Outline Preemption games under L´ evy uncertainty Svetlana Boyarchenko and Sergei Levendorski˘ i University of Texas at Austin; University of Leicester March 9, 2012 Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 1 / 49
Outline Outline Introduction 1 Deterministic environment 2 Stochastic environment 3 Preemption under L´ evy uncertainty 4 Subgame perfect equilibria 5 Conclusion 6 Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 1 / 49
Introduction Motivation Stopping time games are a special case of stochastic games Stopping time games have important applications in economics and finance Main part of the research on stopping games is focused on existence of the value of the game and optimal stopping strategies Deriving optimal stopping strategies and pricing game options is not less important, especially in economics and finance Preemption games are a subset of stopping time games In preemption games, there is a region called the preemption zone, where each of the players prefers to be the leader (be the first to move) In the preemption zone, it may happen that it is optimal to move for one player, but not for both, hence there is a coordination problem Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 2 / 49
Introduction Aims of the paper For a stochastic preemption game with two players characterize subgame perfect equilibria study effects of jumps on equilibria study effects of players’ asymmetry For simplicity, consider one-factor jump-diffusion models (L´ evy models) Technical challenges In entry-exit problems with strategic interactions under non-Gaussian uncertainty and other stopping games, non-monotone payoffs are the rule rather than exception. In our model, the value function of the low cost player is not only non-monotone, but, for some equilibria, it is also discontinuous. Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 3 / 49
Introduction Our paper builds on the following Existing models 1. Fudenberg and Tirole (1985) – preemption games in continuous time, deterministic environment, symmetric firms 2. Dixit and Pindyck (1994) – preemption games in continuous time, Gaussian uncertainty, symmetric firms 3. Thijssen et al. (2006) – preemption games in continuous time, imperfect information, symmetric firms 4. Pawlina and Kort (2006) – preemption games in continuous time, Gaussian uncertainty, asymmetric firms (sunk cost asymmetry) 5. Thijssen et al. (2002) – preemption games in continuous time, non-Gaussian uncertainty, symmetric firms Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 4 / 49
Deterministic environment Deterministic environment Two firms consider a single investment opportunity to enhance their profit flow time t ≥ 0 is continuous; r > 0 – discount rate; π n i n j – instantaneous profit of firm i ∈ { 1 , 2 } , where, for k ∈ { i , j } , � 0 if firm k has not invested n k = . 1 if firm k has invested Assume π 10 > π 00 ∨ ∨ . π 11 > π 01 c i ( t ) - the present value of the cost of investment at time t of firm i ; c i ( t ) e rt � ′ < 0; c i ( t ) e rt � ′′ > 0 ∀ t . � � Additional assumptions guarantee that investment is not optimal at t = 0; investment happens in finite time. Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 5 / 49
Deterministic environment Let T i be the time of investment of firm i Value functions at time t = 0; T i < T j � T i � T j � ∞ V i ( T i , T j ) π 00 e − rt dt + π 10 e − rt dt − c i ( T i ) + π 11 e − rt dt = 0 T i T j � T i � T j � ∞ V j ( T i , T j ) π 00 e − rt dt + π 01 e − rt dt + π 11 e − rt dt − c j ( T j ) = 0 T i T j Value functions at time t = 0; T i = T j = T s � T s � ∞ π 00 e − rt dt + π 11 e − rt dt − c i ( T s ) . P i ( T s ) = 0 T s If the firms precommit to investment times then optimal values T i < T j < T s can be easily found from the FOC. Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 6 / 49
Deterministic environment Value functions: t–the current state V i lead ( t ) – the value at entry of firm i if this firm is the leader V i f ( t ) – the value of firm i if this firm is the follower preempted at t P i ( t ) – the value at entry of firm i if the firms enter simultaneously � V i ( t , T j ) , if t ≤ T j , V i lead ( t ) = P i ( t ) , if t ≤ T j . � V i ( t , T i ) , if t ≤ T i , V i f ( t ) = P i ( t ) , if t ≤ T i . Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 7 / 49
