The convergence rate from discrete to continuous optimal investment stopping problem
aa r X i v : . [ q -f i n . M F ] A p r The convergence rate from discrete to continuous optimalinvestment stopping problem
Dingqian Sun ∗ Abstract
We study the optimal investment stopping problem in both continuous and discrete case,where the investor needs to choose the optimal trading strategy and optimal stopping timeconcurrently to maximize the expected utility of terminal wealth. Based on the work [9] withan additional stochastic payoff function, we characterize the value function for the continu-ous problem via the theory of quadratic reflected backward stochastic differential equation(BSDE for short) with unbounded terminal condition. In regard to discrete problem, we getthe discretization form composed of piecewise quadratic BSDEs recursively under Markovianframework and the assumption of bounded obstacle, and provide some useful prior estimatesabout the solutions with the help of auxiliary forward-backward SDE system and Malliviancalculus. Finally, we obtain the uniform convergence and relevant rate from discretely to contin-uously quadratic reflected BSDE, which arise from corresponding optimal investment stoppingproblem through above characterization.
Keywords : optimal investment stopping problem, utility maximization, quadratic reflectedBSDE, discretely reflected BSDE, convergence rate : 60G40, 65C30, 93E20.
In this paper, we consider a small trader in an incomplete financial market who can invest inrisky stocks and a riskless asset and is also granted the right to stop the whole investment duringthe finite trading time interval [0 , T ] to obtain corresponding payoff. The objective of the investor isto maximize her/his exponential utility of terminal wealth, which includes both the profit or loss oninvestment and the final payoff, by choosing the optimal trading strategy and optimal stopping timesimultaneously. For the continuous case, the investor is allowed to stop the investment, which islike exercising an American option, at any time before T . While for the discrete case, the investerwill be restricted to given discrete exercise time, where the payoff can be regarded as a kind ofBermudan option.Such utility maximization problems of mixed optimal stopping/control type have been initiallystudied in [10], which involved both consumption and final wealth under continuous framework ∗ School of Mathematical Sciences, Fudan University, Shanghai, China, 200433. This work was completed duringthe visit in the University of Warwick as a joint Ph.D. student under the guidance of Dr.Liang and partially supportedby China Scholarship Council; National Science Foundation of China (No. 11631004); and Science and TechnologyCommission of Shanghai Municipality (No. 14XD1400400).
Email:[email protected] , τ ] with fixed τ ∈ [0 , T ], it will then become the usual exponential utility maximization problemwhich has been widely discussed before, see [8], [9], [16] and [17]. To be specific, when the terminalpayoff at τ is bounded, the problem has been completely solved in [8] with the help of quadraticBSDE with bounded terminal data. It turns out that the value function of such problem can becharacterized by the solution to a particular BSDE, whose generator is of quadratic growth in z -variable. Related theory of quadratic BSDE can be traced back to [12] with bounded terminalvalue, where the existence and uniqueness of solutions were estabilished. Then it was extendedto unbounded case to obtain the existence in [2], and subsequently the uniqueness with convexgenerators in [3], [5] and [6]. Recently, [9] generalized the previous work, the exponential utilitymaximization problem with bounded payoff, to the unbounded framework on the basis of abovedevelopment and studied utility indifference valuation of derivatives with unbounded payoffs asapplication.Inspired by the above connection, we adjust the order of optimization and decompose ourproblem with extra payoff function into original utility maximization framework, which then reducedto a pure optimal stopping problem, and further obtain the value function in terms of the solution toa quadratic reflected BSDE, where the generator has almost the same form as in utility maximizationproblem in [9]. While the existence and uniqueness of solutions to such quadratic reflected BSDEhave been developed, see [13] for bounded terminal value and obstacle and [14], [1] for unboundedcases, the main difficulty left is to represent the solution to reflected BSDE via the supremum ofsolutions to a collection of BSDEs, which have the same quadratic generator as the former, i.e., Y t = sup τ ∈ [ t,T ] Y t ( τ ) , t ∈ [0 , T ] in subsection 2.3. Since here the group of BSDEs have different timehorizon [0 , τ ] and terminal value g τ and thus corresponding different pairs of solutions ( Y ( τ ) , Z ( τ )),we can not directly apply the optimal stopping representation of reflected BSDEs (see Proposition2.3 in [7] for reference), but need to further use the comparison theorem and uniqueness in quadraticBSDE to prove such characterization, see Theorem 2.4 for more details.Regarding the discrete problem, we need to restructure the framework and proceed under Marko-vian system for the sake of following convergence analysis between the two forms. We first give apractical example to illustrate how we get the Markovian structure arising from previous continuousproblem. While due to the additon of stochastic factor, the generator we considered herein will bemore complicated than that in previous section, i.e., f ( t, x, z ) of quadratic growth in z and satis-fying locally Lipschitz condition with respect to both x and z , which is generalized in Assumption3.1. Then when restricted the exercise time to some given discrete time points, we can deducerecursively from the comparison result of BSDEs to get the backward discretization form, whichis composed of piecewise BSDEs and actually a so-called discretely reflected BSDE, see subsection3.2 for the form and related properties.The main result of this paper is the convergence analysis and relevant rate from discrete to con-tinuous optimal investment stopping problem. Thanks to previous discussion and characterization,2e can now transform the problem into the convergence from discretely to continuously reflectedBSDE, which has been studied when the generator is uniformly Lipschitz in all the variables, see[15] based on the Euler scheme of forward SDE and section 3 in [4]. Whereas originating from theutility maximization problem, we are facing reflected BSDEs with generator of quadratic growth,which brings us new difficulties during estimation and thus we have to restrict ourselves to the caseof bounded and Lipschitz obstacle, and also the deterministic diffusion term in forward SDE at thisstage.Firstly, with the help of the properties of quadratic BSDE and reflected BSDE with boundedterminal value, we can inductively prove the boundness of ˆ Y Π in discretization form and the rela-tionship ˆ Y Π ≤ Y , which makes it possible to implement the usual techniques using to deal withBSDE of quadratic growth.Moreover, in order to handle the additional term coming from reflection, we need further prop-erties about ˆ Z Π appearing in piecewise BSDEs of the discretization form. We establish the con-nections between our discretization form and an auxiliary forward-backward SDE system definedon each time interval [ t i − , t i ] with different terminal functions { u Π i } ≤ i ≤ n . Recall the existingresults in Markovian FBSDE system that the solution Z to quadratic BSDE with bounded andLipschitz terminal g ( X T ) is controlled by C ( K g + 1), see [18], where K g is the Lipschitz constantof g . And then in [19], the prior estimate on Z is generalized to the superquadratic case withunbounded terminal condition and also the case with random diffusion term in forward SDE andbounded terminal condition. While unfortunately, neither of them can cover the situation in ourassumptions since here the locally Lipschitz coefficient of x involves z . However, motivated by theproof of these results, we can make use of the BMO property of Z and the representation derivedfrom Mallivian calculus to fill this gap and get the explicit bound of Z . Together with the uniformLipschitz continuity of terminal functions { u Π i } in auxiliary forward-backward SDE system, we canobtain the boundness of ˆ Z Π in discretization form at last.Finally, we give the complete proof of the uniform convergence from discretely to continuouslyquadratic reflected BSDE and obtain the convergence rate as follows when the obstacle g is Lipschitz(and also the double rate if g in C b ):max i n E " sup t ∈ [ t i − ,t i ] | ˆ Y Π t − Y t | + E "Z T | ˆ Z Π t − Z t | dt C | Π | and max i n E " sup t ∈ [ t i − ,t i ] | ˆ K Π t − K t | C | Π | . The paper is organized as follows. We discuss the continuous optimal investment stopping prob-lem in section 2 and give the characterization of value function in terms of the solution to quadraticreflected BSDE. In section 3, we focus on Markovian framework and put forward the assumptionsbased on a practical example, and further obtain the discretization form for corresponding discreteproblem. Then in section 4, after providing some auxiliary results regarding the discretization formwith the aid of a forward-backward SDE system, we finally provide the convergence result of thetwo forms and section 5 concludes the paper. 3
Continuous optimal investment stopping problem
We fix a finite time horizon [0 , T ] with
T >
0. Let B be a m -dimensional standard Brownianmotion defined on a complete probability space (Ω , F , P ), and {F t } t > be the augumented naturalfiltration of B which satisfies the usual conditions.Let P denote the progressively measurable σ -field on [0 , T ] × Ω. Consider a financial market consisting of one risk-free bond with interest rate zero and d m stocks. In the case d < m , we face an incomplete market. The price process of the i th stock isdescribed as dS it S it = b it dt + σ it dB t , i = 1 , ..., d, where b i (resp. σ i ) is an R -valued (resp. R m -valued) predictable bounded stochastic process. The R d × m -valued volatility matrix has full rank, that is, σ t σ trt is invertible P -a.s., for any t ∈ [0 , T ].Furthermore, we assume the R m -valued risk premium process defined as θ t = σ trt ( σ t σ trt ) − b t , t ∈ [0 , T ]is also bounded. For i = 1 , ..., d , let π it denote the amount of money invested in stock i at time t , and then the number of shares should be π it S it . An R d -valued predictable process π = ( π t ) t T is called a self-financing trading strategy if R π dSS is well defined, for example, R T | π trt σ t | dt < ∞ , P -a.s., which means the investor trades dynamically among the risk-free bond and the risky assetswith her/his initial capital and no extra investment or withdrawal during the investment.The wealth process with initial capital x and trading strategy π satisfies the equation X πt = x + d X i =1 Z t π iu S iu dS iu = x + Z t π tru σ u ( dB u + θ u du ) , t ∈ [0 , T ] . Suppose there is an additional adapted process ( g t ) t T defined as the payoff at each time t andrecall that the investor has the right to stop at any time during the trading interval [0 , T ], whichmeans, if the investor chooses to stop at τ ∈ [0 , T ], then the total wealth of the investor is X τ + g τ .Here, g τ > π to maximize theexpected utility of total wealth, which is in exponential form with the parameter α >
0, i.e. V (0 , x ) = sup τ ∈ [0 ,T ] sup π ∈U ad [0 ,τ ] E [ U α ( X τ + g τ )]= sup τ ∈ [0 ,T ] sup π ∈U ad [0 ,τ ] E (cid:20) − exp (cid:18) − α (cid:18) x + Z τ π tru dS u S u + g τ (cid:19)(cid:19)(cid:21) . Here V (0 , x ) is called the value function at initial time 0 and U ad [0 , τ ] is the admissible strategy seton [0 , τ ], given by Definition 1 in [9] .More generally, we can consider this mixed optimal stopping/control problem in dynamic form V ( t, X t ) = sup τ ∈ [ t,T ] sup π ∈U ad [ t,τ ] E (cid:20) − exp (cid:18) − α (cid:18) X t + Z τt π tru dS u S u + g τ (cid:19)(cid:19) (cid:12)(cid:12)(cid:12) F t (cid:21) , (2.1)4or all t T . Here X t is the initial wealth when we start at the initial time t . We first present the existence and uniqueness results of quadratic reflected BSDEs with theterminal data and obstacle satisfying exponential integrability, which were perfectly proved in [1],and we will use the results to further solve the optimal investment stopping problem in continuoussetting to implement the utility maximization.A reflected BSDE with generator f , lower obstacle g and terminal condition g T (here we onlyconsider this special case) is an equation of the form g t Y t = g T + Z Tt f ( s, Z s ) ds − Z Tt Z trs dB s + K T − K t , t ∈ [0 , T ] , (2.2)satisfying the flat-off condition: Z T ( Y t − g t ) dK t = 0 . (2.3)Recall the generator f : [0 , T ] × Ω × R m → R is a P ⊗ B ( R m ) measurable function and the obstacle g is an R -valued continuous adapted process.Let E λ,λ ′ [0 , T ] denote all the R -valued continuous adapted processes ( Y t ) t T such that E [ e λY −∗ + e λ ′ Y + ∗ ] < ∞ , where Y ±∗ , sup t ∈ [0 ,T ] ( Y t ) ± and E p [0 , T ] , E p,p [0 , T ]. H p ([0 , T ]; R m ) denotes all R m -valued predictable processes ( Z t ) t T with E ( R T | Z t | R m dt ) p < ∞ and K p [0 , T ] denotes all the R -valued continuous adapted processes ( K t ) t T , which are increasing with K =0 and E | K T | p < ∞ . Assumption 2.1
The obstacle g satisfies the exponential integrable condition: E h e λαg −∗ + e λ ′ αg + ∗ i < ∞ , for some λ, λ ′ > with λ + λ ′ < . Assumption 2.2
The obstacle g satisfies the arbitrary exponential integrable condition: E h e p | g ∗ | i < ∞ , ∀ p > . Theorem 2.3
Suppose that Assumption 2.1 holds with parameters λ and λ ′ . Then, the quadraticreflected BSDE (2.2) and (2.3) with generator f ( t, z ) = − α π t ∈C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ trt π t − (cid:18) α θ t − z (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − z tr θ t + 12 α | θ t | (2.4) admits a unique solution ( Y, Z, K ) ∈ T p ∈ (1 , λλ ′ λ + λ ′ ) E λα,λ ′ α [0 , T ] × H p ([0 , T ]; R m ) × K p [0 , T ] . Here C is a closed and convex set in the definition of admissible strategy satisfying ∈ C , which the strategycan take values in.In addition, if g satisfies Assumption 2.2, then the unique solution belongs to E p [0 , T ] × H p ([0 , T ]; R m ) × K p [0 , T ] for all p ∈ [1 , ∞ ) , i.e., E " e pγY ∗ + Z T | Z s | ds ! p + K pT < ∞ . roof. One can easily check that f with the form (2.4) satisfies − α | z | f ( t, z ) − z tr θ t + 12 α | θ t | , (2.5)and is concave in z , i.e., it satisfies Assumptions (H1) and (H3) in [1]. Consequently, we can getthe existence and uniqueness directly from Theorem 3.2 and 4.1 there. (cid:3) Now we can characterize the value function of the optimal problem by using the solution to theabove reflected BSDE.
Theorem 2.4
Suppose that g satisfies Assumption 2.2 and let ( Y, Z, K ) be the unique solution toquadratic reflected BSDE (2.2) and (2.3) with generator (2.4). Then, the value function (2.1) ofthe continuous optimal investment stopping problem can be given by V ( t, X t ) = − exp( − α ( X t + Y t )) , ∀ t ∈ [0 , T ] . Proof.
