Optimal investment in illiquid market with search frictions and transaction costs
OOPTIMAL INVESTMENT IN ILLIQUID MARKETWITH SEARCH FRICTIONS AND TRANSACTION COSTS
JIN HYUK CHOI, TAE UNG GANG
Abstract.
We consider an optimal investment problem to maximize expected utility of the termi-nal wealth, in an illiquid market with search frictions and transaction costs. In the market model,an investor’s attempt of transaction is successful only at arrival times of a Poisson process, and theinvestor pays proportional transaction costs when the transaction is successful. We characterizethe no-trade region describing the optimal trading strategy. We provide asymptotic expansionsof the boundaries of the no-trade region and the value function, for small transaction costs. Theasymptotic analysis implies that the effects of the transaction costs are more pronounced in themarket with less search frictions.
Keywords : stochastic control, optimal investment, illiquidity, transaction costs, search frictions1.
Introduction
Understanding the effects of liquidity on optimal investment is one of the main topics in math-ematical finance and financial economics. According to [46], stated simply, liquidity is the ease oftrading a security. Various sources of illiquidity include exogenous transaction costs, search fric-tions such as the difficulty of locating a counterparty with whom to trade, and price impacts dueto private information. This paper studies an optimal investment problem in the market modelwith two different types of illiquidity: search frictions and transaction costs.Under the assumption of the perfect liquidity, the pioneering papers [36, 37] formulate theoptimal investment problem (so-called Merton’s portfolio problem) with a geometric Brownianmotion and a CRRA (constant relative risk aversion) investor, and show that the optimal solutionis to keep the constant fraction of wealth invested in the risky asset. With the same assumptionof perfect liquidity, more general stochastic processes and utility functions have been considered toobtain more general characterizations of the optimal investment strategies (e.g., [26, 27, 30, 29, 24]).In the optimal investment problems, the assumption of perfect liquidity can be relaxed by con-sidering search frictions in the market. As Table 1 in [3] shows, many financial assets are illiquid inthe sense that it is difficult to find a counterparty who is willing to trade. One way to incorporatethis type of illiquidity, search frictions, into the optimal investment problem is to impose somerestrictions on trade times. In the classical Merton framework, [43] considers an investor who isallowed to change portfolio only at times which are multiple of a constant h >
0, and [44, 35, 3]assume that an illiquid asset can only be traded on the arrival of a randomly occurring trading
Department of mathematics, Ulsan National Institute of Science and Technology; [email protected],[email protected]. For instance, [32, 4, 19, 12] study how asymmetric information effects price impact and optimal trading strategyin equilibrium. [2, 21, 41, 42] consider optimal order execution problems with exogenously given price impacts. Here, perfect liquidity assumption means that assets can be traded in any quantity and at any moment in time,without any transaction costs. a r X i v : . [ q -f i n . M F ] J a n PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 2 opportunity that is represented by jump times of a Poisson process. [39, 13] consider an optimalinvestment/consumption problem under the assumption that the asset price is observed only atthe random trade times. [20] complicates the model by using random intensity of trade times,regime-switching, and liquidity shocks. In [17], a risky asset can be traded only on deterministictime intervals and the investor pays proportional transaction costs.Transaction costs (such as order processing fees or transaction taxes) are another source ofmarket illiquidity, and the optimal investment problems with transaction costs have been extensivelystudied in mathematical finance community. [34, 18, 45] study the model in [37] with the assumptionthat proportional transaction costs are levied on each transaction, and show (with different level ofmathematical rigorosity) that it is optimal to keep the fraction of wealth invested in the risky assetin an interval so called no-trade region . The boundaries of the no-trade region are characterized interms of the free-boundaries determined by the HJB (Hamilton-Jacobi-Bellman) equation of thecontrol problem. The models with transaction costs and multiple risky assets have been studied(e.g., [1, 33, 38, 9] for costs on all assets and [16, 7, 23, 11] for costs on only one assets) to characterizethe value function or no-trade region. Because the HJB equations in the models with proportionaltransaction costs do not have explicit solutions, the asymptotic analysis for small transaction costshas been studied (e.g., [45, 25, 22, 10] for models with a single risky asset, and [8, 40, 7, 11] formodels with multiple risky assets). More general stochastic processes have been also considered inthe framework of optimal investment with transaction costs (e.g., [14, 15, 5]).In this paper, we merge the aforementioned frameworks and analyze an optimal investmentproblem in a market model with both search frictions and transaction costs. We consider theclassical Merton’s portfolio problem with a log-utility investor, whose goal is to maximize theexpected utility of wealth at the terminal time
T >
0. We assume that the investor’s attemptof trading is successful only when a Poisson process with intensity λ jumps (as in [44, 35]), andthe investor needs to pay proportional transaction costs for successful trading (as in [18, 45]). Weshow that there exists a unique classical solution of the HJB equation and provide the standardverification argument. As in the aforementioned models for transaction costs, the optimal tradingstrategy is characterized by a no-trade region: if the investor can trade at time t ∈ [0 , T ), thenthere are two constants 0 ≤ y ( t ) ≤ ¯ y ( t ) ≤ y ( t ) , ¯ y ( t )]. Strict concavityof the value function uniquely determines the boundaries y ( t ) and ¯ y ( t ) of the no-trade region.We provide asymptotic expansions of the value function and the no-trade boundaries for smalltransaction costs. We differentiate the coefficients of the first-order terms with respect to the searchfriction parameter λ , and examine the signs of these quantities as λ → ∞ . This analysis impliesthat the effects of the transaction costs are more pronounced (more widening effect of the no-traderegion and more diminishing effect of the value function) in the market with less search frictions. The remainder of the paper is organized as follows. Section 2 describes the model. In Section3, we provide the verification argument and the strict concavity of the value function. In Section4, we characterize the optimal trading strategy in terms of the no-trade region, and present someproperties of the no-trade region. In Section 5, we provide asymptotic analysis for small transactioncosts. Proof of the technical lemmas can be found in Appendix. To be more specific, let 0 < λ < λ and consider two markets M and M with search friction parameters λ and λ , respectively. In words, M has less search frictions than M . Our result implies that if we increase thetransaction costs, the widening speed of the no-trade region in M is faster than that in M . Similarly, if we increasethe transaction costs, the diminishing speed of the optimal value in M is faster than that in M . PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 3 The Model
Let (Ω , F , ( F t ) t ≥ , P ) be a filtered probability space satisfying the usual conditions. Under thefiltration, we assume that ( B t ) t ≥ is a standard Brownian motion and ( P t ) t ≥ is a Poisson processwith intensity λ >
0. We assume that B and P are independent processes.We consider a market that has two different types of illiquidity: (i) the investor’s attempt oftrading is successful only when the Poisson process ( P t ) t ≥ jumps, and (ii) the investor needs topay proportional transaction costs for successful trading.To be more specific, we consider a financial market consisting of a bond and a stock, whose priceprocesses ( S (0) t ) t ≥ and ( S t ) t ≥ are given by the following stochastic differential equations (SDEs): dS (0) t = S (0) t rdt (2.1) dS t = S t ( µdt + σdB t ) , (2.2)where µ, r, σ, S (0)0 , S are constants and σ, S (0)0 , S are assumed to be strictly positive.The proportional transaction costs are described by two constants (cid:15) ∈ [0 ,
1) and ¯ (cid:15) ∈ [0 , ∞ ): theinvestor gets (1 − (cid:15) ) S t for selling one share of the stock, and pays (1 + ¯ (cid:15) ) S t for purchasing one shareof the stock.Let W (1) t be the amount of wealth invested in the stock and W (0) t be the amount of wealth inthe bond, at time t ≥
0. If the investor tries to obtain the stock worth M s at time s ∈ [0 , t ], then W (1) t = w (1)0 + (cid:90) t W (1) s − (cid:0) µds + σdB s (cid:1) + (cid:90) t M s dP s ,W (0) t = w (0)0 + (cid:90) t W (0) s − rds + (cid:90) t (cid:0) (1 − (cid:15) ) M − s − (1 + ¯ (cid:15) ) M + s (cid:1) dP s , (2.3)where we use notation x ± = max { , ± x } for x ∈ R , and the constants w (1)0 ≥ w (0)0 ≥ M t ) t ≥ is called admissible if it is a predictable process and the corre-sponding total wealth process W := W (0) + W (1) is nonnegative all the time. This nonnegativitycondition is equivalent to W (0) t ≥ W (1) t ≥ t ≥
0, because the rebalancing times arediscrete. Therefore, an admissible strategy M satisfies − W (1) t − ≤ M t ≤ W (0) t − (cid:15) , t ≥ . (2.4)For an admissible strategy M and the corresponding solutions W (1) and W (0) of the SDEs (2.3),let X t := W (1) t /W t be the fraction of the total wealth invested in the stock market at time t . Then,the inequalities in (2.4) imply that 0 ≤ X t ≤ W, X ): dW t = ( r (1 − X t − ) + µX t − ) W t − dt + σX t − W t − dB t − (cid:0) ¯ (cid:15)M + t + (cid:15)M − t (cid:1) dP t ,dX t = X t − (1 − X t − ) (cid:0) ( µ − r − σ X t − ) dt + σdB t (cid:1) + (cid:32) M t + (cid:0) ¯ (cid:15)M + t + (cid:15)M − t (cid:1) X t − W t − − ¯ (cid:15)M + t − (cid:15)M − t (cid:33) dP t , (2.5) Therefore, bigger λ implies more frequent trading opportunities (less search frictions), on average. Indeed, for s > t and A = { W (1) t < , W (0) t > } or { W (1) t > , W (0) t < } , we observe that P ( W s < | A ) ≥ P ( P t = P s and W s < | A ) > . PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 4 where the initial conditions are W = w := w (1)0 + w (0)0 and X = x := w (1)0 /w . We assumethat the initial total wealth is strictly positive, w >
0. The nonnegativity of w (1)0 and w (1)0 imply0 ≤ x ≤ T > ( M t ) t ∈ [0 ,T ] E [ln( W T )] , (2.6)where the supremum is taken over all admissible trading strategies. Remark . If λ = ∞ and ¯ (cid:15) = (cid:15) = 0, then our model becomes the classical Merton’s portfolioproblem. The case of ¯ (cid:15) = (cid:15) = 0 and λ < ∞ is studied in [35].3. Value function
Let V be the value function of the control problem (2.6): V ( t, x, w ) = sup ( M s ) s ∈ [ t,T ] E [ln( W T ) |F t ] (cid:12)(cid:12)(cid:12) ( X t ,W t )=( x,w ) . (3.1)As usual, the scaling property of the wealth process and the property of log-utility enable us toconjecture the form of the value function as V ( t, x, w ) = ln( w ) + v ( t, x )for a function v (we verify this in Theorem 3.3). Then, the HJB equation for (3.1) becomes v ( T, x ) , v t + x (1 − x )( µ − r − σ x ) v x + σ x (1 − x ) v xx + ( µ − r ) x + r − σ x − λv + λ sup y ∈ [0 , (cid:16) v ( t, y ) − ln (cid:16) (cid:15)y (cid:15)x (cid:17) { x
There exists a unique v ∈ C ([0 , T ] × [0 , ∩ C , ([0 , T ] × (0 , that satisfies thefollowings: (i) v satisfies the HJB equation (3.2) for ( t, x ) ∈ (0 , T ) × (0 , .(ii) For x ∈ { , } , the map t (cid:55)→ v ( t, x ) is continuously differentiable on [0 , T ] and satisfies v ( T, x ) , v t ( t, x ) + ( µ − r ) x + r − σ x − λv ( t, x )+ λ sup y ∈ [0 , (cid:16) v ( t, y ) − ln (cid:16) (cid:15)y (cid:15)x (cid:17) { x
Remark . In Lemma 3.1, we present the differential equations for x ∈ (0 ,
1) and x ∈ { , } as(3.2) and (3.3). The reason we separate these two cases is that v x and v xx may not be continuouslyextended to the endpoints x ∈ { , } . PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 5
Theorem 3.3.
Let V be the value function in (3.1) and v be as in Lemma 3.1. Then, for ( t, x ) ∈ [0 , T ] × [0 , , V ( t, x, w ) = ln( w ) + v ( t, x ) . (3.4) Proof.