Deterministic environment Value functions – symmetric firms V i lead (t) V i f (t) P i (t) 0 T s t T p T j T i Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 8 / 49
Deterministic environment Value functions – asymmetric firms, c 2 ( t ) = kc 1 ( t ) , k > 1 V 1 lead (t) V 1 f (t) V 2 lead (t) V 2 f (t) t 1 t 2 t 3 t 4 t Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 9 / 49
Deterministic environment Strategies and equilibrium concepts: Fudenberg and Tirole (1985) A simple strategy of player i in the game starting at t is a pair of real valued functions ( G i , α i ) : [ t , ∞ ) × [ t , ∞ ) → [0 , 1] × [0 , 1] , satisfying certain properties. For any t , G i ( t ) is the cumulative probability that i invested by time t ; α i ( t ) is the intensity with which the player moves right before G i ( t ) jumps. Nash equilibrium is defined in a standard way. If simple strategies satisfy a certain intertemporal consistency condition, they are called closed-loop . A perfect equilibrium is a Nash equilibrium in closed-loop strategies. Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 10 / 49
Stochastic environment Uncertainty (Ω , F , ( F t ) 0 ≤ t < ∞ , P ) is a filtered probability space X = { X t } t ≥ 0 is a L´ evy process on R Can be defined by the generating triplet ( σ 2 , µ, F ( dx )), where σ 2 is the variance of the Brownian motion (BM) component, µ ∈ R , and F ( dx ) is the L´ evy density. For ( α, β ) ⊂ R \ 0, F (( α, β )) dt is the probability of a jump from 0 into ( α, β ) during time interval dt . Alternatively, a L´ evy process can be uniquely defined by L´ evy exponent: � e β X t � = e t Ψ( β ) . E If X is the BM, Ψ( β ) = σ 2 2 β 2 + µβ. Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 11 / 49
Stochastic environment L´ evy-Khintchine formula and example: DEJD The L´ evy-Khintchine formula expresses the L´ evy exponent in terms of generating triplet. If the jump part is a finite variation process, then the L´ evy-Khintchine formula can be written in the form Ψ( β ) = σ 2 � 2 β 2 + b β + ( e β y − 1) F ( dy ) , (1) R \ 0 Double-exponential jump-diffusion (DEJD) model: σ > 0, F ( dy ) = c + λ + e − λ + y 1 (0 , + ∞ ) ( y ) dy + c − ( − λ − ) e − λ − y 1 ( −∞ , 0) ( y ) dy , (2) where λ − < 0 < λ + and c ± ≥ 0 If c + = 0, then there are no positive jumps (process is spectrally negative). Substituting (2) into (1): Ψ( β ) = σ 2 c + β c − β 2 β 2 + b β + λ + − β + λ − − β . (3) Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 12 / 49
Stochastic environment Main objects (i) τ + h , the first entrance time of X into the semi-infinite interval [ h , + ∞ ); (ii) the normalized expected present value (EPV) operator �� + ∞ � ( E q f )( x ) = E x qe − qt f ( X t ) dt 0 (iii) the supremum and infimum processes X t = sup 0 ≤ s ≤ t X s and X t = inf 0 ≤ s ≤ t X s ; and the EPV operators under supremum and infimum processes �� + ∞ �� + ∞ � � ( E + q f )( x ) = E x qe − qt f ( X t ) dt ( E − q f )( x ) = E x qe − qt f ( X t ) dt , ; 0 0 (iv) the notation κ ± E ± q e β x � � | x =0 ; q ( β ) = (v) the operator version of the Wiener-Hopf factorization formula which states that E q = E + q E − q = E − q E + q , as operators in spaces of measurable semi-bounded functions Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 13 / 49
Stochastic environment Example Brownian motion � + ∞ e − β + y u ( x + y ) dy , E + q u ( x ) = β + 0 � 0 e − β − y u ( x + y ) dy , E − q u ( x ) = − β − −∞ where β − < 0 < β + are the roots of q − σ 2 2 β 2 − µβ = 0 . Double-exponential jump-diffusion model: � + ∞ e − β + E + � a + j β + j y u ( x + y ) dy , q u ( x ) = j 0 j =1 , 2 � 0 e − β − E − � a − j ( − β − j y u ( x + y ) dy , q u ( x ) = j ) −∞ j =1 , 2 2 < λ − < β − 1 < λ + < β + where β − 1 < 0 < β + 2 are the roots of q − Ψ( β ) = 0 , and a ± j > 0 are constants. Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 14 / 49
Stochastic environment Standing assumptions Assumption 1. No-bubble condition �� ∞ � e − qt e X t dt < ∞ ⇔ q − Ψ(1) > 0 E 0 Assumption 2. X is a L´ evy process satisfying (ACP)-property, with non-trivial supremum and infimum processes. Assumption 3. Monotonicity of density of negative jumps and complete monotonicity of density of positive jumps (Bernstein condition) (ACP)-property: For any f ∈ L ∞ ( R ), E q f is continuous. A sufficient condition is: for some t > 0, the transition measure P t is absolutely continuous. Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 15 / 49
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