For any fixed t ∈ [0 , T ] and τ ∈ [ t, T ], we first solve the optimal control problem for thetime interval [ t, τ ] by considering the following quadratic BSDE Y t ( τ ) = g τ + Z τt f ( s, Z s ( τ )) ds − Z τt Z trs ( τ ) dB s , (2.6)where the generator f has the same form as in reflected BSDE, i.e., satisfies (2.4). For convenience,we will note the above equation as BSDE ( f, g τ ) thereafter. Additionally, we denote the solutionto this BSDE as ( Y . ( τ ) , Z . ( τ )) in order to emphasize its dependence on the terminal time τ andcorresponding terminal value g τ . Then we can represent the latter part of the value function bydynamic programming principle as follows,sup π ∈U ad [ t,τ ] E (cid:20) − exp (cid:18) − α (cid:18) X t + Z τt π tru dS u S u + g τ (cid:19)(cid:19) (cid:12)(cid:12)(cid:12) F t (cid:21) = − exp( − α ( X t + Y t ( τ ))) , (2.7)based on the existing result in [9], see Theorem 6. In turn, the original mixed optimal stop-ping/control problem becomes V ( t, X t ) = sup τ ∈ [ t,T ] [ − exp( − α ( X t + Y t ( τ )))] = − exp − α X t + sup τ ∈ [ t,T ] Y t ( τ ) !! , (2.8)and we only need to show that Y t = sup τ ∈ [ t,T ] [ Y t ( τ )] for any t ∈ [0 , T ].First, for any 0 t τ T , let Y t , Y τ + Z τt f ( s, Z s ) ds − Z τt Z trs dB s and we have Y t = Y t + K τ − K t . Recalling that ( Y ( τ ) , Z ( τ )) satisfies (2.6) on [ t, τ ] with the samegenerator as ( Y , Z ) and their terminal values satisfy Y τ > g τ , we can then deduce that Y t > Y t ( τ )6ia the comparison theorem of quadratic BSDE, see Theorem 5 in [3], where the proof and resultcan be easily adapted to the case of concave generator with quadratic growth from below. Moreover,since K is an increasing process, we have K τ − K t > Y t = Y t + K τ − K t > Y t ( τ ) for any τ ∈ [ t, T ], which gives rise to Y t > sup τ ∈ [ t,T ] [ Y t ( τ )].The idea of the following proof comes from representation of the solution to reflected BSDE,which is corresponded to an optimal stopping problem, see [7]. For any t ∈ [0 , T ], define D t , inf { s ∈ [ t, T ] : Y s = g s } and since Y T = g T , we can obtain t D t T . Considering the reflectedBSDE on interval [ t, D t ], Y t = Y D t + Z D t t f ( s, Z s ) ds − Z D t t Z trs dB s + K D t − K t , by the continuity of K and the flat-off condition (2.3), we have K D t = K t (which means K s ≡ K t for any s ∈ [ t, D t ]) and then ( Y., Z. ) becomes the solution to BSDE ( f, Y D t ) on [ t, D t ]. In themeanwhile, note that ( Y. ( D t ) , Z. ( D t )) is the solution to BSDE ( f, g D t ) on [ t, D t ] and the definitionof D t futher yields Y D t = g D t . Thus by the uniqueness of solution to quadratic BSDE, see [5], wehave Y. = Y. ( D t ) on [ t, D t ], and specifically Y t = Y t ( D t ), which completes the proof. (cid:3) Remark 2.5
We need to note here that for convenience, what we discussed in this paper is quadraticreflected BSDE with lower obstacle, whose solution we have proved in the above theorem can becharacterized by the supremum of the solutions to a collection of BSDEs with the same generator.Therefore, we require consistency of the supremum whether it is taking inside or outside the expo-nential in the expression of value function (2.8). To this end, when quoting the result in [9], wehave to change the sign of Y appearing in the value function as (2.7) and then the correspondinggenerator of BSDE. Actually, denoting the generator there as F , one can readily check that theysatisfy f ( t, z ) = − F ( t, − z ) and that is why we are considering concave generator in this section. From this section, we will concentrate on Markovian framework, that is, the following decoupledforward-backward SDE with reflection X t = x + Z t b ( s, X s ) ds + Z t σ ( s ) dB s ,Y t = g ( X T ) + Z Tt f ( s, X s , Z s ) ds − Z Tt Z trs dB s + K T − K t , t ∈ [0 , T ] ,Y t > g ( X t ) and Z T ( Y t − g ( X t )) dK t = 0 . (3.1)For the functions that appear in the above system, we have following general assumptions. Assumption 3.1 b, σ, g and f are deterministic functions that satisfy: ( a ) b : [0 , T ] × R → R and σ : [0 , T ] → R m are continuous functions and there exists constants M b , K b and M σ such that ∀ t ∈ [0 , T ] , ∀ x, x ′ ∈ R , | b ( t, x ) | M b (1 + | x | ) , | b ( t, x ) − b ( t, x ′ ) | K b | x − x ′ | , | σ ( t ) | M σ . b ) f : [0 , T ] × R × R m → R and g : R → R are continuous functions and there exists constants M f , K x , K z , K g and M g such that ∀ t ∈ [0 , T ] , ∀ x, x ′ ∈ R and ∀ z, z ′ ∈ R m , | f ( t, x, z ) | M f + α | z | , | f ( t, x, z ) − f ( t, x ′ , z ) | K x (1 + | z | ) | x − x ′ | , | f ( t, x, z ) − f ( t, x, z ′ ) | K z (1 + | z | + | z ′ | ) | z − z ′ | , | g ( x ) − g ( x ′ ) | K g | x − x ′ | , | g ( x ) | M g . Let S ∞ [0 , T ] denote the set of R -valued progressively measurable bounded processes and S p [0 , T ]denote the space of all R -valued adapted processes ( Y t ) t ∈ [0 ,T ] such that E [sup t T | Y t | p ] < ∞ .Then under the above assumptions, we know the decoupled system (3.1) with bounded termi-nal condition and bounded obstacle has a unique solution ( X, Y, Z, K ) ∈ S [0 , T ] × S ∞ [0 , T ] × H ([0 , T ]; R m ) × K [0 , T ]. For more details of this result, we refer to [13]. We will see from a special case with subspace portfolio constraints in this subsection that howwe can get the above Markovian structure from the previous general problem. Here for simplicity,we consider a market with a single stock whose coefficients depend on a single stochastic factordriven by a 2-dim Brownian motion, that is, m = 2, d = 1 and dS t S t = b ( t, V t ) dt + σ ( t, V t ) dB ,t ,dV t = η ( V t ) dt + ( κ , κ ) (cid:18) dB ,t dB ,t (cid:19) , (3.2)where κ , κ are two positive constants satisfying | κ | + | κ | = 1. We assume that b, σ and η areuniformly bounded and Lipschitz with respect to x , and furthermore, σ > δ for some δ >
0. Thenthe wealth process is dX t = π t dS t S t = π t [ b ( t, V t ) dt + σ ( t, V t ) dB ,t ] . Setting C = R and θ ( t, V t ) , b ( t,V t ) σ ( t,V t ) , we know from the above assumptions that θ is also bothbounded and Lipschitz. Supposing further that the payoff has the form as a function of stochasticfactor V , that is, g ( V · ), the reflected BSDE (2.2) will then become g ( V t ) Y t = g ( V T ) + Z Tt f ( s, Z s ) ds − Z Tt Z trs dB s + K T − K t , t ∈ [0 , T ] , (3.3)where the generator in (2.4) reduces to f ( t, z ) = f ( t, ( z , z )) = − α | z | − z θ ( t, V t ) + 12 α | θ ( t, V t ) | . If we regard the equation of stochastic factor (3.2) as the forward SDE and let ¯ f ( t, x, z ) , − α | z | − z θ ( t, x ) + α | θ ( t, x ) | , then ¯ f is now a deterministic function and (3.3) becomes g ( V t ) Y t = g ( V T ) + Z Tt ¯ f ( s, V s , Z s ) ds − Z Tt Z trs dB s + K T − K t , t ∈ [0 , T ] . (3.4)8ombined with (3.2), they constitute a Markovian system as (3.1) and one can easily check that ¯ f satisfies the above Assumption.In order to avoid confusion about the notations, we will still use b and σ to denote the coefficientsof forward SDE and ( X, Y, Z, K ) the solution of forward-backward SDE with reflection in thefollowing discussion, and consider the discrete problem and subsequent convergence under thegeneralized Assumption 3.1.