Without loss of generality, we prove V (0 , x , w ) = ln( w )+ v (0 , x ). Let M be an admissibletrading strategy and ( W, X ) be the corresponding solution of (2.5). Let τ n := inf { t ≥ P t = n } .If X τ n = 0 (resp., X τ n = 1), then for τ n ≤ t < τ n +1 , X t = 0 , dW t = rW t dt (cid:0) resp., X t = 1 , dW t = µW t dt + σW t dB t (cid:1) . (3.5)We apply Ito’s formula to (cid:0) ln( W t ) + v ( t, X t ) (cid:1) t ∈ [0 ,T ] and use (2.5) and (3.5) to obtainln( W T ) − ln( w ) − v (0 , x )= (cid:90) T (cid:18)(cid:16) v t ( t, x ) + x (1 − x )( µ − r − σ x ) v x ( t, x ) + σ x (1 − x ) v xx ( t, x )+ ( µ − r ) x + r − σ x (cid:17) · {
1] such thatˆ y ( t, x ) ∈ argmax y ∈ [0 , (cid:16) v ( t, y ) − ln (cid:16) (cid:15)y (cid:15)x (cid:17) { x For t ∈ [0 , T ) , the maps ( a, b ) (cid:55)→ ˜ V ( t, a, b ) and x (cid:55)→ v ( t, x ) are strictly concave. Indeed, the SDE in (2.5) without dP t term has a unique (explicit) solution. The unique solution of (2.5) can beobtained by patching the unique solutions on time intervals between the jump times of the Poisson process, with thejump size described by the coefficient of dP t term. PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 7 Proof. Without loss of generality, we prove the statement for t = 0 case. Let ( a , b ) (cid:54) = ( a , b )be elements of [0 , ∞ ) \ { (0 , } and θ ∈ (0 , a , b ) and( a , b ), and we denote them by ˆ M and ˆ M , respectively. Then the structure of the SDE in(2.3) implies that M θ := (1 − θ ) ˆ M + θ ˆ M is an admissible trading strategy with initial position( a θ , b θ ) := ((1 − θ ) a + θa , (1 − θ ) b + θb ) and satisfies (1 − θ ) W a ,b , ˆ M T + θW a ,b , ˆ M T ≤ W a θ ,b θ ,M θ T . (3.13)We also observe that P (cid:16) W a ,b , ˆ M T (cid:54) = W a ,b , ˆ M T (cid:17) ≥ P (cid:16) W a ,b , ˆ M T (cid:54) = W a ,b , ˆ M T and P T = 0 (cid:17) = P ( P T = 0) · P (cid:16) W a ,b , ˆ M T (cid:54) = W a ,b , ˆ M T (cid:12)(cid:12) P T = 0 (cid:17) = e − λT · P (cid:18) ( a − a ) e rT + ( b − b ) e ( µ − σ ) T + σB T (cid:54) = 0 (cid:19) = e − λT > , (3.14)where the second equality is due to the independence of B and P , and the last equality is due to( a , b ) (cid:54) = ( a , b ) and the fact that B T is a continuous random variable. By these observations, weobtain the strict concavity of ˜ V :(1 − θ ) ˜ V (0 , a , b ) + θ ˜ V (0 , a , b ) = E (cid:104) (1 − θ ) ln (cid:16) W a ,b , ˆ M T (cid:17) + θ ln (cid:16) W a ,b , ˆ M T (cid:17)(cid:105) < E (cid:104) ln (cid:16) (1 − θ ) W a ,b , ˆ M T + θW a ,b , ˆ M T (cid:17)(cid:105) ≤ E (cid:104) ln (cid:16) W a θ ,b θ ,M θ T (cid:17)(cid:105) ≤ ˜ V (0 , a θ , b θ ) , where the first inequality is due to the strict concavity of logarithm and (3.14), and the secondinequality is from (3.13). Therefore, the map ( a, b ) (cid:55)→ ˜ V (0 , a, b ) is strictly concave. This alsoimplies that the map x (cid:55)→ ˜ V (0 , − x, x ) is strictly concave. Finally, Theorem 3.3 and the relation(3.12) connect ˜ V and v as ˜ V (0 , − x, x ) = V (0 , x, 1) = v (0 , x ) , and we conclude that the map x (cid:55)→ v (0 , x ) is strict concave. (cid:3) Optimal strategy In this section, we show that the optimal strategy can be characterized in terms of the no-traderegion . We start with the construction of the candidate boundary points y and ¯ y of the no-traderegion in the following lemma. Lemma 4.1. For t ∈ [0 , T ) , there exist ≤ y ( t ) ≤ ¯ y ( t ) ≤ such that (cid:8) y ( t ) (cid:9) = argmax y ∈ [0 , (cid:16) v ( t, y ) − ln(1 + ¯ (cid:15)y ) (cid:17) , (cid:8) ¯ y ( t ) (cid:9) = argmax y ∈ [0 , (cid:16) v ( t, y ) − ln(1 − (cid:15)y ) (cid:17) . The inequality (3.13) becomes strict on the event (cid:110) ω ∈ Ω : ∃ t ∈ [0 , T ] such that ∆ P t ( ω ) = 1 and ˆ M t ( ω ) ˆ M t ( ω ) < (cid:111) . PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 8 To be more specific, the following statements hold:(i) The map y (cid:55)→ v ( t, y ) − ln(1 + ¯ (cid:15)y ) strictly increases (decreses) on y ∈ [0 , y ( t )] ( y ∈ [ y ( t ) , ).If < y ( t ) < , then y ( t ) is the unique solution of the equation v x ( t, x ) = ¯ (cid:15) (cid:15)x .(ii) The map y (cid:55)→ v ( t, y ) − ln(1 − (cid:15)y ) strictly increases (decreses) on y ∈ [0 , ¯ y ( t )] ( y ∈ [¯ y ( t ) , ).If < ¯ y ( t ) < , then ¯ y ( t ) is the unique solution of the equation v x ( t, x ) = − (cid:15) − (cid:15)x .Proof. Let t ∈ [0 , T ) be fixed. We consider the map z ∈ [0 , (cid:15) ] (cid:55)→ ˜ V ( t, − (1 + ¯ (cid:15) ) z, z ), where ˜ V is defined in (3.11). Theorem 3.3 and (3.12) imply that this map is differentiable (we denote thepartial derivatives as ˜ V a and ˜ V b ), and the map is strictly concave due to Proposition 3.4. Therefore,its derivative D ( t, z ) := − (1 + ¯ (cid:15) ) ˜ V a ( t, − (1 + ¯ (cid:15) ) z, z ) + ˜ V b ( t, − (1 + ¯ (cid:15) ) z, z ) (4.1)is strictly increasing in z ∈ (0 , (cid:15) ), and there exists a unique z ( t ) ∈ [0 , (cid:15) ] such that (cid:40) D ( t, z ) > z ∈ (0 , z ( t )) ,D ( t, z ) < z ∈ ( z ( t ) , (cid:15) ) . (4.2)Obviously, in case z ( t ) ∈ (0 , (cid:15) ), z ( t ) is the unique solution of D ( t, z ) = 0.By Theorem 3.3 and (3.12), we observe that for y ∈ [0 , v ( t, y ) − ln(1 + ¯ (cid:15)y ) = V (cid:16) t, y, (cid:15)y (cid:17) = ˜ V (cid:16) t, − y (cid:15)y , y (cid:15)y (cid:17) . We take derivative with respect to y above and use (4.1) to obtain ∂∂y (cid:16) v ( t, y ) − ln(1 + ¯ (cid:15)y ) (cid:17) = (cid:15)y ) D (cid:16) t, y (cid:15)y (cid:17) . (4.3)Now we define y ( t ) := z ( t )1 − ¯ (cid:15)z ( t ) ∈ [0 , y (cid:55)→ y (cid:15)y is strictly increasing on [0 , z ( t ) in (4.2) implies that D (cid:16) t, y (cid:15)y (cid:17) > y ∈ (cid:0) , y ( t ) (cid:1) ,D (cid:16) t, y (cid:15)y (cid:17) < y ∈ (cid:0) y ( t ) , (cid:1) . (4.4)Also, in case y ( t ) ∈ (0 , y ( t ) is the unique solution of D (cid:16) t, y (cid:15)y (cid:17) = 0. From (4.3) and (4.4), weconclude that statement (i) holds and (cid:8) y ( t ) (cid:9) = argmax y ∈ [0 , (cid:16) v ( t, y ) − ln(1 + ¯ (cid:15)y ) (cid:17) . By the same way, we conclude that max y ∈ [0 , (cid:0) v ( t, y ) − ln(1 − (cid:15)y ) (cid:1) has the unique maximizerdenoted by ¯ y ( t ) and statement (ii) holds.It only remains to check the inequality y ( t ) ≤ ¯ y ( t ). If ¯ y ( t ) = 1, then y ( t ) ≤ ¯ y ( t ) is obvious. If¯ y ( t ) < 1, then v x ( t, ¯ y ( t )) ≤ − (cid:15) − (cid:15) ¯ y ( t ) ≤ ¯ (cid:15) (cid:15) ¯ y ( t ) by (ii). From (4.3) and (4.4), we obtain y ( t ) ≤ ¯ y ( t ). (cid:3) In the next theorem, we explicitly characterize the optimizer ˆ y in (3.8) in terms of y and ¯ y inLemma 4.1. Theorem 4.2. For t ∈ [0 , T ) , the argmax in (3.8) is a singleton, and ˆ y has the following expression: ˆ y ( t, x ) = y ( t ) , if x ∈ (cid:2) , y ( t ) (cid:1) x, if x ∈ (cid:2) y ( t ) , ¯ y ( t ) (cid:3) ¯ y ( t ) , if x ∈ (¯ y ( t ) , 1] (4.5) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 9 t T y ( t ) y ( t ) y ∞ 10 no - trade regionsell regionbuy region x y ( t,x ) y ( t ) y ( t ) y ∞ y ( t ) y ( t ) buy no - trade sell0 Figure 1. The left graph shows y ( t ) and ¯ y ( t ) as functions of t , and the right graph describes ˆ y ( t, x ) as afunction of x for fixed t = 0 . 5. In both graphs, the dashed line is the Merton fraction y ∞ . The parametersare µ = 0 . , r = 0 . , σ = 1 , λ = 3 , (cid:15) = ¯ (cid:15) = 0 . 