An incomplete equilibrium with a stochastic annuity
aa r X i v : . [ q -f i n . M F ] S e p AN INCOMPLETE EQUILIBRIUM WITH A STOCHASTICANNUITY
KIM WESTON AND GORDAN ŽITKOVIĆ
Abstract.
We prove the global existence of an incomplete, continuous-timefinite-agent Radner equilibrium in which exponential agents optimize theirexpected utility over both running consumption and terminal wealth. Themarket consists of a traded annuity, and, along with unspanned income, themarket is incomplete. Set in a Brownian framework, the income is drivenby a multidimensional diffusion, and, in particular, includes mean-revertingdynamics.The equilibrium is characterized by a system of fully coupled quadraticbackward stochastic differential equations, a solution to which is proved toexist under Markovian assumptions. Introduction
We prove the existence of a Radner equilibrium in an incomplete, continuous-timefinite-agent market setting. The economic agents act as price takers in a fullycompetitive setting and maximize exponential utility from running consumptionand terminal wealth. An annuity in one-net supply is traded on a financial market,and it pays a constant running and terminal dividend to its shareholders. Theagents choose between consuming their income and dividend streams or investingin the annuity.Although our setting and the income dynamics are quite general, our financialmarket looks relatively simple at first glance. The only available asset is the annuity,and the agents’ only choice at any given moment is how much to consume, keepingin mind that the only way to transfer wealth from one time to the next is throughthe annuity. This apparent simplicity is quite misleading, since the scarcity of theavailable traded assets leads to market incompleteness, a notorious difficulty inequilibrium analysis. Indeed, the fewer assets the agents have at their disposal, the
Date : September 18, 2018.2010
Mathematics Subject Classification.
Primary: 91B51. Secondary: 60H30. JEL Classifi-cation: D52, G12.
Key words and phrases.
Incomplete markets, Radner equilibrium, Annuity, BSDE, Systems ofBSDE.The authors are grateful to Kasper Larsen for helpful discussions. The first author acknowl-edges the support by the National Science Foundation under Grant No. DMS-1606253. The secondauthor acknowledges the support by the National Science Foundation under Grant No. DSM-1815017 (2018-2021). Any opinions, findings and conclusions or recommendations expressed inthis material are those of the authors and do not necessarily reflect the views of the NationalScience Foundation (NSF). less efficient the market becomes and the harder it becomes to use the standardtools such as the representative agent approach. In our case, this lack of assets ispushed to its limit.Admittedly, it would be more realistic to consider markets with several assets, bothrisky and riskless, where the incompleteness is derived from the constraints on eachasset’s ability to incorporate all the risk present in the environment. We believethat the exploration of such problems is one of the most interesting and importantareas of future research in this area. Unfortunately, the formidable mathematicaldifficulties present in virtually all such problems leave them outside the scope ofthe techniques available to us today.One of the advantages of our model is its ability to incorporate various incomestream dynamics, including unspanned mean-reverting income streams (which havebeen studied extensively for their empirical relevance; see, e.g., [Wan04, Wan06,Coc14]). To the best of our knowledge, our model is the first with exponentialagents to incorporate unspanned mean-reverting income in equilibrium and provethe existence of such a equilibrium. The general income streams we study lead tostochastic annuity dynamics, which prevent a money market account from beingreplicated by trading in the annuity in equilibrium.Our approach crucially relies on the presence of a traded annuity. We also needutility functions of exponential type and a Markovian assumption on the dynamicsof the income streams in order to obtain conveniently structured individual agentproblems, amenable to a BSDE analysis. Even so, the analysis involves a non-standard Ansatz for the value function, as we need to formally treat the asset price A as a quantity that, in standard models, plays the role of a money market account.We are not the first to introduce a traded annuity into an equilibrium model (see,e.g., [VV99, Cal01, CLM12, CL14, Wes18]). Our contribution is to recognize therole of a traded annuity price in the individual agent value functions, even whengeneral income streams render the annuity dynamics computationally intractable.The backward stochastic differential equation (BSDE)/PDE-system approach toincomplete market equilibria dates back to [Žit12, Zha12, CL15, KXŽ15, XŽ18],with the early work relying on a smallness- type assumption on some ingredientof the model (the time-horizon, size of the endowment, etc.) The mathematicalanalysis of the present paper is quite involved and relies heavily on some recentresults of [XŽ18], which overcome smallness conditions and treat the existence andstability of solutions to quadratic systems of BSDE. Moreover, the applicability ofthose results in our setting is not at all immediate and is contingent on a numberof a-priori estimates specific to our model. Notation and conventions.