We continue to consider the optimal investment stopping problem in a discrete setting, whichmeans the investor is only allowed to stop the investment process at given discrete time pointsΠ , { t i , i = 0 , , . . . , n | t < t < t < · · · < t n = T } . Denote D [ t, T ] , [ t, T ] ∩ Π and∆ t i = t i − t i − for i = 1 , . . . , n , and let | Π | , max i n ∆ t i . The corresponding value function fordiscrete problem becomessup τ ∈ D [ t,T ] sup π ∈U ad [ t,τ ] E (cid:20) − exp (cid:18) − α (cid:18) X t + Z τt π tru dS u S u + g ( X τ ) (cid:19)(cid:19) (cid:12)(cid:12)(cid:12) F t (cid:21) = sup τ ∈ D [ t,T ] [ − exp( − α ( X t + Y t ( τ )))]= − exp (cid:20) − α (cid:18) X t + max τ ∈ D [ t,T ] Y t ( τ ) (cid:19)(cid:21) , where Y · ( τ ) satisfies the BSDE Y t ( τ ) = g ( X τ ) + Z τt f ( s, X s , Z s ( τ )) ds − Z τt Z trs ( τ ) dB s , t ∈ [0 , τ ] . Define ˆ Y Π t , max τ ∈ D [ t,T ] Y t ( τ ) = max τ ∈ D [ t,T ] E [ g ( X τ ) + R τt f ( s, X s , Z s ( τ )) ds |F t ] for any t ∈ [0 , T ]. Then the value function in discrete case turns out to be V Π ( t, X t ) = − exp[ − α ( X t + ˆ Y Π t )],which indicates that we only need to focus on the difference between ˆ Y Π and Y . Thanks to thecomparison result of quadratic BSDEs, we can characterize ˆ Y Π inductively and it is actually a so-called discretely reflected BSDE, which means that reflection only operates at specific time pointsΠ. We will depict the processes ¯ Y Π and ( ˆ Y Π , ˆ Z Π , ˆ K Π ) recursively as follows and in order to simplifythe notation, we will proceed with the case m = 1, while one can easily generalize the results to m -dimension: • ˆ Y Π t n = ¯ Y Π t n = g ( X T ); • For i = n, n − , · · · , t ∈ [ t i − , t i ), ( ¯ Y Π , ˆ Z Π ) is the solution to quadratic BSDE:¯ Y Π t = ˆ Y Π t i + Z t i t f ( r, X r , ˆ Z Π r ) dr − Z t i t ˆ Z Π r dB r ; (3.5) • For i = n, n − , · · · ,
1, define ˆ Y Π t = ¯ Y Π t for any t ∈ ( t i − , t i ) and ˆ Y Π t i − = ¯ Y Π t i − ∨ g ( X t i − ); • Let ˆ K Π0 , i = 1 , , · · · , n , t ∈ ( t i − , t i ], and define ˆ K Π t ≡ ˆ K Π t i , P ij =1 ( ˆ Y Π t j − − ¯ Y Π t j − ).9ince ˆ K Π t i ∈ F t i − for any 1 i n , we know that ˆ K Π is {F t } -predictable. In addition, we candeduce from definition that ˆ K Π t i − ˆ K Π t i − = ˆ Y Π t i − − ¯ Y Π t i − , which leads toˆ Y Π t i − = ˆ Y Π t i + Z t i t i − f ( r, X r , ˆ Z Π r ) dr − Z t i t i − ˆ Z Π r dB r + ˆ K Π t i − ˆ K Π t i − , (3.6)and that is why it is called discretely reflected BSDE. Lemma 3.2
Let Assumption 3.1 hold. Then, we have ( i ) both ¯ Y Π and ˆ Y Π are bounded by M g + M f T , uniformly in Π ; ( ii ) ¯ Y Π t ˆ Y Π t Y t for all t ∈ [0 , T ] . Proof. ( i ) For the first claim, since k ¯ Y Π k ∞ k ˆ Y Π t n k ∞ + M f ∆ t n = k g ( X T ) k ∞ + M f ∆ t n M g + M f ∆ t n M g + M f T on [ t n − , t n ) by Corollary 2.2 in [12] and | ˆ Y Π t n − | | ¯ Y Π t n − | ∨ | g ( X t n − ) | M g + M f ∆ t n , the conclusion holds for the first interval [ t n − , t n ) and also for t = t n .Then for the next interval [ t n − , t n − ), using the Corollary again we have k ¯ Y Π k ∞ k ˆ Y Π t n − k ∞ + M f ∆ t n − M g + M f (∆ t n − + ∆ t n ) M g + M f T on [ t n − , t n − ), and similarly | ˆ Y Π t n − | | ¯ Y Π t n − | ∨| g ( X t n − ) | M g + M f (∆ t n − + ∆ t n ).By analogy, we can finally obtain k ¯ Y Π k ∞ ∨ k ˆ Y Π k ∞ M g + M f ( P nj = i ∆ t j ) M g + M f T on[ t i − , t i ) for any i = 1 , . . . , n , i.e., k ¯ Y Π k ∞ and k ˆ Y Π k ∞ are bounded by M g + M f T on the wholeinterval [0 , T ], and the bound is obviously independent of Π.( ii ) Observing that ¯ Y Π and ˆ Y Π may not be equal only on Π, one can easily get the first inequalityby definition. As for the second one, we first have ˆ Y Π t n = Y t n = g ( X T ). Assume ˆ Y Π t i Y t i holds.Then, similarly as the arguments in the proof of Theorem 2.4, comparing (3.5) and Y t = Y t i + Z t i t f ( s, X s , Z s ) ds − Z t i t Z s dB s + K t i − K t , the comparison result of quadratic BSDE with bounded terminals and the fact K is increasingfurther yield that ¯ Y Π t Y t for any t ∈ [ t i − , t i ). Moreover, since g ( X t ) is the lower obstacle of Y t ,we have ˆ Y Π t i − ¯ Y Π t i − ∨ g ( X t i − ) Y t i − , and thus ˆ Y Π t Y t for t ∈ [ t i − , t i ). Then by induction,we can conclude the second inequality. (cid:3) In consideration of the connections we have built respectively for the continuous and discreteoptimal investment stopping problem in previous sections, we may now lay emphasis on the con-vergence from discretely to continuously quadratic reflected BSDE.
We will introduce a forward-backward SDE system on each interval [ t i − , t i ] instead of analyzingthe discretization form directly. Define u Π n ( x ) = g ( x ) = ˜ Y T and for i = n, n − , · · · ,
1, let ( ˜
Y , ˜ Z )be the solution to the BSDE defined piecewise by˜ Y t = u Π i ( X t i ( t i − , x )) + Z t i t f ( r, X r ( t i − , x ) , ˜ Z r ) dr − Z t i t ˜ Z r dB r , t ∈ [ t i − , t i ) , (4.1)10here X · ( t i − , x ) represents the solution to forward SDE in (3.1) starting from ( t i − , x ). Let u Π i − ( x ) = ˜ Y t i − ( x ) ∨ g ( x ) and notice that here we write as the form ˜ Y t i − ( x ) in order to show thedependence of ˜ Y on the initial value x of the SDE. Lemma 4.1
By the definition of the collection of functions { u Π i } i n , we have ˆ Y Π t i = u Π i ( X t i ) .Here X t i , X t i (0 , x ) . Proof.
We will prove this lemma by induction. Firstly, for i = n , ˆ Y Π t n = g ( X T ) = u Π n ( X t n ). Ifwe assume that the result holds for i , i.e., ˆ Y Π t i = u Π i ( X t i ), then when it comes to i −
1, we haveˆ Y Π t i − = ¯ Y Π t i − ∨ g ( X t i − ) and u Π i − ( X t i − ) = ˜ Y t i − ( X t i − ) ∨ g ( X t i − ) separately.Comparing (3.5) and (4.1) when x = X t i − and noticing that X r , X r (0 , x ) = X r ( t i − , X t i − )for any r ∈ [ t i − , t i ], we know that the two BSDEs have the same generator. Especially, wehave X t i ( t i − , X t i − ) = X t i and then u Π i ( X t i ( t i − , X t i − )) = u Π i ( X t i ) = ˆ Y Π t i by assumption, whichmeans the BSDEs have the same terminal value as well, which is bounded as proved. Then bythe uniqueness of quadratic BSDE with bounded terminal condition, see [12], we can concludethat ¯ Y Π t = ˜ Y t ( X t i − ) on [ t i − , t i ), and specifically, ¯ Y Π t i − = ˜ Y t i − ( X t i − ). Consequently, we obtainˆ Y Π t i − = u Π i − ( X t i − ) by taking maximum with g ( X t i − ) on both sides, which completes the proof. (cid:3) Let us introduce the following more general forward-backward SDE on [0 , T ] for later use, X t = x + Z t b ( s, X s ) ds + Z t σ ( s ) dB s ,Y t = g ( X T ) + Z Tt f ( s, X s , Z s ) ds − Z Tt Z s dB s , (4.2)and give a crucial estimate of Z in next lemma. Lemma 4.2
Suppose Assumption 3.1 holds. Then, there exists a version of Z such that ∀ t ∈ [0 , T ] , | Z t | exp( KT )[ M σ exp(2 K b T ) K g + 1] , where K , M σ K x exp(2 K b T ) . The proof is given in the appendix. Since the locally Lipschitz condition of f with respect to x in Assumption 3.1 involves z , we can not use the existing result, like in [18] and [19], directly.Fortunately, we know that with bounded terminal value, the martingale Z ∗ B belongs to the spaceof BMO martingales, which can essentially help us to prove the boundness of Z and further givethe explicit form.Next, let us give a useful lemma called discrete backward Gronwall Inequality, which will playan important role in the following content. Lemma 4.3
Let Π and ∆ t i define as above. Suppose that { a i , b i } ni =1 satisfy a i > , b i > , and a i − e C ∆ t i a i + b i for i = 2 , · · · , n , then max i n a i e CT " a n + n X i =1 b i . roof. By backward induction, we have a n − e C ∆ t n a n + b n e C ∆ t n [ a n + b n ] ,a n − e C ∆ t n − a n − + b n − e C (∆ t n − +∆ t n ) [ a n + b n + b n − ] ,. . . . . . thus one can easily get that a i e C ( T − t i ) a n + n X j = i +1 b j , i = 1 , · · · , n, which completes the proof. (cid:3) Now we can consider further property of the collection of functions { u Π i } i n . Lemma 4.4
Suppose Assumption 3.1 holds. Then u Π i is bounded and Lipschitz continuous, uni-formly in Π and i . Proof.