05, and T = 1. where y ( t ) and ¯ y ( t ) are uniquely determined in Lemma 4.1.Proof. We may rewrite the maximization in (3.8) asmax y ∈ [0 , (cid:16) v ( t, y ) − ln (cid:16) (cid:15)y (cid:15)x (cid:17) { x If ¯ (cid:15) = (cid:15) = 0, then Lemma 4.1 implies that y ( t ) = ¯ y ( t ), hence the no-trade region becomes asingleton. For ¯ (cid:15) > 0, one may expect that the investor would not want to buy the stock at timesclose to the terminal time T due to the transaction costs. If ¯ (cid:15) = 0 and the Merton fraction µ − rσ isgreater than zero, then one may expect that the investor would want to hold strictly positive sharesof the stock all the time. Our next task is to examine and prove this type of intuitive statements.For detailed analysis, we first provide stochastic representations of v and v x . We apply theFeynman-Kac formula (i.e., see Theorem 5.7.6 in [28]) and the expression of the optimizer ˆ y inTheorem 4.2 to Lemma 3.1, and obtain the following representation for v : v ( t, x ) = (cid:90) Tt e − λ ( s − t ) E (cid:20) ( µ − r ) Y ( t,x ) s + r − σ (cid:16) Y ( t,x ) s (cid:17) + λ L ( s, Y ( t,x ) s ) (cid:21) ds, (4.8)where for ( s, z ) ∈ [ t, T ) × [0 , Y ( t,x ) s := x · exp (cid:0)(cid:0) µ − r − σ (cid:1) ( s − t ) + σ ( B s − B t ) (cid:1) x · exp (cid:0)(cid:0) µ − r − σ (cid:1) ( s − t ) + σ ( B s − B t ) (cid:1) + (1 − x ) ,L ( s, y ) := v ( s, ˆ y ( s, y )) − ln (cid:16) (cid:15) ˆ y ( s,y )1+¯ (cid:15)y (cid:17) { y< ˆ y ( s,y ) } − ln (cid:16) − (cid:15) ˆ y ( s,y )1 − (cid:15)y (cid:17) { y> ˆ y ( s,y ) } . (4.9)The representation of v x is given in the following lemma. Lemma 4.3. The function L in (4.9) is continuously differentiable with respect to y , L y ( t, x ) = ¯ (cid:15) (cid:15)x , x ∈ (0 , y ( t )] v x ( t, x ) , x ∈ ( y ( t ) , ¯ y ( t )) − (cid:15) − (cid:15)x , x ∈ [¯ y ( t ) , for ( t, x ) ∈ [0 , T ) × (0 , , (4.10) and v x ( t, x ) has the following representation: for ( t, x ) ∈ [0 , T ) × (0 , , v x ( t, x ) = (cid:90) Tt e − λ ( s − t ) E (cid:104)(cid:16) ∂∂x Y ( t,x ) s (cid:17) (cid:16) µ − r − σ Y ( t,x ) s + λ L y ( s, Y ( t,x ) s ) (cid:17)(cid:105) ds. (4.11) Proof. Combining (4.5) and (4.9), we rewrite L as L ( t, x ) = v ( t, y ( t )) − ln (cid:16) (cid:15) y ( t )1+¯ (cid:15)x (cid:17) , x ∈ (0 , y ( t )] v ( t, x ) , x ∈ ( y ( t ) , ¯ y ( t )) v ( t, ¯ y ( t )) − ln (cid:16) − (cid:15) ¯ y ( t )1 − (cid:15)x (cid:17) , x ∈ [¯ y ( t ) , . (4.12)We take derivative with respect to x above and obtain the expression (4.10) for x ∈ (0 , \{ y ( t ) , ¯ y ( t ) } . If 0 < y ( t ) < < ¯ y ( t ) < v x ( t, y ( t )) = ¯ (cid:15) (cid:15)y ( t ) (resp., v x ( t, ¯ y ( t )) = − (cid:15) − (cid:15) ¯ y ( t ) ). Therefore, we conclude that L is continuously differentiable with respect to y and (4.10)is valid.Since − (cid:15) − (cid:15)x < v x ( t, x ) < ¯ (cid:15) (cid:15)x for x ∈ ( y ( t ) , ¯ y ( t )), we observe that for ( t, x ) ∈ [0 , T ) × (0 , − (cid:15) − (cid:15) ≤ L y ( t, x ) ≤ ¯ (cid:15). (4.13)We also observe that for ( t, x ) ∈ [0 , T ) × (0 , e − (cid:12)(cid:12)(cid:12) r − µ + σ (cid:12)(cid:12)(cid:12) ( s − t ) − σ | B s − B t | ≤ ∂∂x Y ( t,x ) s ≤ e (cid:12)(cid:12)(cid:12) r − µ + σ (cid:12)(cid:12)(cid:12) ( s − t )+ σ | B s − B t | . (4.14)Now we take derivative with respect to x in (4.8). The mean value theorem and the dominatedconvergence theorem, together with the inequalities (4.13) and (4.14), allow us to take derivativeinside of the expectation. By (4.10) and the chain rule, we obtain the representation (4.11). (cid:3) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 11 Using these representations, we extract some properties about the boundaries y ( t ) and ¯ y ( t ) ofthe no-trade region. Proposition 4.4. (Properties of no-trade region) Let y ∞ := µ − rσ denote the Merton fraction.(i) If ¯ (cid:15) > ( (cid:15) > , resp.), then there exists t ∈ [0 , T ) such that y ( t ) = 0 ( ¯ y ( t ) = 1 , resp.) for t ∈ [ t , T ) .(ii) If < y ∞ ( y ∞ < , resp.), then ¯ y ( t ) > ( y ( t ) < , resp.) for t ∈ [0 , T ) .(iii) If < y ∞ < and at least one of ¯ (cid:15) and (cid:15) is strictly positive, then y ( t ) < ¯ y ( t ) for t ∈ [0 , T ) .(iv) If < y ∞ and ¯ (cid:15) = 0 ( y ∞ < and (cid:15) = 0 , resp.), then y ( t ) > ( ¯ y ( t ) < , resp.) for t ∈ [0 , T ) .Proof. By (4.14), we have 0 < E (cid:104) ∂∂x Y ( t,x ) s (cid:105) ≤ e ( | r − µ | + σ )( s − t ) , (4.15)and we apply this inequality and (4.13) to the expression (4.11) to obtain c ( T − t ) ≤ v x ( t, x ) ≤ ¯ c ( T − t ) for x ∈ (0 , , (4.16)where c and ¯ c are constants that only depend on µ, r, σ, λ, ¯ (cid:15), (cid:15), T .(i) Suppose that ¯ (cid:15) > (cid:15) > c ≤ 0, then we choose t = 0 and easily observe thatlim x ↓ ∂∂x (cid:16) v ( t, x ) − ln(1 + ¯ (cid:15)x ) (cid:17) ≤ t ∈ [ t , T ) . (4.17)If ¯ c > 0, then (4.16) implies that (4.17) holds with t = (cid:0) T − ¯ (cid:15) ¯ c (cid:1) + . Therefore, in any case, wecan choose t ∈ [0 , T ) that satisfies (4.17). By Lemma 4.1 and (4.17), we conclude y ( t ) = 0 for t ∈ [ t , T ).(ii) Suppose that 0 < y ∞ (the case of y ∞ < t ∈ [0 , T ) be fixed.To check ¯ y ( t ) > 0, we observe from (4.11) and the dominated convergence theorem thatlim x ↓ v x ( t, x ) = (cid:90) Tt e − λ ( s − t ) E (cid:20) lim x ↓ (cid:16) ∂∂x Y ( t,x ) s (cid:17) (cid:16) µ − r − σ Y ( t,x ) s + λL y ( s, Y ( t,x ) s ) (cid:17)(cid:21) ds = (cid:90) Tt e − λ ( s − t ) E (cid:20) e ( µ − r − σ )( s − t )+ σ ( B s − B t ) (cid:21) · (cid:18) µ − r + λ (cid:16) ¯ (cid:15) · { y ( s ) > } + lim x ↓ v x ( s, x ) · { y ( s )=0 < ¯ y ( s ) } − (cid:15) · { ¯ y ( s )=0 } (cid:17)(cid:19) ds ≥ (cid:90) Tt e ( µ − r − λ )( s − t ) ( µ − r − λ(cid:15) ) ds, (4.18)where the second equality is from (4.10) and (4.14), and the inequality is due to the fact that − (cid:15) − (cid:15)x < v x ( t, x ) for x < ¯ y ( t ). Obviously, the last integral in (4.18) is nonnegative if µ − r − λ(cid:15) ≥ µ − r − λ(cid:15) < 0, then µ − r − λ < (cid:15) ∈ [0 , 1) and we observe that (cid:90) Tt e ( µ − r − λ )( s − t ) ( µ − r − λ(cid:15) ) ds = µ − r − λ(cid:15)µ − r − λ (cid:16) e ( µ − r − λ )( T − t ) − (cid:17) > − (cid:15), where we use y ∞ > x ↓ ∂∂x (cid:16) v ( t, x ) − ln(1 − (cid:15)x ) (cid:17) > , and we conclude ¯ y ( t ) > PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 12 (iii) Suppose that 0 < y ∞ < (cid:15) > (cid:15) > y ( t ) < y ( t ) > 0. Therefore, there are only two possibilities: y ( t ) = 0 or 0 < y ( t ) < y ( t ) = 0, we immediately obtain y ( t ) < ¯ y ( t ) since ¯ y ( t ) > 0. In case 0 < y ( t ) < 1, byLemma 4.1, we have v x ( t, y ( t )) = ¯ (cid:15) (cid:15) y ( t ) > − (cid:15) − (cid:15) y ( t ) , and this inequality, together with (4.3) and (4.4), implies y ( t ) < ¯ y ( t ).(iv) Suppose that 0 < y ∞ and ¯ (cid:15) = 0 (the case of y ∞ < (cid:15) = 0 can be treated similarly).Lemma 4.1 implies that y ( t ) > x ↓ v x ( t, x ) > . (4.19)The second equality in (4.18) can be written aslim x ↓ v x ( t, x ) = (cid:90) Tt e ( µ − r − λ )( s − t ) (cid:18) µ − r + λ lim x ↓ v x ( s, x ) · { y ( s )=0 < ¯ y ( s ) } (cid:19) ds, (4.20)where we substitute ¯ (cid:15) = 0 and use ¯ y ( s ) > t ∗ as t ∗ := inf (cid:26) t ∈ [0 , T ) : lim x ↓ v x ( s, x ) > s ∈ [ t, T ) (cid:27) . (4.21)Due to (4.16) and the strict positivity of µ − r , for t close enough to T , the integrand in (4.20)is strictly positive and lim x ↓ v x ( t, x ) > 0. Therefore, the set in (4.21) is non-empty and t ∗ < T .Suppose that t ∗ > 0. For 0 ≤ δ ≤ t ∗ , we apply (4.21) and (4.19) to expression (4.20) and obtainlim x ↓ v x ( t ∗ − δ, x ) = (cid:90) Tt ∗ e ( µ − r − λ )( s − t ∗ + δ ) ( µ − r ) ds + (cid:90) t ∗ t ∗ − δ e ( µ − r − λ )( s − t ∗ + δ ) (cid:18) µ − r + λ lim x ↓ v x ( s, x ) · { y ( s )=0 < ¯ y ( s ) } (cid:19) ds. The first integral above is greater than a strictly positive number independent of δ , and the secondintegral can be made arbitrary close to zero as δ ↓ 0, due to (4.16). Therefore, there exists δ ∗ ∈ (0 , t ∗ ]such that lim x ↓ v x ( t ∗ − δ, x ) > δ ∈ [0 , δ ∗ ]. Then, lim x ↓ v x ( s, x ) > s ∈ [ t ∗ − δ ∗ , T ),and this contradicts to the definition of t ∗ in (4.21). Therefore, we conclude that t ∗ = 0, and now(4.19) implies that y ( t ) > t ∈ (0 , T ). Lastly, y (0) > (cid:3) Straightforward interpretations of Proposition 4.4 are as follows:(i) The existence of the transaction costs for selling (buying, resp.) the stock makes the investornot to sell (buy, resp.) the stock when it is close to the terminal time. For short period of time,the benefit of rebalancing is small.(ii) If 0 < y ∞ ( y ∞ < 1, resp), then the investor never rebalances to the zero-holding of the stock(bond, resp.). However, if the initial holding of the stock (bond, resp.) is zero, then the investormay not try to leave the state of zero-holding of the stock (bond, resp.), depending on the size ofthe transaction costs.