ (cid:18)Z Ω | X | p dc (cid:19) p (cid:18)Z Ω | Y | q dc (cid:19) q . The following is the main theorem.
Theorem 3.2.
Let X ∈ L (Ω , F T , P ) be a monotone function. Then we have ¯ E [ X ] = Z Ω X d ¯ c, E [ X ] = Z Ω X dc. roof. Since E [ X ] = − ¯ E [ − X ], we only prove that ¯ E [ X ] = R Ω X d ¯ c . Let X ∈ L (Ω , F T , P ) be a monotone random variable. Let f be a simple function. Let ǫ > (cid:12)(cid:12)(cid:12)(cid:12) ¯ E [ X ] − Z Ω X d ¯ c (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ¯ E [ X ] − ¯ E [ f ] | + (cid:12)(cid:12)(cid:12)(cid:12) ¯ E [ f ] − Z Ω f d ¯ c (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω f d ¯ c − Z Ω X d ¯ c (cid:12)(cid:12)(cid:12)(cid:12) . (3.5)Since simple functions are dense in L (Ω , F T , P ), there exists an increasingsimple function f ր X satisfying both k X − f k L < e − k T · ǫ (cid:18)Z Ω | f − X | d ¯ c (cid:19) = (cid:18)Z ∞ ¯ c ( | f − X | > x ) dx (cid:19) < ǫ . Since the ¯ E is L -continuous for the comonotonic random variables X and f byLemma 3.2, the first term of the right hand side of (3.5) is less than ǫ/
3. Theequation (3.1) implies that the second term of the right hand side of (3.5) iszero.The capacity ¯ c is submodular by Lemma 3.1 and so Theorem 3.1 impliesthat the third term of the right hand side of (3.5) becomes (cid:12)(cid:12)(cid:12)(cid:12)Z Ω f d ¯ c − Z Ω X d ¯ c (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω | f − X | d ¯ c ≤ (cid:18)Z Ω | f − X | d ¯ c (cid:19) (cid:18)Z Ω d ¯ c (cid:19) ≤ (cid:18)Z Ω | f − X | d ¯ c (cid:19) < ǫ . So we obtain (cid:12)(cid:12)(cid:12)(cid:12) ¯ E [ X ] − Z Ω X d ¯ c (cid:12)(cid:12)(cid:12)(cid:12) < ǫ. Therefore, the proof is done.We will show that there exists Q ∈ P such that the minimax expectationtakes a maximum or minimum. Lemma 3.3.
The set of densities D := (cid:26) dQdP (cid:12)(cid:12)(cid:12) Q ∈ P (cid:27) is weakly compact in L (Ω , F T , P ) . roof. As in the proof of Lemma 3.2, we can prove E "(cid:18) dQdP (cid:19) ≤ e k T . So we have dQdP ∈ L (Ω , F T , P ). Thus we have D ⊂ L (Ω , F T , P ) .We want to show that D is weakly closed in L (Ω , F T , P ). Suppose that thesequence ( Z n ) in D converges weakly to Z . I.e., f ( Z n ) → f ( Z ) for all f ∈ ( L ) ∗ , where ( L ) ∗ is the set of continuous dual functionals of L . We want to show Z ∈ D .For X ∈ L (Ω , F , P ), define the linear functional J X as J X ( Z ) := E [ XZ ] ∀ Z ∈ D . (3.6)By the H¨older’s inequality, we have | J X ( Z ) | ≤ E [ | XZ | ] ≤ (cid:18)Z | X | dP (cid:19) / · (cid:18)Z | Z | dP (cid:19) / < + ∞ . So J X is bounded and thus continuous on L .By the assumption, we have J X ( Z n ) → J X ( Z ) as n → ∞ . That is, lim n →∞ Z XdQ n = lim n →∞ E [ XZ n ] = E [ XZ ] = Z XdQ.