The first assertion in regard to the boundness actually can be proved following the sameprocedure as in Lemma 3.2. Now let us prove by induction that each u Π i is Lipschitz continuous.Clearly, u Π n = g is Lipschitz by assumption and the Lipschitz constant is L n , K g . Assuming that u Π i is Lipschitz with constant L i , then we need to show the result for u Π i − .For any x , x ∈ R , t ∈ [ t i − , t i ), denote ( ˜ Y j , ˜ Z j ) as the solution to (4.1) with initial value x j , j = 1 ,
2. Regarding (4.1) as the system (4.2) on [ t i − , t i ] with the terminal value function u Π i and by the Lipschitz and boundness assumption, we can get from Lemma 4.2 that | ˜ Z jt | C (1 + L i )on [ t i − , t i ], where C denotes the constant which may depend on T and all the constants appearingin the Assumption except K g and may vary from line to line. Consider the difference between thetwo solutions˜ Y t − ˜ Y t = u Π i ( X t i ( t i − , x )) − u Π i ( X t i ( t i − , x )) − Z t i t ( ˜ Z r − ˜ Z r ) dB r + Z t i t h f ( r, X r ( t i − , x ) , ˜ Z r ) − f ( r, X r ( t i − , x ) , ˜ Z r ) i dr, t ∈ [ t i − , t i ) , (4.3)and define V it = ( f ( t,X t ( t i − ,x ) , ˜ Z t ) − f ( t,X t ( t i − ,x ) , ˜ Z t )˜ Z t − ˜ Z t { ˜ Z t = ˜ Z t } , t ∈ [ t i − , t i ]; V it i − , t ∈ [0 , t i − ) . Noting that | V it | K z (1 + | ˜ Z t | + | ˜ Z t | ) C (1 + L i ) for all t ∈ [0 , t i ], we can rewrite (4.3) as˜ Y t − ˜ Y t = u Π i ( X t i ( t i − , x )) − u Π i ( X t i ( t i − , x )) − Z t i t ( ˜ Z r − ˜ Z r ) dB r + Z t i t ( ˜ Z r − ˜ Z r ) V ir dr + Z t i t h f ( r, X r ( t i − , x ) , ˜ Z r ) − f ( r, X r ( t i − , x ) , ˜ Z r ) i dr = u Π i ( X t i ( t i − , x )) − u Π i ( X t i ( t i − , x )) − Z t i t ( ˜ Z r − ˜ Z r ) dB Q i r + Z t i t h f ( r, X r ( t i − , x ) , ˜ Z r ) − f ( r, X r ( t i − , x ) , ˜ Z r ) i dr. V it is bounded on [0 , t i ], we can define an equivalent martingalemeasure Q i on F t i by d Q i d P = E t i ( R · V ir dB r ), then we have that B Q i t , B t − R t V ir dr is a standardBrownian motion under Q i . Since ˜ Z j is bounded on [ t i − , t i ], we can obtain | ˜ Y t − ˜ Y t | E Q i t (cid:12)(cid:12) u Π i ( X t i ( t i − , x )) − u Π i ( X t i ( t i − , x )) (cid:12)(cid:12) + E Q i t (cid:20)Z t i t (cid:12)(cid:12)(cid:12) f ( r, X r ( t i − , x ) , ˜ Z r ) − f ( r, X r ( t i − , x ) , ˜ Z r ) (cid:12)(cid:12)(cid:12) dr (cid:21) L i E Q i t | X t i ( t i − , x ) − X t i ( t i − , x ) | + E Q i t (cid:20)Z t i t K x (1 + | ˜ Z r | ) | X r ( t i − , x ) − X r ( t i − , x ) | dr (cid:21) [ L i + C (1 + L i )∆ t i ] e K b ∆ t i | x − x | , t ∈ [ t i − , t i ) , where the last inequality above comes from standard estimate of forward SDE with deterministicdiffusion term. Thus, we have | ˜ Y t i − − ˜ Y t i − | ( L i e K ∆ t i + K ∆ t i ) | x − x | by letting K = C + K b and K = Ce K b T . According to the definition of u Π i − and the inequality | a ∨ b − a ∨ b | | a − a | ∨ | b − b | , we have | u Π i − ( x ) − u Π i − ( x ) | | ˜ Y t i − − ˜ Y t i − | ∨ | g ( x ) − g ( x ) | [( L i e K ∆ t i + K ∆ t i ) ∨ L n ] | x − x | . Therefore, we have proved that u Π i − is Lipschitz continuous and the Lipschitz constant satisfies L i − ( L i e K ∆ t i + K ∆ t i ) ∨ L n , for i = 2 , · · · , n . Now it suffices to show that ( L i ) i n areuniformly bounded. Noting that L i − ∨ L n ( L i ∨ L n ) e K ∆ t i + K ∆ t i , one can apply Lemma 4.3directly to obtainmax i n L i max i n L i ∨ L n e K T ( L n + K T ) = e K T ( K g + K T ) . (cid:3) At last, we can use the above subsidiary lemmas to obtain the boundness of ˆ Z Π appearing inthe discretization form. Lemma 4.5
Suppose Assumption 3.1 holds. Then, we have ˆ Z Π is bounded on [0 , T ] , uniformly in Π . Proof.
Applying Lemma 4.2 to the auxiliary forward-backward SDE system (4.1), we can get | ˜ Z t | exp( KT )[ M σ exp(2 K b T ) L i + 1] on [ t i − , t i ], where K is defined the same as in the previouslemma and the bound is independent of the initial value of the system (4.1). In turn, reviewingLemma 4.1, we could obtain that ( ˜ Y , ˜ Z ) and ( ¯ Y Π , ˆ Z Π ) coincide on [ t i − , t i ) by setting x = X t i − in (4.1), which indicates that | ˆ Z Π t | exp( KT )[ M σ exp(2 K b T ) L i + 1] on [ t i − , t i ). Then by Lemma4.4, the uniform boundness of L i guarantees that ˆ Z Π is bounded on the whole [0 , T ] and the bounddoes not rely on Π. (cid:3) Now we are ready to give the main result of this paper.13 heorem 4.6
Let Assumption 3.1 hold. Then, we have the following estimate with q = : sup t ∈ [0 ,T ] E (cid:2) | ¯ Y Π t − Y t | (cid:3) + sup t ∈ [0 ,T ] E h | ˆ Y Π t − Y t | i + E "Z T | ˆ Z Π t − Z t | dt C | Π | q , (4.4)max i n E " sup t ∈ [ t i − ,t i ] | ˆ Y Π t − Y t | + sup t ∈ [ t i − ,t i ] | ¯ Y Π t − Y t | C | Π | q (4.5) and sup t ∈ [0 ,T ] E h | ˆ K Π t − K t | i + max i n E " sup t ∈ [ t i − ,t i ] | ˆ K Π t − K t | C | Π | q . (4.6) In addition, if we further assume that g is C b , which means it is twice differentiable and allderivatives are uniformly bounded, we can obtain all the above estimates with q = 1 . Proof.