(iii) If 0 < y ∞ < < y ∞ ( y ∞ < 1, resp.) and there is no cost for buying (selling, resp.) the stock, then justholding the bond (stock, resp.) and setting zero balance in the stock (bond, resp.) is suboptimal,even with the search frictions and transaction costs. PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 13 Remark . Obviously, if y ∞ ≤ y ∞ ≥ 1, resp.), then there is no reason for buying (selling, resp.)the stock, so y ( t ) = 0 (¯ y ( t ) = 1, resp.) for t ∈ [0 , T ).5. Asymptotic analysis In this section, we provide asymptotic analysis for small transaction costs. To be specific, wefocus on the first order approximation of the no-trade region and the value function with respectto the transaction cost parameter around zero. To consider non-trivial cases (see Remark 4.5), weassume that the Merton fraction y ∞ is between zero and one, and we set ¯ (cid:15) = (cid:15) for convenience. Assumption 5.1. In this section, we assume that 0 < y ∞ < (cid:15) = (cid:15) = (cid:15) for (cid:15) ∈ [0 , Notation 5.2. (Current section only) (i) To emphasize their dependence on the transaction cost parameter (cid:15) , we denote v, y, ¯ y, ˆ y, L, L y by v (cid:15) , y (cid:15) , ¯ y (cid:15) , ˆ y (cid:15) , L (cid:15) , L (cid:15)y . In particular, when (cid:15) = 0, they are denoted by v , y , ¯ y , ˆ y , L , L y .(ii) Under Assumption 5.1, Lemma 4.1 and Proposition 4.4 imply that0 < ˆ y ( t, x ) = y ( t ) = ¯ y ( t ) < t, x ) ∈ [0 , T ) × [0 , . (5.1)In words, ˆ y ( t, x ) is independent of variable x (just a function of t ) and its value equals y ( t ) and¯ y ( t ). For convenience, we abuse notation and write ˆ y ( t ) for ˆ y ( t, x ) (i.e., ˆ y ( t ) = y ( t ) = ¯ y ( t )).We start with the technical lemma that is used in the proof of the asymptotic result. Lemma 5.3. (i) Let (cid:15) = 0 . For t ∈ [0 , T ) , v x ( t, ˆ y ( t )) = 0 , (5.2)sup x ∈ (0 , v xx ( t, x ) < . (5.3) (ii) Let F : [0 , T ) × (0 , → R be defined as F ( t, x ) := λ (cid:90) Tt e − λ ( s − t ) E (cid:104)(cid:16) ∂∂x Y ( t,x ) s (cid:17) · sgn (cid:16) ˆ y ( s ) − Y ( t,x ) s (cid:17)(cid:105) ds. (5.4) Then, for t ∈ [0 , T ) , − < F ( t, ˆ y ( t )) < . (5.5) (iii) Suppose that x ∈ (0 , and lim (cid:15) ↓ x (cid:15) = x . Then, for t ∈ [0 , T ) , lim (cid:15) ↓ v (cid:15)x ( t, x (cid:15) ) = v x ( t, x ) , (5.6)lim (cid:15) ↓ v (cid:15)xx ( t, x (cid:15) ) = v xx ( t, x ) . (5.7) Proof. See Appendix. (cid:3) The following theorem provides the first order approximation of the no-trade boundaries. Theorem 5.4. For t ∈ [0 , T ) , y (cid:15) ( t ) = ˆ y ( t ) − F ( t, ˆ y ( t )) − v xx ( t, ˆ y ( t )) · (cid:15) + o ( (cid:15) ) , (5.8)¯ y (cid:15) ( t ) = ˆ y ( t ) − F ( t, ˆ y ( t )) + 1 v xx ( t, ˆ y ( t )) · (cid:15) + o ( (cid:15) ) , (5.9) where F is defined in (5.4) . In particular, for small enough (cid:15) > , we have < y (cid:15) ( t ) < ˆ y ( t ) < ¯ y (cid:15) ( t ) < . (5.10) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 14 Proof. For ( t, x ) ∈ [0 , T ) × (0 , (cid:12)(cid:12) v (cid:15)x ( t, x ) − v x ( t, x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:90) Tt e − λ ( s − t ) E (cid:104)(cid:16) ∂∂x Y ( t,x ) s (cid:17) L (cid:15)y ( s, Y ( t,x ) s ) (cid:105) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ (cid:90) Tt e − λ ( s − t ) E (cid:104)(cid:16) ∂∂x Y ( t,x ) s (cid:17)(cid:105) ds · (cid:15) − (cid:15) ≤ λT · e ( µ − r + σ ) T · (cid:15) − (cid:15) , (5.11)where the first inequality is due to (4.13) and the positivity of ∂∂x Y ( t,x ) s in (4.14), and the secondinequality is due to (4.15) and Assumption 5.1.Proposition 4.4 and Assumption 5.1 imply that y (cid:15) ( t ) < y (cid:15) ( t ) > t ∈ [0 , T ). ByLemma 4.1, if 0 < y (cid:15) ( t ) < 1, then v (cid:15)x ( t, y (cid:15) ( t )) = (cid:15) (cid:15) · y (cid:15) ( t ) , and if y (cid:15) ( t ) = 0, then − (cid:15) − (cid:15)x < v (cid:15)x ( t, x ) < (cid:15) (cid:15)x for x ∈ (0 , ¯ y (cid:15) ( t )). In any case, we have (cid:12)(cid:12) v (cid:15)x ( t, y (cid:15) ( t )) (cid:12)(cid:12) ≤ (cid:15) − (cid:15) , (5.12)where v (cid:15)x ( t, 0) := lim x ↓ v (cid:15)x ( t, x ) is well-defined (see (4.18) for details) for the case of y (cid:15) ( t ) = 0. Weuse the mean value theorem and (5.2) to obtaininf x ∈ (0 , (cid:12)(cid:12) v xx ( t, x ) (cid:12)(cid:12) · (cid:12)(cid:12) y (cid:15) ( t ) − ˆ y ( t ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) v x ( t, y (cid:15) ( t )) − v x ( t, ˆ y ( t )) (cid:12)(cid:12) ≤ (cid:12)(cid:12) v x ( t, y (cid:15) ( t )) − v (cid:15)x ( t, y (cid:15) ( t )) (cid:12)(cid:12) + (cid:12)(cid:12) v (cid:15)x ( t, y (cid:15) ( t )) (cid:12)(cid:12) . (5.13)The above inequality, together with (5.3), (5.11), and (5.12), we conclude that y (cid:15) ( t ) − ˆ y ( t ) = O ( (cid:15) ) for t ∈ [0 , T ) . (5.14)By the same way, we also obtain¯ y (cid:15) ( t ) − ˆ y ( t ) = O ( (cid:15) ) for t ∈ [0 , T ) . (5.15)The expression of L (cid:15)y in (4.10), together with (5.14) and (5.15), implies the following limit:lim (cid:15) ↓ L (cid:15)y ( t, x ) (cid:15) = (cid:40) , if x ∈ (0 , ˆ y ( t )) − , if x ∈ (ˆ y ( t ) , 1) for t ∈ [0 , T ) . (5.16)For ( s, x ) ∈ ( t, T ) × (0 , P (cid:16) Y ( t,x ) s = ˆ y ( t ) (cid:17) = 0 and (5.16) producelim (cid:15) ↓ L (cid:15)y ( s, Y ( t,x ) s ) (cid:15) = sgn (cid:16) ˆ y ( t ) − Y ( t,x ) s (cid:17) almost surely. (5.17)For ( t, x ) ∈ [0 , T ) × (0 , v (cid:15)x ( t, x ) − v x ( t, x ) (cid:15) = λ (cid:90) Tt e − λ ( s − t ) E (cid:20)(cid:16) ∂∂x Y ( t,x ) s (cid:17) L (cid:15)y ( s,Y ( t,x ) s ) (cid:15) (cid:21) ds −−→ (cid:15) ↓ F ( t, x ) . (5.18)The inequality 0 < ˆ y ( t ) < < y (cid:15) ( t ) < (cid:15) > 0. Hence,by Lemma 4.1, we obtain v (cid:15)x ( t, y (cid:15) ( t )) = (cid:15) (cid:15) y (cid:15) ( t ) for small enough (cid:15) > 0. (5.19)By the mean value theorem, there exists k ( (cid:15) ) such that v (cid:15)x ( t, y (cid:15) ( t )) − v (cid:15)x ( t, ˆ y ( t )) = v (cid:15)xx ( t, k ( (cid:15) )) (cid:0) y (cid:15) ( t ) − ˆ y ( t ) (cid:1) and lim (cid:15) ↓ k ( (cid:15) ) = ˆ y ( t ) . (5.20) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 15 ϵ y ϵ ( t ) y ϵ ( t ) y ( t ) ϵ v ( t, y ( t )) v ϵ ( t, y ( t )) Figure 2. The left graph shows y (cid:15) ( t ) and ¯ y (cid:15) ( t ) as functions of (cid:15) , where the dashed lines are the linearapproximations of them in Theorem 5.4, i.e., ˆ y ( t ) − F ( t, ˆ y ( t )) − v xx ( t, ˆ y ( t )) · (cid:15) and ˆ y ( t ) − F ( t, ˆ y ( t )) − v xx ( t, ˆ y ( t )) · (cid:15) . The rightgraph describes v (cid:15) ( t, ˆ y ( t )) as a function of (cid:15) , where the dashed line is the linear approximation of it inTheorem 5.5, i.e., v ( t, ˆ y ( t )) − (cid:16) G ( t ) + λ (cid:82) Tt G ( s ) ds (cid:17) · (cid:15) . In both graphs, the parameters are µ = 0 . , r =0 . , σ = 1 , λ = 3 , T = 1, and t = 0 . Now we obtain (5.8) as follows:lim (cid:15) ↓ y (cid:15) ( t ) − ˆ y ( t ) (cid:15) = lim (cid:15) ↓ v (cid:15)x ( t, y (cid:15) ( t )) − v (cid:15)x ( t, ˆ y ( t )) (cid:15) · v (cid:15)xx ( t, k ( (cid:15) ))= lim (cid:15) ↓ v (cid:15)xx ( t, k ( (cid:15) )) (cid:18) 11 + (cid:15) y (cid:15) ( t ) − v (cid:15)x ( t, ˆ y ( t )) − v x ( t, ˆ y ( t )) (cid:15) (cid:19) = − F ( t, ˆ y ( t )) − v xx ( t, ˆ y ( t )) , (5.21)where the first equality is from (5.20), the second equality is due to (5.19) and (5.2), and the thirdequality is due to (5.18) and (5.7) with the limit in (5.20). We also obtain (5.9) by the same way.Due to (5.3) and (5.5), we observe that − F ( t, ˆ y ( t )) − v xx ( t, ˆ y ( t )) < − F ( t, ˆ y ( t ))+1 v xx ( t, ˆ y ( t )) > 0. Therefore, (5.8)and (5.9) imply that (5.10) holds for small enough (cid:15) > (cid:3) The following theorem provides the first order approximation of the value function. Theorem 5.5. For t ∈ [0 , T ) , v (cid:15) ( t, ˆ y ( t )) = v ( t, ˆ y ( t )) − (cid:18) G ( t ) + λ (cid:90) Tt G ( s ) ds (cid:19) · (cid:15) + o ( (cid:15) ) , (5.22) where G : [0 , T ] → R is defined as G ( t ) := λ (cid:90) Tt e − λ ( s − t ) E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t, ˆ y ( t )) s − ˆ y ( s ) (cid:12)(cid:12)(cid:12)(cid:105) ds (5.23) Proof. For (cid:15) > 0, we define a function ψ (cid:15) : [0 , T ) → R as ψ (cid:15) ( t ) := v (cid:15) ( t, ˆ y ( t )) − v ( t, ˆ y ( t )) (cid:15) . (5.24)Using (4.8) and (4.12), we obtain ψ (cid:15) ( t ) = λ (cid:90) Tt e − λ ( s − t ) ψ (cid:15) ( s ) ds + λ (cid:90) Tt e − λ ( s − t ) E (cid:104) I ( t,s,(cid:15) )1 + I ( t,s,(cid:15) )2 + I ( t,s,(cid:15) )3 (cid:105) ds, (5.25) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 16 where the random variables I ( t,s,(cid:15) )1 , I ( t,s,(cid:15) )2 , I ( t,s,(cid:15) )3 are defined as I ( t,s,(cid:15) )1 := (cid:18) v (cid:15) ( s,y (cid:15) ( s )) − v (cid:15) ( s, ˆ y ( s )) (cid:15) − (cid:15) ln (cid:18) (cid:15) y (cid:15) ( s )1+ (cid:15) Y ( t, ˆ y t )) s (cid:19)(cid:19) · (cid:110) Y ( t, ˆ y t )) s ≤ y (cid:15) ( s ) (cid:111) ,I ( t,s,(cid:15) )2 := (cid:18) v (cid:15) ( s,Y ( t, ˆ y s )) s ) − v (cid:15) ( s, ˆ y ( s )) (cid:15) (cid:19) · (cid:110) y (cid:15) ( s ) 1. Therefore,1 (cid:110) Y ( t, ˆ y t )) s ≤ y (cid:15) ( s ) (cid:111) ≤ (cid:8) We rewrite (5.25) as J (cid:15) ( t ) = ψ (cid:15) ( t ) − λ (cid:90) Tt e − λ ( s − t ) ψ (cid:15) ( s ) ds = − ddt (cid:18)(cid:90) Tt e − λ ( s − t ) ψ (cid:15) ( s ) ds (cid:19) . We integrate both sides above to obtain (cid:90) Tt e − λ ( s − t ) ψ (cid:15) ( s ) ds = (cid:90) Tt J (cid:15) ( s ) ds. (5.35)Using (5.35), the equation (5.25) becomes ψ (cid:15) ( t ) = J (cid:15) ( t ) + λ (cid:90) Tt J (cid:15) ( s ) ds. (5.36)Finally, (5.34) and (5.36) imply (5.22). (cid:3) Corollary 5.6. For ( t, x ) ∈ [0 , T ) × (0 , , v (cid:15) ( t, x ) = v ( t, x ) − (cid:32) G ( t ) + λ (cid:90) Tt G ( s ) ds − (cid:90) x ˆ y ( t ) F ( t, η ) dη (cid:33) · (cid:15) + o ( (cid:15) ) , (5.37) where F and G are defined in (5.4) and (5.23) .Proof. The inequality (5.11) and the limit (5.18) allow us to use the dominated convergence theo-rem, and we obtain lim (cid:15) ↓ (cid:90) x ˆ y ( t ) v (cid:15)x ( t, η ) − v x ( t, η ) (cid:15) dη = (cid:90) x ˆ y ( t ) F ( t, η ) dη. Using the above limit and Theorem 5.5, we obtain v (cid:15) ( t, x ) − v ( t, x ) (cid:15) = v (cid:15) ( t, ˆ y ( t )) − v ( t, ˆ y ( t )) (cid:15) + (cid:90) x ˆ y ( t ) v (cid:15)x ( t, η ) − v x ( t, η ) (cid:15) dη −−→ (cid:15) ↓ − (cid:18) G ( t ) + λ (cid:90) Tt G ( s ) ds (cid:19) + (cid:90) x ˆ y ( t ) F ( t, η ) dη. The above limit is equivalent to (5.37). (cid:3) Not surprisingly, Theorem 5.4 and Theorem 5.5 indicate that the no-trade region widens and thevalue function diminishes as the transaction cost parameter (cid:15) increases. See Figure 2 for numericalillustrations. Our next task is to investigate some intertwined effects of the search frictions andtransaction costs on the no-trade region and value function. To be specific, for two different searchfriction parameters λ < λ , we would like to compare the magnitude of the widening (deminishing,resp.) effects of the transaction costs on the no-trade region (the value function, resp.). Figure 3numerically describes this comparison result. The following technical lemma turns out to be usefulto the proof of this comparison. Lemma 5.7. (i) There is a constant C such that (cid:12)(cid:12) ˆ y ( t ) − y ∞ (cid:12)(cid:12) ≤ Cλ for ( t, λ ) ∈ [0 , T ) × [1 , ∞ ) . (5.38) (ii) ∂∂λ v x ( t, x ) and ∂∂λ v xx ( t, x ) exist for ( t, x ) ∈ [0 , T ] × (0 , , and lim λ →∞ λ ∂∂λ v x ( t, x ) (cid:12)(cid:12)(cid:12) x =ˆ y ( t ) = 0 , lim λ →∞ λ ∂∂λ v xx ( t, x ) (cid:12)(cid:12)(cid:12) x =ˆ y ( t ) = σ , lim λ →∞ λv xx ( t, ˆ y ( t )) = − σ , lim λ →∞ λv xxx ( t, ˆ y ( t )) = 0 . (5.39) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 18 ϵ y ϵ - y ϵ λ = λ = ϵ v ϵ - v λ = λ = Figure 3. The left graph compares ¯ y (cid:15) ( t ) − y (cid:15) ( t ) for λ = 0 . λ = 3 (solid line), asfunctions of (cid:15) . The right graph compares v ( t, ˆ y ( t )) − v (cid:15) ( t, ˆ y ( t )) for λ = 0 . λ = 3(solid line), as functions of (cid:15) . The other parameters are µ = 0 . , r = 0 . , σ = 1 , T = 1, and t = 0 . (iii) ∂∂λ ˆ y ( t ) exists for t ∈ [0 , T ) , and there is a constant C such that (cid:12)(cid:12) ∂∂λ ˆ y ( t ) (cid:12)(cid:12) ≤ Cλ for ( t, λ ) ∈ [0 , T ) × [1 , ∞ ) . (5.40) (iv) ∂∂λ E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t, ˆ y ( t )) s − ˆ y ( s ) (cid:12)(cid:12)(cid:12)(cid:105) exists for ≤ t ≤ s ≤ T , and there is a constant C such that (cid:12)(cid:12)(cid:12) ∂∂λ (cid:16) E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t, ˆ y ( t )) s − ˆ y ( s ) (cid:12)(cid:12)(cid:12)(cid:105)(cid:17)(cid:12)(cid:12)(cid:12) ≤ Cλ for ( t, s, λ ) ∈ [0 , T ) × [ t, T ] × [1 , ∞ ) . (5.41) (v) There is a constant C such that E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t,x ) s − x (cid:12)(cid:12)(cid:12)(cid:105) √ s − t ≤ C for ( t, s, x ) ∈ [0 , T ) × ( t, T ] × (0 , , (5.42) and for t ∈ [0 , T ) , lim s ↓ t E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t,x ) s − x (cid:12)(cid:12)(cid:12)(cid:105) √ s − t = σ (cid:113) π x (1 − x ) . (5.43) Proof. See Appendix. (cid:3) Proposition 5.8. For t ∈ [0 , T ) , there exists Λ( t ) > such that Λ( t ) < λ < λ implies (cid:40) (cid:0) ¯ y (cid:15) ( t ) − y (cid:15) ( t ) (cid:1) (cid:12)(cid:12) λ = λ < (cid:0) ¯ y (cid:15) ( t ) − y (cid:15) ( t ) (cid:1) (cid:12)(cid:12) λ = λ (cid:0) v ( t, ˆ y ( t )) − v (cid:15) ( t, ˆ y ( t )) (cid:1) (cid:12)(cid:12) λ = λ < (cid:0) v ( t, ˆ y ( t )) − v (cid:15) ( t, ˆ y ( t )) (cid:1) (cid:12)(cid:12) λ = λ (cid:41) for small enough (cid:15) > .Proof. Theorem 5.4 implies that¯ y (cid:15) ( t ) − y (cid:15) ( t ) = − v xx ( t, ˆ y ( t )) · (cid:15) + o ( (cid:15) ) . Therefore, to prove the first statement, it is enough to show that ∂∂λ (cid:0) v xx ( t, ˆ y ( t )) (cid:1) > λ . Indeed, using (ii) and (iii) in Lemma 5.7, we obtain λ ∂∂λ (cid:0) v xx ( t, ˆ y ( t )) (cid:1) = λ ∂∂λ v xx ( t, x ) (cid:12)(cid:12) x =ˆ y ( t ) + λv xxx ( t, ˆ y ( t )) · λ ∂∂λ ˆ y ( t ) −−−→ λ →∞ σ > . (5.44) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 19 To prove the second statement, it is enough check ∂∂λ (cid:16) G ( t ) + λ (cid:82) Tt G ( s ) ds (cid:17) > λ , according to Theorem 5.5. For this purpose, we focus on proving the following:lim λ →∞ √ λ (cid:18) ∂∂λ (cid:18) G ( t ) + λ (cid:90) Tt G ( s ) ds (cid:19)(cid:19) > . (5.45)By (iv) in Lemma 5.7, we can take derivative inside of the expectations and obtain √ λ (cid:18) ∂∂λ (cid:18) G ( t ) + λ (cid:90) Tt G ( s ) ds (cid:19)(cid:19) = λ ∂∂λ G ( t ) + (cid:90) Tt (cid:16) λ G ( s ) + λ ∂∂λ G ( s ) (cid:17) ds, (5.46)where the partial derivative ∂∂λ G ( t ) can be written as ∂∂λ G ( t ) = (cid:90) Tt e − λ ( s − t ) (cid:16) (1 − λ ( s − t )) E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t, ˆ y ( t )) s − ˆ y ( s ) (cid:12)(cid:12)(cid:12)(cid:105) + λ ∂∂λ E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t, ˆ y ( t )) s − ˆ y ( s ) (cid:12)(cid:12)(cid:12)(cid:105)(cid:17) ds. Observe that (i) and (v) in Lemma 5.7 imply E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t, ˆ y ( t )) s − ˆ y ( s ) (cid:12)(cid:12)(cid:12)(cid:105) ≤ C (cid:0) √ s − t + λ (cid:1) for ( t, s, λ ) ∈ [0 , T ) × ( t, T ] × [1 , ∞ ) , (5.47)for a constant C . We use (5.38), (5.41), (5.43), and (D.4) to obtain the following limits:lim λ →∞ λ G ( s ) = σ √ y ∞ (1 − y ∞ ) , lim λ →∞ λ ∂∂λ G ( s ) = − σ √ y ∞ (1 − y ∞ ) for s ∈ [0 , T ) . (5.48)We apply (5.41), (5.47), and (D.5) to the expressions of G and ∂∂λ G to obtain the boundedness: (cid:12)(cid:12)(cid:12) λ G ( s ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) λ ∂∂λ G ( s ) (cid:12)(cid:12)(cid:12) ≤ C for ( s, λ ) ∈ [0 , T ] × [1 , ∞ ) (5.49)Finally, (5.48) and (5.49) enable us to apply the dominated convergence theorem to (5.46):lim λ →∞ √ λ (cid:18) ∂∂λ (cid:18) G ( t ) + λ (cid:90) Tt G ( s ) ds (cid:19)(cid:19) = σ √ y ∞ (1 − y ∞ )( T − t ) > . (5.50)Therefore, we conclude (5.45) and the proof is done. (cid:3) Acknowledgement This work was supported by the National Research Foundation of Korea (NRF) grant fundedby the Korea government (MSIT) (No. 2020R1C1C1A01014142). References [1] M. Akian, J. Menaldi, and A. 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Proof of Lemma 3.1 Since the parabolic type PDE (3.2) is not uniformly elliptic, we change variable as x = h ( z ) := e z e z and consider the PDE for v ( t, h ( z )).To handle the nonlinear term in the PDE, we first consider the following map φ from C b ([0 , T ] × R )(equipped with the uniform norm) to itself: φ ( f )( t, z ) := (cid:90) Tt e − λ ( s − t ) E (cid:104) K f ( s, Z ( t,z ) s ) (cid:105) ds, (A.1)where for ( s, z ) ∈ [ t, T ] × R , Z ( t,z ) s := z + (cid:16) µ − r − σ (cid:17) ( s − t ) + σ ( B s − B t ) ,K f ( s, z ) := ( µ − r ) h ( z ) + r − σ h ( z ) + λ sup ζ ∈ R (cid:16) f ( s, ζ ) − ln (cid:16) (cid:15)h ( ζ )1+¯ (cid:15)h ( z ) (cid:17) { z<ζ } − ln (cid:16) − (cid:15)h ( ζ )1 − (cid:15)h ( z ) (cid:17) { z>ζ } (cid:17) . (A.2)We observe that for f, g ∈ C b ([0 , T ] × R ), (cid:107) φ ( f ) − φ ( g ) (cid:107) ∞ ≤ λ (cid:90) Tt e − λ ( s − t ) (cid:32) sup ζ ∈ R | f ( s, ζ ) − g ( s, ζ ) | (cid:33) ds ≤ (cid:16) − e − λT (cid:17) (cid:107) f − g (cid:107) ∞ , (A.3)where the first inequality is due to the triangular inequality for supremum. Therefore, the map φ in (A.1) is a contraction map and there exists a unique ˆ u ∈ C b ([0 , T ] × R ) such that φ (ˆ u ) = ˆ u , bythe Banach fixed point theorem. Claim : For all δ ∈ (0 , K ˆ u ∈ C δ ,δ ([0 , T ] × R ).