Since Z n ∈ D , there exist θ ( n ) t and Q θ ( n ) t ∈ P satisfying Z n = dQ θ ( n ) dP = exp − Z T | θ ( n ) s | ds + Z T θ ( n ) s dB s ! . Let lim n →∞ θ ( n ) t = θ t . Then we have Z ′ = lim n →∞ Z n = exp − Z T | θ s | ds + Z T θ s dB s ! . So we have Z T XZdP = Z T XZ ′ dP ∀ X ∈ L (Ω , F T , P ) . Therefore, it becomes Z = Z ′ a.e. and thus Z ∈ D . It is proven that D is aweakly compact set. 11 heorem 3.3 (James’ Theorem) . A weakly closed subset D of a Banach space L (Ω , F T , P ) is weakly compact if and only if each continuous linear functionalon L (Ω , F T , P ) attains a maximum or a minimum on D . By James’ Theorem, the linear functional J X as in (3.6) attains a maximumon D . That is, there exists Q ∗ ∈ P such thatsup Q ∈P E Q [ ξ ] = E Q ∗ [ ξ ] ξ ∈ L (Ω , F T , P ) . To specify Q ∗ ∈ P , we need Lemma 3.4 which gives the restriction to thegenerator g of BSDE (3.8), in addition to Theorem 2.6. Let { S t } be the solutionof the following stochastic differential equation, S t = S + Z t η ( t, S t ) dt + Z t σ ( t, S t ) dB t , t ∈ [0 , T ] , (3.7)where η , σ : [0 , T ] × ℜ → ℜ are continuous in ( t, S ) and Lipschitz continuous in S . Lemma 3.4 ([1]) . Let { S t } be the solution of (3.7). Let Φ be the monotonefunction such that Φ( S T ) ∈ L (Ω , F T , P ) . Let ( y t , z t ) be the solution of thefollowing BSDE y t = Φ( S T ) + Z Tt θ s | z s | − Z Tt z s dB s . (3.8) Then the followings hold, z t σ ( t, S t ) ≥ , a.e. t ∈ [0 , T ) , if Φ is an increasing function z t σ ( t, S t ) ≤ , a.e. t ∈ [0 , T ) , if Φ is a decreasing function. Suppose that Φ is an increasing function. Then for | θ s | ≤ k , by Theorem 3.4,the solution ( y t , z t ) of (3.8) becomes the unique solution of the form of BSDE y ( θ ) t = Φ( S T ) + Z Tt θ s z ( θ ) s ds − Z Tt z ( θ ) s dB s (3.9)= Φ( S T ) − Z Tt z ( θ ) s d ¯ B θs , where ¯ B θt = B t − R t θ s ds .Let ( y ( k ) t , z ( k ) t ) be the unique solution of the following BSDE y ( k ) t = Φ( S T ) + Z Tt k | z ( k ) s | ds − Z Tt z ( k ) s dB s . (3.10)As we did at the end of Section 2, we have y ( k ) t ≥ y ( θ ) t for all t ∈ [0 , T ] byapplying the Comparison Theorem for BSDEs to (3.9) and (3.10). Therefore,we get y ( k )0 = E Q k [Φ( S T )] ≥ y ( θ )0 = E Q θ [Φ( S T )]12here Q k and Q θ are respectively defined as dQ k dP = e − R T k ds + R T kdB s = e − k T + kB T , dQ θ dP = e − R T θ s ds + R T θ s dB s . Thus we have E Q k [Φ( S T )] = sup Q θ ∈P E Q θ [Φ( S T )] := ¯ E [Φ( S T )] , since Q k ∈ P and | θ t | ≤ k . In the similar fashion, we can also show that E Q − k [Φ( S T )] = inf Q ∈P E Q [Φ( S T )] := E [Φ( S T )] , where Q − k is defined as dQ − k dP = e − k T − kB T . Now suppose that Φ is a decreasing function. Then − Φ is an increasing function.So we have ¯ E [Φ( S T )] = −E [ − Φ( S T )] = − E Q − k [ − Φ( S T )] = E Q − k [Φ( S T )] , E [Φ( S T )] = − ¯ E [ − Φ( S T )] = − E Q k [ − Φ( S T )] = E Q k [Φ( S T )] . Acknowledgment
This work was supported by the research grant of Sungshin Women’s Uni-versity in 2018.
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