The whole proof is divided into three steps.
Step 1.
Firstly, we claim the following estimatemax i n E | ¯ Y Π t i − Y t i | + E "Z T | ˆ Z Π t − Z t | dt C | Π | q . (4.7)Recall the discretization form (3.5) and the reflected forward-backward SDE (3.1), and noticethat they are based on the same forward SDE. Denote ∆ Y = Y − ¯ Y Π , ∆ ˆ Y = Y − ˆ Y Π and∆ Z = Z − ˆ Z Π . Apply Itˆo’s formula to ψ (∆ Y t ) for an increasing C function ψ yet to be determinedlater, and we have for t ∈ [ t i − , t i ), ψ (∆ Y t ) = ψ (∆ ˆ Y t i ) + Z t i t ψ ′ (∆ Y s )( f ( s, X s , Z s ) − f ( s, X s , ˆ Z Π s )) ds − Z t i t ψ ′ (∆ Y s )∆ Z s dB s + Z t i t ψ ′ (∆ Y s ) dK s − Z t i t ψ ′′ (∆ Y s ) | ∆ Z s | ds. (4.8)We deduce from Lemma 4.5 and Assumption 3.1 that (cid:12)(cid:12)(cid:12) f ( s, X s , Z s ) − f ( s, X s , ˆ Z Π s ) (cid:12)(cid:12)(cid:12) K z (1 + | Z s | + | ˆ Z Π s | ) | ∆ Z s | K z (1 + 2 M z ) | ∆ Z s | + K z | ∆ Z s | , (4.9)where M z denotes the uniform bound of ˆ Z Π . Plugging the last inequality into (4.8) and using theassumption that ψ is increasing, we have from Lemma 3.2 that | ∆ ˆ Y t | | ∆ Y t | , and ψ (∆ Y t ) ψ (∆ Y t i ) − Z t i t ψ ′ (∆ Y s )∆ Z s dB s + Z t i t K z (1 + 2 M z ) ψ ′ (∆ Y s ) | ∆ Z s | ds + Z t i t (cid:20) K z ψ ′ (∆ Y s ) − ψ ′′ (∆ Y s ) (cid:21) | ∆ Z s | ds + Z t i t ψ ′ (∆ Y s ) dK s ψ (∆ Y t i ) − Z t i t ψ ′ (∆ Y s )∆ Z s dB s + Z t i t K z M z ) | ψ ′ (∆ Y s ) | ds + Z t i t (cid:20) K z ψ ′ (∆ Y s ) + K z − ψ ′′ (∆ Y s ) (cid:21) | ∆ Z s | ds + Z t i t ψ ′ (∆ Y s ) dK s , (4.10)14here the last inequality comes from H¨older’s Inequality.We now choose ψ with the following form ψ ( x ) = 12 K z ( e K z x − K z x − , such that K z ψ ′ + K z − ψ ′′ = 0, and it is straightforward to check that ψ is a C ∞ function, increasingon [0 , ∞ ) and satisfies ψ (0) = 0. Furthermore, recalling Lemma 3.2 and the boundness of Y as thesolution to forward-backward SDE (3.1) with reflection and denoting M ∞ , k ¯ Y Π k ∞ + k Y k ∞ , wecan then get the following properties of ψ on [0 , M ∞ ]:( a ) | ψ ′ ( x ) | C ψ ( x ) , ( b ) K z | x | ψ ( x ) , ( c ) ψ ′ ( x ) C x, (4.11)where C = 4 K z e K z M ∞ and C = C / C , K z (1 + 2 M z ) C and Λ t , e ˜ Ct . Applying Itˆo’s formula again to Λ t ψ (∆ Y t ) and notingthat ∆ Y t > t ψ (∆ Y t ) + K z Z t i t Λ s | ∆ Z s | ds Λ t i ψ (∆ Y t i ) − Z t i t ˜ C Λ s ψ (∆ Y s ) ds + K z M z ) Z t i t Λ s | ψ ′ (∆ Y s ) | ds − Z t i t Λ s ψ ′ (∆ Y s )∆ Z s dB s + Z t i t Λ s ψ ′ (∆ Y s ) dK s . Noting (4.11)-(a), we further haveΛ t ψ (∆ Y t ) + K z Z t i t Λ s | ∆ Z s | ds Λ t i ψ (∆ Y t i ) − Z t i t Λ s ψ ′ (∆ Y s )∆ Z s dB s + Z t i t Λ s ψ ′ (∆ Y s ) dK s . (4.12)In view of (4.11)-(c), the integrand of the last term in (4.12) is estimated as follows:Λ s ψ ′ (∆ Y s ) dK s C Λ s ∆ Y s dK s = C Λ s ( Y s − ¯ Y Π s ) dK s , ∀ s ∈ [ t i − , t i ] . (4.13)Then, the flat-off condition in (3.1), (3.5) and the definition of ˆ Y Π further yield that( Y s − ¯ Y Π s ) dK s =( g ( X s ) − ¯ Y Π s ) dK s = (cid:20) g ( X s ) − E F s (cid:18) ˆ Y Π t i + Z t i s f ( r, X r , ˆ Z Π r ) dr (cid:19)(cid:21) dK s = E F s (cid:20) g ( X s ) − ˆ Y Π t i − Z t i s f ( r, X r , ˆ Z Π r ) dr (cid:21) dK s E F s (cid:20) g ( X s ) − g ( X t i ) − Z t i s f ( r, X r , ˆ Z Π r ) dr (cid:21) dK s . (4.14)Next, we will consider two cases respectively in order to get finer convergence result when wehave additional regularity assumption about the obstacle function g . Let Assumption 3.1 hold inboth cases. We will utilize some standard estimates of forward SDE and use C to denote a universalconstant that only depends on K g , K b , M b and M σ at this stage.15 ase I. If g is Lipschitz, we have E F s [ g ( X s ) − g ( X t i )] K g E F s | X s − X t i | C | Π | (1 + | X s | ) , ∀ s ∈ [ t i − , t i ] . Case II. If g is further in C b , applying Itˆo’s formula to g ( X t ) gives that g ( X s ) − g ( X t i ) = − Z t i s (cid:20) g ′ ( X r ) b ( r, X r ) + 12 g ′′ ( X r ) | σ ( r ) | (cid:21) dr − Z t i s g ′ ( X r ) σ ( r ) dB r . Supposing both | g ′ | and | g ′′ | are bounded by K g and taking conditional expectation, together withthe assumptions of b and σ , we can obtain that E F s [ g ( X s ) − g ( X t i )] C E F s (cid:20)Z t i s (1 + | X r | ) dr (cid:21) = C Z t i s (1 + E F s | X r | ) dr C ( t i − s )(1 + | X s | ) C | Π | (1 + | X s | ) , ∀ s ∈ [ t i − , t i ] . Combining the two cases together and letting q = when we have Lipschitz obstacle funtionand q = 1 when considering C b obstacle with more regularity, we get E F s [ g ( X s ) − g ( X t i )] C | Π | q (1 + | X s | ) C | Π | q (cid:20) E F s (cid:18) sup t T | X t | (cid:19)(cid:21) , ∀ s ∈ [ t i − , t i ] . Plugging it back to (4.14) and making use of Lemma 4.5, we have( Y s − ¯ Y Π s ) dK s E F s (cid:20) g ( X s ) − g ( X t i ) − Z t i s f ( r, X r , ˆ Z Π r ) dr (cid:21) dK s (cid:20) C | Π | q (cid:18) E F s (cid:18) sup t T | X t | (cid:19)(cid:19) + Z t i s (cid:16) M f + α | ˆ Z Π r | (cid:17) dr (cid:21) dK s h C | Π | q (cid:0) E F s [ X ] (cid:1) + (cid:16) M f + α | M z | (cid:17) | Π | i dK s . (4.15)Here we denote X , sup t T | X t | , which is a F T -measurable and square-integrable random vari-able. Note that we will let the constant C in the following further depend on T, M z , M ∞ , E |X | , E | K T | and all the constants appearing in Assumption 3.1, which may vary from line to line as before. Inturn, plugging the above estimate into (4.13) and taking expectation, we can obtain that E (cid:20)Z t i t Λ s ψ ′ (∆ Y s ) dK s (cid:21) C E (cid:20)Z t i t Λ s ( Y s − ¯ Y Π s ) dK s (cid:21) C | Π | q Λ t i E (cid:20)Z t i t (cid:0) E F s [ X ] (cid:1) dK s (cid:21) = C | Π | q Λ t i E [(1 + X ) ( K t i − K t )] . Since ψ ′ ( x ) = e K z x − , M ∞ ] and Z ∈ H ([0 , T ]; R m ), taking expectation onboth sides of (4.12) gives that for any t ∈ [ t i − , t i ], E (cid:20) Λ t ψ (∆ Y t ) + K z Z t i t Λ s | ∆ Z s | ds (cid:21) E [Λ t i ψ (∆ Y t i )] + C | Π | q Λ t i E [(1 + X ) ( K t i − K t )] , (4.16)16hich further implies E (cid:2) ψ (∆ Y t i − ) (cid:3) e ˜ C ∆ t i { E [ ψ (∆ Y t i )] + C | Π | q E [(1 + X ) ( K t i − K t i − )] } e ˜ C ∆ t i E [ ψ (∆ Y t i )] + C | Π | q E [(1 + X ) ( K t i − K t i − )]by letting t = t i − and noting e ˜ C ∆ t i Λ T . Applying Lemma 4.3 again and noticing the fact that∆ Y t n = 0, we can further obtainmax i n E [ ψ (∆ Y t i )] e ˜ CT " C | Π | q n X i =1 E [(1 + X ) ( K t i − K t i − )] = C | Π | q E [(1 + X ) K T ] C | Π | q [ E (1 + |X | ) + E | K T | ] C | Π | q . (4.17)Thus, we can conclude from (4.11)-(b) and Lemma 3.2 thatmax i n E | ∆ ˆ Y t i | max i n E | ∆ Y t i | C | Π | q . Setting t = t i − again in (4.16) and taking summation from i = 1 to n on both sides give rise to E "Z T | ∆ Z s | ds n X i =1 E "Z t i t i − Λ s | ∆ Z s | ds C " E [Λ t n ψ (∆ Y t n )] + C | Π | q Λ T n X i =1 E [(1 + X ) ( K t i − K t i − )] C | Π | q , through the same arguments as in (4.17) and the fact ψ (0) = 0. Consequently, (4.7) follows, and itis easy to check the other part of (4.4). Step 2.
Taking supremum over [ t i − , t i ] on both sides of (4.12), we can observe that E " sup t ∈ [ t i − ,t i ] Λ t ψ (∆ Y t ) E [Λ t i ψ (∆ Y t i )] + E " sup t ∈ [ t i − ,t i ] (cid:12)(cid:12)(cid:12)(cid:12)Z t i t Λ s ψ ′ (∆ Y s )∆ Z s dB s (cid:12)(cid:12)(cid:12)(cid:12) + E " sup t ∈ [ t i − ,t i ] Z t i t Λ s ψ ′ (∆ Y s ) dK s . (4.18)For the second term in the above inequality, applying B-D-G inequality and using (4.11)-(a)and Young’s Inequality, we obtain E " sup t ∈ [ t i − ,t i ] (cid:12)(cid:12)(cid:12)(cid:12)Z t i t Λ s ψ ′ (∆ Y s )∆ Z s dB s (cid:12)(cid:12)(cid:12)(cid:12) C E "Z t i t i − | Λ s ψ ′ (∆ Y s )∆ Z s | ds / C E " sup t ∈ [ t i − ,t i ] Λ t ψ (∆ Y t ) Z t i t i − Λ s | ∆ Z s | ds / E " sup t ∈ [ t i − ,t i ] Λ t ψ (∆ Y t ) + C E "Z t i t i − Λ s | ∆ Z s | ds .
17n turn, it follows that the third term is equal to E "Z t i t i − Λ s ψ ′ (∆ Y s ) dK s C | Π | q Λ t i E [(1 + X )( K t i − K t i − )] C | Π | q Λ T E [(1 + X ) K T ] . Plugging them back into (4.18) gives E " sup t ∈ [ t i − ,t i ] Λ t ψ (∆ Y t ) E [Λ t i ψ (∆ Y t i )] + C E "Z t i t i − Λ s | ∆ Z s | ds + C | Π | q Λ T E [(1 + X ) K T ] T E [ ψ (∆ Y t i )] + C Λ T E "Z T | ∆ Z s | ds + C | Π | q Λ T E [(1 + X ) K T ] C E [ ψ (∆ Y t i )] + C | Π | q . Then by the result of the first step, we deduce thatmax i n E " sup t ∈ [ t i − ,t i ] Λ t ψ (∆ Y t ) C max i n E [ ψ (∆ Y t i )] + C | Π | q C | Π | q , and thus (4.5) follows. Step 3.
Now we need to check the assertion related to K . From (3.1), (3.5) and (3.6), we haveˆ K Π t = ˆ K Π t i = ˆ Y Π0 − [ ¯ Y Π t { t = t i } + ˆ Y Π t { t = t i } ] − Z t f ( r, X r , ˆ Z Π r ) dr + Z t ˆ Z Π r dB r ,K t = Y − Y t − Z t f ( r, X r , Z r ) dr + Z t Z r dB r . Denote ∆ K , K − ˆ K Π . It then follows that∆ K t = ∆ ˆ Y − [∆ Y t { t = t i } + ∆ ˆ Y t { t = t i } ] − Z t [ f ( r, X r , Z r ) − f ( r, X r , ˆ Z Π r )] dr + Z t ∆ Z r dB r . Applying B-D-G inequality and moment inequality, together with the results proved in theformer two steps and the estimate (4.9), we then deduce that18 " sup t ∈ [ t i − ,t i ] | ∆ K t | E | ∆ ˆ Y | + E " sup t ∈ [ t i − ,t i ] (cid:16) | ∆ Y t | + | ∆ ˆ Y t | (cid:17) + E " sup t ∈ [ t i − ,t i ] (cid:12)(cid:12)(cid:12)(cid:12)Z t ∆ Z r dB r (cid:12)(cid:12)(cid:12)(cid:12) + E " sup t ∈ [ t i − ,t i ] (cid:12)(cid:12)(cid:12)(cid:12)Z t [ f ( r, X r , Z r ) − f ( r, X r , ˆ Z Π r )] dr (cid:12)(cid:12)(cid:12)(cid:12) [ E | ∆ ˆ Y | ] + " E sup t ∈ [ t i − ,t i ] (cid:16) | ∆ Y t | + | ∆ ˆ Y t | (cid:17)! + E "(cid:18)Z t i | ∆ Z r | dr (cid:19) + E (cid:20)Z t i K z (1 + 2 M z + | ∆ Z r | ) | ∆ Z r | dr (cid:21) C | Π | q + C (cid:18) E (cid:20)Z t i (1 + | ∆ Z r | ) dr (cid:21)(cid:19) (cid:18) E (cid:20)Z t i | ∆ Z r | dr (cid:21)(cid:19) C | Π | q + C | Π | q ( T + | Π | q ) C | Π | q . Thus we obtain sup t ∈ [0 ,T ] E h | ˆ K Π t − K t | i max i n E " sup t ∈ [ t i − ,t i ] | ˆ K Π t − K t | C | Π | q , and the proof is complete. (cid:3) We have characterized the continuous and discrete optimal investment stopping problem sepa-rately and provided the convergence result, which comes down to the convergence from discretely tocontinuously quadratic reflected BSDE, via the tools of quadratic BSDE with bounded terminals.While at present, we need the bounded assumption due to technical restriction when we try to applythe method in Lipschitz case to Quadratic, and what we discussed here is actually a specific formof quadratic generator without y involved, since the BSDE essentially originates from the utilitymaximization problem. We may further consider the real discrete scheme for the quadratic reflectedBSDE which will indicate the way to solve the optimal investment stopping problem numerically,as well as generalize the settings about generator and terminal value in future research. A Proof of Lemma 4.2
Proof.