(Proof of Claim): We first check that K ˆ u is bounded. Since 0 < h ( z ) < 1, we observe that | K ˆ u ( t, z ) | ≤ | µ | + 2 | r | + σ + λ (cid:16) (cid:107) ˆ u (cid:107) ∞ + ln (cid:16) (cid:15) − (cid:15) (cid:17)(cid:17) , (A.4) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 22 i.e., (cid:107) K ˆ u (cid:107) ∞ < ∞ . Therefore, to prove the claim, it is enough to check that K ˆ u is uniformly Lipschitzwith respect to z variable and -H¨older continuous with respect to t variable. For ∆ > 0, we obtainthe following inequalities: | K ˆ u ( t, z + ∆) − K ˆ u ( t, z ) |≤ ( | µ − r | + σ )∆ + λ sup ζ ∈ R (cid:12)(cid:12)(cid:12) { z +∆ <ζ } · ln (cid:16) (cid:15)h ( z +∆)1+¯ (cid:15)h ( z ) (cid:17) + 1 { z<ζ ≤ z +∆ } · ln (cid:16) (cid:15)h ( ζ )1+¯ (cid:15)h ( z ) (cid:17)(cid:12)(cid:12)(cid:12) + λ sup ζ ∈ R (cid:12)(cid:12)(cid:12) { z +∆ >ζ } · ln (cid:16) − (cid:15)h ( z +∆)1 − (cid:15)h ( z ) (cid:17) − { z ≤ ζ 1. We can treat∆ < K ˆ u is uniformly Lipschitz with respect to z .For ∆ > 0, using (A.5) and the mean value theorem, we observe that E (cid:12)(cid:12)(cid:12) e λ ∆ K ˆ u ( s, Z ( t +∆ ,z ) s ) − K ˆ u ( s, Z ( t,z ) s ) (cid:12)(cid:12)(cid:12) ≤ λe λT (cid:107) K ˆ u (cid:107) ∞ ∆ + E (cid:12)(cid:12)(cid:12) K ˆ u ( s, Z ( t +∆ ,z ) s ) − K ˆ u ( s, Z ( t,z ) s ) (cid:12)(cid:12)(cid:12) ≤ λe λT (cid:107) K ˆ u (cid:107) ∞ ∆ + (cid:16) | µ − r | + σ + λ (cid:16) ¯ (cid:15) + (cid:15) − (cid:15) (cid:17)(cid:17) E (cid:12)(cid:12)(cid:12) Z ( t +∆ ,z ) s − Z ( t,z ) s (cid:12)(cid:12)(cid:12) ≤ λe λT (cid:107) K ˆ u (cid:107) ∞ ∆ + (cid:16) | µ − r | + σ + λ (cid:16) ¯ (cid:15) + (cid:15) − (cid:15) (cid:17)(cid:17) (cid:18)(cid:12)(cid:12)(cid:12) µ − r − σ (cid:12)(cid:12)(cid:12) ∆ + σ (cid:113) π ∆ (cid:19) , (A.6)where the last inequality is due to E (cid:12)(cid:12)(cid:12) Z ( t +∆ ,z ) s − Z ( t,z ) s (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) µ − r − σ (cid:12)(cid:12)(cid:12) ∆ + σ E | B t +∆ − B t | = (cid:12)(cid:12)(cid:12) µ − r − σ (cid:12)(cid:12)(cid:12) ∆ + σ (cid:113) π ∆ . Using ˆ u = φ (ˆ u ) and triangular inequality, we obtain | K ˆ u ( t + ∆ , z ) − K ˆ u ( t, z ) | ≤ λ sup ζ ∈ R | ˆ u ( t + ∆ , ζ ) − ˆ u ( t, ζ ) | = λ sup ζ ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Tt +∆ e − λ ( s − t − ∆) E (cid:104) K ˆ u ( s, Z ( t +∆ ,ζ ) s ) (cid:105) ds − (cid:90) Tt e − λ ( s − t ) E (cid:104) K ˆ u ( s, Z ( t,ζ ) s ) (cid:105) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ sup ζ ∈ R (cid:18) (cid:90) Tt +∆ e − λ ( s − t ) E (cid:12)(cid:12)(cid:12) e λ ∆ K ˆ u ( s, Z ( t +∆ ,ζ ) s ) − K ˆ u ( s, Z ( t,ζ ) s ) (cid:12)(cid:12)(cid:12) ds + (cid:90) t +∆ t e − λ ( s − t ) E (cid:12)(cid:12)(cid:12) K ˆ u ( s, Z ( t,ζ ) s ) (cid:12)(cid:12)(cid:12) ds (cid:19) ≤ C ∆ , where the generic constant C only depends on the market parameters and (cid:107) ˆ u (cid:107) ∞ and the lastinequality is due to (A.4) and (A.6). We can treat ∆ < K ˆ u is -H¨older continuous with respect to t variable.(End of the proof of Claim).Let δ ∈ (0 , 1) be fixed. Then, the above claim and Theorem 9.2.3 in [31] ensure that thereexists a unique function u ∈ C δ , δ ((0 , T ) × R ) such that it satisfies the following PDE on( t, z ) ∈ (0 , T ) × R . (cid:40) u ( T, z ) , u t + ( µ − r − σ ) u z + σ u zz − λu + K ˆ u (A.7) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 23 Since u admits a unique continuous extension on [0 , T ] × R (i.e., see chapter 8.5 in [31]), we let u ∈ C δ , δ ([0 , T ] × R ). By the Feynman-Kac formula (i.e., see Theorem 5.7.6 in [28]), the solution u of the parabolic PDE (A.7) has the stochastic representation u = φ (ˆ u ), where φ is defined in(A.1). Since ˆ u is chosen as the unique fixed point of the map φ , we conclude that u = ˆ u .Our next task is to define u ( t, ±∞ ) for t ∈ [0 , T ]. Using lim z →∞ h ( Z ( t,z ) s ) = 1 and lim z →−∞ h ( Z ( t,z ) s ) =0 almost surely, we obtain lim z →∞ K u ( s, Z ( t,z ) s ) = µ − σ + λ sup ζ ∈ R (cid:16) u ( s, ζ ) − ln (cid:16) − (cid:15)h ( ζ )1 − (cid:15) (cid:17)(cid:17) lim z →−∞ K u ( s, Z ( t,z ) s ) = r + λ sup ζ ∈ R ( u ( s, ζ ) − ln (1 + ¯ (cid:15)h ( ζ ))) a.s. (A.8)The above convergence and (cid:107) K u (cid:107) ∞ < ∞ enable us to apply the dominated convergence theorem:lim z →±∞ u ( t, z ) = lim z →±∞ φ ( u )( t, z ) = lim z →±∞ (cid:90) Tt e − λ ( s − t ) E (cid:104) K u ( s, Z ( t,z ) s ) (cid:105) ds = (cid:82) Tt e − λ ( s − t ) (cid:16) µ − σ + λ sup ζ ∈ R (cid:16) u ( s, ζ ) − ln (cid:16) − (cid:15)h ( ζ )1 − (cid:15) (cid:17)(cid:17)(cid:17) ds, for z → ∞ (cid:82) Tt e − λ ( s − t ) (cid:0) r + λ sup ζ ∈ R ( u ( s, ζ ) − ln (1 + ¯ (cid:15)h ( ζ ))) (cid:1) ds, for z → −∞ . Therefore, we can continuously extend u to z = ±∞ and u ( t, ∞ ) and u ( t, −∞ ) are defined by theabove limit. We observe that for z ∈ {∞ , −∞} , u ( t, z ) satisfies0 = u t ( t, z ) + ( µ − r ) h ( z ) + r − σ h ( z ) − λu ( t, z )+ λ sup ζ ∈ R (cid:16) u ( t, ζ ) − ln (cid:16) (cid:15)h ( ζ )1+¯ (cid:15)h ( z ) (cid:17) { z<ζ } − ln (cid:16) − (cid:15)h ( ζ )1 − (cid:15)h ( z ) (cid:17) { z>ζ } (cid:17) , (A.9)where the function h is continuously extended as h ( ∞ ) := 1 and h ( −∞ ) := 0.Now we define v as v ( t, x ) := u ( t, h − ( x )) for ( t, x ) ∈ [0 , T ] × [0 , v is well-defined because h : R ∪ {∞ , −∞} → [0 , 1] is bijective. We observe that for ( t, x ) ∈ (0 , T ) × (0 , 1) and z = h − ( x ), v t ( t, x ) = u t ( t, z ) ,x (1 − x ) v x ( t, x ) = u z ( t, z ) ,x (1 − x ) v xx ( t, x ) = u zz ( t, z ) − (1 − x ) u z ( t, z ) . (A.10)The PDE for u in (A.7) with ˆ u replaced by u and the equalities in (A.10) produce the PDE for v , which is (3.2). Therefore, statement (i) is valid.To check statement (ii), we observe that v ( t, 0) = u ( t, −∞ ) and v ( t, 1) = u ( t, ∞ ). Then, thecontinuous differentiability of v ( t, 0) and v ( t, 1) with respect to t is followed by that of u ( t, −∞ )and u ( t, ∞ ), and (A.9) produces (3.3).Finally, statement (iii) is a direct consequence of (A.10) and u ∈ C δ , δ ([0 , T ] × R ). Appendix B. Proof of Lemma 5.3 (i) When (cid:15) = 0, Lemma 4.1 and (5.1) produce (5.2).To prove (5.3), we first check that Y ( t,x ) s in (4.9) safisfies dY ( t,x ) s = Y ( t,x ) s (1 − Y ( t,x ) s ) (cid:16) ( µ − r − σ Y ( t,x ) s ) ds + σdB s (cid:17) . (B.1) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 24 Then application of Ito’s formula produces that for ( s, x ) ∈ [ t, T ) × (0 , (cid:16) Y ( t,x ) s − x (cid:17) = (cid:90) st Y ( t,x ) u (1 − Y ( t,x ) u ) (cid:16) Y ( t,x ) u − x )( µ − r − σ Y ( t,x ) u ) + σ Y ( t,x ) u (1 − Y ( t,x ) u ) (cid:17) du + (cid:90) st σ ( Y ( t,x ) u − x ) Y ( t,x ) u (1 − Y ( t,x ) u ) dB u . Since 0 < Y ( t,x ) s < 1, the local martingale part above is a true martingale and we obtain ∂∂s (cid:18) E (cid:20)(cid:16) Y ( t,x ) s − x (cid:17) (cid:21)(cid:19) = E (cid:104) Y ( t,x ) s (1 − Y ( t,x ) s ) (cid:16) Y ( t,x ) s − x )( µ − r − σ Y ( t,x ) s ) + σ Y ( t,x ) s (1 − Y ( t,x ) s ) (cid:17)(cid:105) = − x (1 − x ) · E (cid:34) ∂∂x (cid:32) Y ( t,x ) s (cid:16) − Y ( t,x ) s (cid:17)(cid:16) µ − r − σ Y ( t,x ) s (cid:17) x (1 − x ) (cid:33)(cid:35) , (B.2)where the second equality is from elementary computations using the definition of Y ( t,x ) s in (4.9).For x ∈ (0 , Y ( t,x ) s − xx (1 − x ) = A ( t,s ) − A ( t,s ) x +(1 − x ) with A ( t,s ) := e ( µ − r − σ )( s − t )+ σ ( B s − B t ) , (B.3)and the above expression is decreasing in x , therefore,1 − A ( t,s ) < Y ( t,x ) s − xx (1 − x ) < A ( t,s ) − < x < . (B.4)When (cid:15) = 0, the representation of v x in (4.11) becomes v x ( t, x ) = (cid:90) Tt e − λ ( s − t ) E (cid:34) Y ( t,x ) s (cid:16) − Y ( t,x ) s (cid:17)(cid:16) µ − r − σ Y ( t,x ) s (cid:17) x (1 − x ) (cid:35) ds, (B.5)We take derivative with respect to x in the above expression. Then, the mean value theorem andthe dominated convergence