As in the literature, we suppose that the functions b, g and f in the forward-backward SDE(4.2) are differentiable with respect to x and z firstly. Thus the solution ( X, Y, Z ) is differentiablewith respect to x and ( ∇ X, ∇ Y, ∇ Z ) satisfies the following SDE and BSDE ∇ X t = 1 + Z t ∇ b ( s, X s ) ∇ X s ds, ∇ Y t = ∇ g ( X T ) ∇ X T − Z Tt ∇ Z s dB s + Z Tt [ ∇ x f ( s, X s , Z s ) ∇ X s + ∇ z f ( s, X s , Z s ) ∇ Z s ] ds. (A.1)19oreover, we can deduce from Assumption 3.1 that the coefficients appearing in the aboveequations satisfy |∇ b ( t, x ) | K b , |∇ g ( x ) | K g , |∇ x f ( t, x, z ) | K x (1 + | z | ) and |∇ z f ( t, x, z ) | K z (1 + 2 | z | ) respectively.Thanks to the Mallivian calculus, it is classical to show that a version of ( Z t ) t ∈ [0 ,T ] is given by( ∇ Y t ( ∇ X t ) − σ ( t )) t ∈ [0 ,T ] . Then, noting that both |∇ X t | and | ( ∇ X t ) − | are bounded by e K b T forany t ∈ [0 , T ], we have the following estimate |∇ x f ( s, X s , Z s ) ∇ X s | K x (1 + | Z s | ) |∇ X s | K x (1 + |∇ Y t ( ∇ X t ) − σ ( t ) | ) e K b T K x e K b T (1 + e K b T M σ |∇ Y t | ) . (A.2)Let K , K x M σ e K b T . Applying Itˆo-Tanaka’s formula to e Kt |∇ Y t | , we obtain e Kt |∇ Y t | = e KT |∇ g ( X T ) ∇ X T | − Z Tt Ke Ks |∇ Y s | ds − Z Tt sgn ( ∇ Y s ) e Ks ∇ Z s dB s + Z Tt sgn ( ∇ Y s ) e Ks [ ∇ x f ( s, X s , Z s ) ∇ X s + ∇ z f ( s, X s , Z s ) ∇ Z s ] ds − Z Tt e Ks dL s , (A.3)where L is a real-valued, adapted, increasing and continuous process known as local time of ∇ Y atlevel 0. The BMO property of Z ∗ B and the fact that |∇ z f ( t, x, z ) | K z (1 + 2 | z | ) guarantee that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z · ∇ z f ( s, X s , Z s ) dB s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) BMO = sup τ ∈ [0 ,T ] E "Z Tτ |∇ z f ( s, X s , Z s ) | ds (cid:12)(cid:12)(cid:12) F τ C τ ∈ [0 ,T ] E "Z Tτ | Z s | ds (cid:12)(cid:12)(cid:12) F τ = C (1 + || Z ∗ B || BMO ) < ∞ , which further implies that E ( R · ∇ z f ( s, X s , Z s ) dB s ) t is a uniformly integrable martingale. In turn,we are able to apply Girsanov theorem and rewrite (A.3) under the equivalent probability Q as e Kt |∇ Y t | e KT |∇ g ( X T ) ∇ X T | − Z Tt Ke Ks |∇ Y s | ds − Z Tt sgn ( ∇ Y s ) e Ks ∇ Z s dB Q s + Z Tt e Ks ( K x e K b T + K |∇ Y s | ) ds e KT |∇ g ( X T ) ∇ X T | − Z Tt sgn ( ∇ Y s ) e Ks ∇ Z s dB Q s + 1 K K x e K b T e KT , where we used the estimate (A.2) and the fact dL t >
0, and B Q t , B t − R t ∇ z f ( s, X s , Z s ) ds isa standard Brownian motion under Q . Then, taking conditional expectation on both sides andnoticing that ∇ Z is actually the second component of the solution to BSDE in (A.1), we obtain e Kt |∇ Y t | E Q (cid:20) e KT |∇ g ( X T ) ∇ X T | + 1 K K x e K b T e KT (cid:12)(cid:12)(cid:12) F t (cid:21) e KT e K b T [ K g + K x /K ] . Z t again, we can finally deduce that for any t ∈ [0 , T ], | Z t | = |∇ Y t ( ∇ X t ) − σ ( t ) | e K b T M σ |∇ Y t | e KT [ e K b T M σ K g + 1] . We conclude the proof by noting that when b, g and f are not differentiable, one can also provethe result by a standard approximation and stability results for BSDEs. (cid:3) References [1] E. Bayraktar, S. Yao. Quadratic reflected BSDEs with unbounded obstacles.
Stochastic Processes andtheir Applications , (4): 1155-1203, 2012.[2] P. Briand and Y. Hu. BSDE with quadratic growth and unbounded terminal value. Probability Theoryand Related Fields , (4): 604618, 2006.[3] P. Briand and Y. Hu. Quadratic BSDEs with convex generators and unbounded terminal conditions. Probability Theory and Related Fields , : 543567, 2008.[4] J.F. Chassagneux. An introduction to the numerical approximation of BSDEs, Lecture notes in Secondschool of CREMMA , 2012.[5] F. Delbaen, Y. Hu and A. Richou. On the uniqueness of solutions to quadratic BSDEs with convexgenerators and unbounded terminal conditions.
Annales de lInstitut Henri Poincare - Probabilites etStatistiques , (2): 559-574, 2011.[6] F. Delbaen, Y. Hu and A. Richou. On the uniqueness of solutions to quadratic BSDEs with convexgenerators and unbounded terminal conditions: The critical case. Discrete and Continuous DynamicalSystems , (11): 5273-5283,2015.[7] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez. Reflected solutions of backwardSDE’s, and related obstacle problems for PDE’s. The Annals of Probability , (2): 702-737, 1997.[8] Y. Hu, P. Imkeller and M. M¨uller. Utility maximization in incomplete markets. The Annals of AppliedProbability , (3): 1691-1712, 2005.[9] Y. Hu, G. Liang and S. Tang. Exponential utility maximization and indifference valuation with un-bounded payoffs. arXiv:1707.00199v3 , 2018.[10] I. Karatzas and H. Wang. Utility maximization with discretionary stopping. SIAM Journal on Controland Optimization , (1): 306-329, 2000.[11] I. Karatzas and H. Wang. A barrier option of American type. Applied Mathematics and Optimization , (3): 259-279, 2000.[12] M. Kobylanski. Backward stochastic differential equations and partial differential equations withquadratic growth. The Annals of Probability , (2): 558-602, 2000.[13] M. Kobylanski, J. P. Lepeltier, M. C. Quenez and S. Torres. Reflected BSDE with superlinear quadraticcoefficient. Probability and Mathematical Statistics , (1): 51-83, 2002.[14] J.P. Lepeltier and M. Xu. Reflected BSDE with quadratic growth and unbounded terminal value, arXiv:0711.0619v1 , 2007.[15] J. Ma and J. Zhang. Representations and regularities for solutions to BSDEs with reflections, StochasticProcesses and their Applications , (4): 539-569, 2005.[16] M-A. Morlais. Utility maximization in a jump market model, Stochastics , (1): 1-27, 2009.
17] M-A. Morlais. Quadratic BSDEs driven by a continuous martingale and applications to the utilitymaximization problem,
Finance and Stochastics , (1): 121-150, 2009.[18] A. Richou. Numerical simulation of BSDEs with drivers of quadratic growth. The Annals of AppliedProbability , (5): 1933-1964, 2011.[19] A. Richou. Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition. Stochastic Processes and their Applications , (9): 3173-3208, 2012.(9): 3173-3208, 2012.