theorem, together with the inequalities (4.14) and (B.4), allow us totake derivative inside of the expectation and obtain that for ( t, x ) ∈ [0 , T ) × (0 , v xx ( t, x ) = (cid:90) Tt e − λ ( s − t ) E (cid:34) ∂∂x (cid:32) Y ( t,x ) s (cid:16) − Y ( t,x ) s (cid:17)(cid:16) µ − r − σ Y ( t,x ) s (cid:17) x (1 − x ) (cid:33)(cid:35) ds (B.6)= − (cid:90) Tt e − λ ( s − t ) ∂∂s (cid:18) E (cid:20)(cid:16) Y ( t,x ) s − xx (1 − x ) (cid:17) (cid:21)(cid:19) ds = − e − λ ( T − t ) E (cid:34)(cid:18) Y ( t,x ) T − xx (1 − x ) (cid:19) (cid:35) − λ (cid:90) Tt e − λ ( s − t ) E (cid:20)(cid:16) Y ( t,x ) s − xx (1 − x ) (cid:17) (cid:21) ds, (B.7)where the second equality is due to (B.2), and the third equality is from integration by parts.Obviously (B.7) implies that v xx ( t, x ) < t, x ) ∈ [0 , T ) × (0 , x ↑ v xx ( t, x ) < x ↓ v xx ( t, x ) < x ↑ v xx ( t, x ) = − e − λ ( T − t ) E (cid:104) (cid:16) − A ( t,T ) (cid:17) (cid:105) − λ (cid:90) Tt e − λ ( s − t ) E (cid:104) (cid:16) − A ( t,s ) (cid:17) (cid:105) ds, lim x ↓ v xx ( t, x ) = − e − λ ( T − t ) E (cid:104) (cid:16) A ( t,T ) − (cid:17) (cid:105) − λ (cid:90) Tt e − λ ( s − t ) E (cid:104) (cid:16) A ( t,s ) − (cid:17) (cid:105) ds, and we conclude that lim x ↑ v xx ( t, x ) < x ↓ v xx ( t, x ) < PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 25 (ii) The SDE for Y ( t,x ) s in (B.1) and 0 < Y ( t,x ) s < s, x ) ∈ [ t, T ) × (0 , ∂∂s (cid:16) E (cid:104) Y ( t,x ) s (cid:105)(cid:17) = E (cid:104) Y ( t,x ) s (cid:16) − Y ( t,x ) s (cid:17) (cid:16) µ − r − σ Y ( t,x ) s (cid:17)(cid:105) (B.8)We divide both sides of (B.8) by x (1 − x ) and take derivative with respect to x . Then, we can putthe derivative inside of the expectation as in the proof of part (i), and obtain E (cid:34) ∂∂x (cid:32) Y ( t,x ) s (cid:16) − Y ( t,x ) s (cid:17)(cid:16) µ − r − σ Y ( t,x ) s (cid:17) x (1 − x ) (cid:33)(cid:35) = ∂∂s (cid:16) E (cid:104) ∂∂x (cid:16) Y ( t,x ) s x (1 − x ) (cid:17)(cid:105)(cid:17) = ∂∂s (cid:32) E (cid:104) ∂∂x Y ( t,x ) s (cid:105) x (1 − x ) − (1 − x ) E (cid:104) Y ( t,x ) s (cid:105) x (1 − x ) (cid:33) . We rearrange the above equation and use (B.8), (B.5), and (B.7) to obtain (cid:90) Tt e − λ ( s − t ) ∂∂s (cid:16) E (cid:104) ∂∂x Y ( t,x ) s (cid:105)(cid:17) ds = x (1 − x ) v xx ( t, x ) + (1 − x ) v x ( t, x ) . (B.9)Now we conclude that F ( t, ˆ y ( t )) < F ( t, ˆ y ( t )) ≤ λ (cid:90) Tt e − λ ( s − t ) E (cid:104) ∂∂x Y ( t,x ) s (cid:105) ds (cid:12)(cid:12)(cid:12) x =ˆ y ( t ) = (cid:16) − e − λ ( T − t ) E (cid:104) ∂∂x Y ( t,x ) T (cid:105) + x (1 − x ) v xx ( t, x ) + (1 − x ) v x ( t, x ) (cid:17) (cid:12)(cid:12)(cid:12) x =ˆ y ( t ) < , where the first inequality is from the definition of F in (5.4) and the positivity of ∂∂x Y ( t,x ) s (see(4.14)), and the equality is due to integration by parts and (B.9), and the last inequality is due tothe positivity of ∂∂x Y ( t,x ) T , v x ( t, ˆ y ( t )) = 0, and v xx ( t, ˆ y ( t )) < F ( t, ˆ y ( t )) > − v (cid:15)x ( t, x (cid:15) ) − v x ( t, x (cid:15) ) = λ (cid:90) Tt e − λ ( s − t ) E (cid:104)(cid:16) ∂∂x Y ( t,x (cid:15) ) s (cid:17) L (cid:15)y ( s, Y ( t,x (cid:15) ) s ) (cid:105) ds. (B.10)In the above expression, when we take limit as (cid:15) ↓ 0, the inequalities (4.13) and (4.14) enable us touse the dominated convergence theorem to conclude thatlim (cid:15) ↓ (cid:0) v (cid:15)x ( t, x (cid:15) ) − v x ( t, x (cid:15) ) (cid:1) = 0 . The above limit and the continuity of v x implies (5.6).To prove (5.7), we first observe that for ( t, x ) ∈ [0 , T ) × (0 , 1) and ( s, z ) ∈ ( t, T ) × (0 , Y ( t,x ) s is given by ϕ ( s, z ; t, x ) := ∂∂z P (cid:16) Y ( t,x ) s ≤ z (cid:17) = exp (cid:18) − σ s − t ) (cid:16) ( r − µ + σ )( s − t )+ln (cid:16) z (1 − x )(1 − z ) x (cid:17)(cid:17) (cid:19) σz (1 − z ) √ π ( s − t ) . (B.11)Then, the expression in (4.11) and ∂∂x Y ( t,x ) s = Y ( t,x ) s (1 − Y ( t,x ) s ) x (1 − x ) imply that v (cid:15)x ( t, x ) − v x ( t, x ) = λ (cid:90) Tt e − λ ( s − t ) E (cid:34) Y ( t,x ) s (cid:16) − Y ( t,x ) s (cid:17) x (1 − x ) L (cid:15)y ( s, Y ( t,x ) s ) (cid:35) ds = λ (cid:90) Tt e − λ ( s − t ) (cid:18)(cid:90) z (1 − z ) x (1 − x ) L (cid:15)y ( s, z ) ϕ ( s, z ; t, x ) dz (cid:19) ds. (B.12) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 26 For x ∈ (0 , ∂∂x (cid:16) z (1 − z ) x (1 − x ) ϕ ( s, z ; t, x ) (cid:17) = (1 − z ) z (cid:16) r − µ +(2 x − ) σ + s − t ln (cid:16) z (1 − x )(1 − z ) x (cid:17)(cid:17) ϕ ( s,z ; t,x )(1 − x ) x σ . (B.13)Assumption 5.1 implies that (cid:12)(cid:12)(cid:12) r − µ + σ (cid:12)(cid:12)(cid:12) ≤ σ , (cid:12)(cid:12) r − µ + (2 x − ) σ (cid:12)(cid:12) ≤ σ for x ∈ (0 , . (B.14)Then, (B.11) and (B.13) produce the following: (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x (cid:16) z (1 − z ) x (1 − x ) ϕ ( s, z ; t, x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) dz ≤ (cid:90) (cid:18) σ + s − t (cid:12)(cid:12)(cid:12)(cid:12) ln (cid:18) z (1 − x )(1 − z ) x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) σ (1 − x ) x √ π ( s − t ) exp (cid:32) − (cid:16) ln (cid:16) z (1 − x )(1 − z ) x (cid:17)(cid:17) σ ( s − t ) + (cid:12)(cid:12)(cid:12) ln (cid:16) z (1 − x )(1 − z ) x (cid:17)(cid:12)(cid:12)(cid:12)(cid:33) dz = (cid:90) ∞−∞ ( σ + s − t | ζ | ) e − ζ σ s − t ) + | ζ | σ (1 − x ) x √ π ( s − t ) · x − x e ζ (1+ x − x e ζ ) dζ ≤ (cid:90) ∞−∞ ( σ + s − t | ζ | ) e − ζ σ s − t ) +4 σ s − t ) σ (1 − x ) x √ π ( s − t ) dζ = √ − x ) x (cid:18) σ √ π ( s − t ) (cid:19) e σ ( s − t ) , (B.15)where the first inequality is due to (B.14), and the first equality is obtained by change of variablesas ζ = ln (cid:16) z (1 − x )(1 − z ) x (cid:17) . The second inequality is due to the inequality of arithmetic and geometricmeans, and the second equality is obtained by direct computations.By (B.15), we conclude that (cid:90) Tt e − λ ( s − t ) (cid:90) (cid:12)(cid:12)(cid:12) ∂∂x (cid:16) z (1 − z ) x (1 − x ) ϕ ( s, z ; t, x ) (cid:17)(cid:12)(cid:12)(cid:12) dz ds < ∞ , (B.16)and the function H : [0 , T ) × (0 , → R given by H ( t, x ) := λ (cid:90) Tt e − λ ( s − t ) (cid:90) ∂∂x (cid:16) z (1 − z ) x (1 − x ) ϕ ( s, z ; t, x ) (cid:17) L (cid:15)y ( s, z ) dz ds (B.17)is well-defined due to (B.16) and the boundedness | L (cid:15)y | ≤ (cid:15) − (cid:15) in (4.13). Then, (B.15) implies that | H ( t, x ) | ≤ √ λe σ T (1 − x ) x (cid:16) T + √ Tσ √ π (cid:17) · (cid:15) − (cid:15) for ( t, x ) ∈ [0 , T ) × (0 , . (B.18)Now, let’s check that H ( t, x ) = v (cid:15)xx ( t, x ) − v xx ( t, x ) . (B.19)Indeed, for ( t, x ) ∈ [0 , T ) × (0 , H ( t, x ) = lim δ → δ (cid:90) x + δx H ( t, η ) dη = lim δ → δ λ (cid:90) Tt e − λ ( s − t ) (cid:90) (cid:16) z (1 − z ) ϕ ( s,z ; t,x + δ )( x + δ )(1 − x − δ ) − z (1 − z ) ϕ ( s,z ; t,x ) x (1 − x ) (cid:17) L (cid:15)y ( s, z ) dz ds = lim δ → δ (cid:0)(cid:0) v (cid:15)x ( t, x + δ ) − v x ( t, x + δ ) (cid:1) − (cid:0) v (cid:15)x ( t, x ) − v x ( t, x ) (cid:1)(cid:1) = v (cid:15)xx ( t, x ) − v xx ( t, x ) , PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 27 where the second equality is due to Fubini’s theorem and the fundamental theorem of calculus, andthe third equality is from (B.12).Finally, we conclude (5.7) by the following observation:lim sup (cid:15) ↓ (cid:12)(cid:12) v (cid:15)xx ( t, x (cid:15) ) − v xx ( t, x ) (cid:12)(cid:12) ≤ lim sup (cid:15) ↓ (cid:12)(cid:12) v (cid:15)xx ( t, x (cid:15) ) − v xx ( t, x (cid:15) ) (cid:12)(cid:12) + lim sup (cid:15) ↓ (cid:12)(cid:12) v xx ( t, x (cid:15) ) − v xx ( t, x ) (cid:12)(cid:12) ≤ lim sup (cid:15) ↓ √ λe σ T (1 − x (cid:15) ) x (cid:15) (cid:16) T + √ Tσ √ π (cid:17) · (cid:15) − (cid:15) = 0 , where the second inequality is due to (B.18), (B.19), and the continuity of v xx . Appendix C. Proof of Lemma 5.7 (i) This result due to equation (2.7) in [35] and the inequality | ˆ y ( t ) − y ∞ | ≤ , T ] × (0 , → R asΓ( t, x ) := E (cid:34) Y (0 ,x ) t (cid:16) − Y (0 ,x ) t (cid:17)(cid:16) µ − r − σ Y (0 ,x ) t (cid:17) x (1 − x ) (cid:35) . (C.1)Note that Γ does not depend on λ . The equations (B.5) and (B.6) can be written as v x ( t, x ) = (cid:90) T − t e − λs Γ( s, x ) ds, v xx ( t, x ) = (cid:90) T − t e − λs Γ x ( s, x ) ds (C.2)We can do the similar argument to obtain representations for v xxx and partial derivatives of v x and v xx with respect to λ , with the observation that Γ , Γ x , Γ xx are continuous & bounded maps on[0 , T ] × (0 , ∂∂λ v x ( t, x ) = − (cid:90) T − t e − λs s Γ( s, x ) ds, ∂∂λ v xx ( t, x ) = − (cid:90) T − t e − λs s Γ x ( s, x ) dsv xx ( t, x ) = (cid:90) T − t e − λs Γ x ( s, x ) ds, v xxx ( t, x ) = (cid:90) T − t e − λs Γ xx ( s, x ) ds. (C.3)Using Y ( t,x ) t = x , direct computations produceΓ(0 , x ) = µ − r − σ x, Γ x (0 , x ) = − σ , Γ xx (0 , x ) = 0 . (C.4)With (C.4) and (5.38), we apply Lemma D.2 to (C.3) and conclude (5.39).(iii) The mean value theorem and (5.2) produceˆ y ,λ + δ ( t ) − ˆ y ,λ ( t ) = − v ,λ + δx ( t, ˆ y ,λ ( t )) − v ,λx ( t, ˆ y ,λ ( t )) v ,λ + δxx ( t,z ( λ,δ )) for z ( λ, δ ) between ˆ y ,λ ( t ) and ˆ y ,λ + δ ( t ) , where we specify the dependence on λ for clarity. Since ∂∂λ v x exists (see (C.3)), the above equalityand (5.3) ensure that ˆ y ,λ ( t ) is differentiable with respect to λ and ∂∂λ ˆ y ( t ) = − ∂∂λ v x ( t, x ) (cid:12)(cid:12) x =ˆ y ( t ) v xx ( t, ˆ y ( t )) . (C.5)Observe that the bounds for ∂∂x Y (0 ,x ) t and Y (0 ,x ) t − xx (1 − x ) in (4.14) and (B.4) do not depend on the variable x . Therefore, the following convergence is uniform on x ∈ (0 , x ( t, x ) = − E (cid:20) σ (cid:16) ∂∂x Y (0 ,x ) t (cid:17) + 2 (cid:16) µ − r − σ Y (0 ,x ) t (cid:17) (cid:18) Y (0 ,x ) t − xx (1 − x ) (cid:19) ∂∂x Y (0 ,x ) t (cid:21) −−→ t ↓ − σ . PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 28 Hence, there is a constant ˜ T ∈ (0 , T ) such that Γ x ( t, x ) ≤ − σ for ( t, x ) ∈ [0 , ˜ T ] × (0 , v xx in (C.3) imply λv xx ( t, ˆ y ( t )) ≤ − σ (cid:16) − e − λ ( T − t ) (cid:17) for ( t, x ) ∈ [ T − ˜ T , T ) × (0 , . (C.6)We obtain (cid:107) Γ xt (cid:107) ∞ < ∞ by using Ito’s formula with (B.1) and the bounds (4.14) and (B.4). Then,(C.3) and (C.4) imply (cid:12)(cid:12) λv xx ( t, ˆ y ( t )) + σ (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:90) T − t e − λs s (cid:16) Γ x ( s, ˆ y ( t )) − Γ x (0 , ˆ y ( t )) s (cid:17) ds + σ e − λ ( T − t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) Γ xt (cid:107) ∞ (cid:16) − e − λ ( T − t ) λ − ( T − t ) e − λ ( T − t ) (cid:17) + σ e − λ ( T − t ) . This implies that there exists a constant ˜Λ (may depend on ˜ T ) such that λv xx ( t, ˆ y ( t )) ≤ − σ for ( t, λ ) ∈ [0 , T − ˜ T ] × [ ˜Λ , ∞ ) . (C.7)Using (C.3) and (C.4), we obtain (cid:12)(cid:12)(cid:12) λ ∂∂λ v x ( t, x ) (cid:12)(cid:12) x =ˆ y ( t ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:90) T − t e − λs s (cid:16) Γ( s, ˆ y ( t )) − Γ(0 , ˆ y ( t )) s (cid:17) ds (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) σ ( y ∞ − ˆ y ( t )) λ (cid:90) T − t e − λs s ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) − e − λ ( T − t ) + λ ( T − t ) (cid:0) λ ( T − t ) (cid:1) e − λ ( T − t ) (cid:17) for ( t, λ ) ∈ [0 , T ] × [1 , ∞ ) , (C.8)where the second inequality is due to (cid:107) Γ t (cid:107) ∞ < ∞ and (5.38).From (C.5), we obtain the boundedness of (cid:12)(cid:12) λ ∂∂λ ˆ y ( t ) (cid:12)(cid:12) :sup ( t,λ ) ∈ [0 ,T − ˜ T ] × [˜Λ , ∞ ) (cid:12)(cid:12) λ ∂∂λ ˆ y ( t ) (cid:12)(cid:12) < ∞ due to (C.7), (C.8), sup x> x (1 + x ) e − x < ∞ ,sup ( t,λ ) ∈ [0 ,T − ˜ T ] × [1 , ˜Λ] (cid:12)(cid:12) λ ∂∂λ ˆ y ( t ) (cid:12)(cid:12) < ∞ due to the continuity on compact set , sup ( t,λ ) ∈ [ T − ˜ T ,T ) × [1 , ∞ ) (cid:12)(cid:12) λ ∂∂λ ˆ y ( t ) (cid:12)(cid:12) < ∞ due to (C.6), (C.8), sup x> x (1+ x ) e − x − e − x < ∞ .Therefore, we conclude (5.40).(iv) The bounds (4.14) and (5.40) enable us to use Leibniz integral rule to obtain ∂∂λ E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t, ˆ y ( t )) s − ˆ y ( s ) (cid:12)(cid:12)(cid:12)(cid:105) = E (cid:20)(cid:18) ∂∂λ ˆ y ( t ) · ∂∂x Y ( t,x ) s (cid:12)(cid:12)(cid:12) x =ˆ y ( t ) − ∂∂λ ˆ y ( s ) (cid:19) · sgn (cid:16) Y ( t, ˆ y ( t )) s − ˆ y ( s ) (cid:17)(cid:21) . The above expression, together with the bounds (4.14) and (5.40), implies (5.41).(v) Explicit computations using the expression of Y ( t,x ) s in (4.9) produce E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t,x ) s − x (cid:12)(cid:12)(cid:12)(cid:105) √ s − t = (cid:90) R x (1 − x ) x +(1 − x ) e ( r − µ + σ 22 ) u − σuz · (cid:12)(cid:12) − e ( r − µ + σ 22 ) u − σuz (cid:12)(cid:12) u · e − z √ π dz (cid:12)(cid:12)(cid:12)(cid:12) u = √ s − t (C.9) ≤ (cid:90) R e | r − µ + σ | u + σu | z | − e −| r − µ + σ | u − σu | z | u · e − z √ π dz (cid:12)(cid:12)(cid:12)(cid:12) u = √ s − t = e ( | r − µ + σ | + σ ) u (cid:18) (cid:82) σu √ e − z √ π dz (cid:19) − e ( −| r − µ + σ | + σ ) u (cid:18) − (cid:82) σu √ e − z √ π dz (cid:19) u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = √ s − t , PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 29 where we use | − e a | ≤ e | a | − e −| a | for a ∈ R to obtain the inequality. The last expression convergesto √ σ √ π as u ↓ 0, therefore, it is bound on u ∈ (0 , √ T ]. Then, we conclude (5.42).Observe that the integrand in (C.9) is bounded by e C | z | · e − z √ π for a constant C independentof ( u, x, z ) ∈ (0 , √ T ] × (0 , × R . Since this bound is integrable with respect to z , we apply thedominated convergence theorem to (C.9) and obtainlim s ↓ t E (cid:104)(cid:12)(cid:12)(cid:12) Y ( t,x ) s − x (cid:12)(cid:12)(cid:12)(cid:105) √ s − t = (cid:90) R x (1 − x ) σ | z | · e − z √ π dz = σ (cid:113) π x (1 − x ) . Therefore, we conclude (5.43). Appendix D. Supplementary Lemmas Lemma D.1. Let F : [0 , T ] × [0 , → R be a continuous function. We define f : [0 , T ] × [0 , → [0 , as f ( t, x ) := max (cid:40) z : z ∈ argmax y ∈ [0 , F ( t, x, y ) (cid:41) , (D.1) then f is upper semicontinuous (which is obviously Borel-measurable).Proof. This type of result is well-known (e.g., see p. 153 in [6]), but we give a short proof here forthe sake of self-containedness.Since F is continuous, argmax y ∈ [0 , F ( t, x, y ) is a nonempty closed subset of [0 , y ∈ [0 , F ( t, x, y ) exists and f is well-defined. Let { ( t n , x n ) } n ∈ N ⊂ [0 , T ] × [0 , t ∞ , x ∞ ) such that lim n →∞ f ( t n , x n ) exists. Then, by definition of f , F ( t n , x n , f ( t ∞ , x ∞ )) ≤ F ( t n , x n , f ( t n , x n )) . (D.2)We let n → ∞ above and using the continuity of F to obtain F ( t ∞ , x ∞ , f ( t ∞ , x ∞ )) ≤ F ( t ∞ , x ∞ , lim n →∞ f ( t n , x n )) . (D.3)This implies that lim n →∞ f ( t n , x n ) ∈ argmax y ∈ [0 , F ( t ∞ , x ∞ , y ), and the definition of f ensures f ( t ∞ , x ∞ ) ≥ lim n →∞ f ( t n , x n ) . Therefore, f is upper semicontinuous. (cid:3) Lemma D.2. Let f ( s, x ) : [0 , t ] × (0 , → R be a continuous function, and g ( λ ) : [1 , ∞ ) → (0 , be a function satisfying lim λ →∞ g ( λ ) = x ∞ ∈ (0 , . Then, for α ∈ { , , , , } and t > , lim λ →∞ λ α +1 (cid:90) t e − λs s α f ( s, g ( λ )) ds = c α · f (0 , x ∞ ) , (D.4) where c = 1 , c = √ π , c = 1 , c = √ π , c = 2 . Also, there exists a constant C such that λ α +1 (cid:90) t e − λs s α ds ≤ C, for ( t, λ, α ) ∈ [0 , ∞ ) × [1 , ∞ ) × { , , , , } . (D.5) Proof. Let η > f on a compact set containing thepoint (0 , x ∞ ), together with lim λ →∞ g ( λ ) = x ∞ ∈ (0 , δ > | f ( s, g ( λ )) − f (0 , x ∞ ) | ≤ η for any ( s, λ ) ∈ [0 , δ ] × (cid:2) δ , ∞ (cid:1) . (D.6) PTIMAL INVESTMENT WITH SEARCH FRICTIONS AND TRANSACTION COSTS 30 Simple computations and (D.6) producelim sup λ →∞ (cid:12)(cid:12)(cid:12) λ (cid:90) t e − λs f ( s, g ( λ )) ds − f (0 , x ∞ ) (cid:12)(cid:12)(cid:12) = lim sup λ →∞ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) δ λe − λs ( f ( s, g ( λ )) ds − f (0 , x ∞ )) ds + (cid:90) tδ λe − λs ( f ( s, g ( λ )) ds − f (0 , x ∞ )) ds − e − λt f (0 , x ∞ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ η. Since η > α = 0. The other cases in(D.4) can be obtained by the same way as above, using the following expressions: λ (cid:90) t e − λs s ds = −√ λte − λt + (cid:90) √ λt e − s ds −−−→ λ →∞ √ π ,λ (cid:90) t e − λs s ds = 1 − (1 + λt ) e − λt −−−→ λ →∞ ,λ (cid:90) t e − λs s ds = − (3+2 λt ) √ λte − λt + 32 (cid:90) √ λt e − s ds −−−→ λ →∞ √ π ,λ (cid:90) t e − λs s ds = 2 − (2 + λt (2 + λt )) e − λt −−−→ λ →∞ . (D.7)One can easily observe that sup x> x α e − x < ∞ for α ∈ { , , , , } . This observation and theexplicit expressions in (D.7) produce the bound (D.5).. This observation and theexplicit expressions in (D.7) produce the bound (D.5).