For
J, d ∈ N , The set of J × d -matrices is denoted by R J × d . The Euclidean space R J is identified with the set of R J × , i.e., vectors in TOCHASTIC ANNUITY IN EQUILIBRIUM 3 R J are columns by default. The i -th row of a matrix Z ∈ R J × d will be denoted by Z i , and |·| will denote the Euclidean norm on either R J × d or R J .We work on a finite time horizon [0 , T ] with T >
0, where F = {F t } t ∈ [0 ,T ] is theusual augmentation of the filtration generated by a d -dimensional Brownian motion B .The stochastic integral with respect to B is taken for R × d -valued (row) processes asif dB were a column of its components, i.e., R σ ( t ) dB t stands for P dj =1 R σ j ( t ) dB j .Similarly, for a process Z with values in R J × d , R Z t dB t is an R J -valued processwhose components are the stochastic integrals of the rows Z i of Z with respect to dB t .For a function defined on a domain in R d , the derivative Du is always assumedto take row-vector values, i.e., Du ( x ) ∈ R d × . If u is R J valued, the Jacobian Du will, as usual, be interpreted as an element of R J × d . The Hessian, D u of ascalar-valued function takes values in R d × d , and we will have no need for Hessiansof vector-valued maps in this paper.To relieve the notation, we omit the time-index from many expressions involvingstochastic processes but keep (and abuse) the notation dt for an integral with respectto the Lebesgue measure.The set of all adapted, continuous and uniformly bounded processes is denoted by S ∞ , and the set of all processes of bounded mean oscillation by BMO (we refer thereader to [Kaz94] for all the necessary background on the BMO processes). Thefamily of all B -integrable processes σ such that R σ dB is in BM O is denoted bybmo.The set of all F -progressively measurable process is denoted by P . P r denotes theset of all c ∈ P with R T | c | r dt < ∞ , a.s. The same notation is used for scalar,vector or matrix-valued processes - the distinction will always be clear from thecontext. 2. The problem
Model primitives.
The model primitives can be divided into three groups. Inthe first one, we describe the uncertain environment underlying the entire economy.In the second, we postulate the form of the dynamics of the traded asset, and in thethird we describe the characteristics of individual agents. A single real consumptiongood is taken as the numeraire throughout.2.1.1.
The state process.
For d ∈ N , we start with an R d -valued state process ξ whose dynamics is given by dξ t = Λ( t, ξ t ) dt + Σ( t, ξ t ) dB t , ξ = x ∈ R d (2.1)where the measurable functions Λ : [0 , T ] × R d → R d and Σ : [0 , T ] × R d → R d × d satisfy the following the regularity assumption: TOCHASTIC ANNUITY IN EQUILIBRIUM 4
Assumption 2.1 (Regularity of the state process) . There exists a constant
K > t, t ′ ∈ [0 , T ] , x, x ′ ∈ R d and z ∈ R d × we have(1) | Λ( t, x ) | ≤ K and | Λ( t, x ) − Λ( t, x ′ ) | ≤ K | x − x ′ | ,(2) | Σ( t, x ) | ≤ K , and | Σ( t, x ) − Σ( t, x ′ ) | ≤ K ( p | t ′ − t | + | x − x ′ | ) and(3) | Σ( t, x ) z | ≥ K | z | . Remark . Under Assumption 2.1 the SDE (2.1) admits a unique strong solution.The full significance of the assumptions above, however, will only be apparent inthe later sections and is related to the ability to use certain existence results forsystems of backward stochastic differential equations.2.1.2.
The traded asset.
Our market consists of a single real asset A in one-netsupply, whose dynamics we postulate to be of the following form: dA t = ( A t µ t − dt + A t σ t dB t , A T = 1 , (2.2)with the processes µ and σ to be determined in the equilibrium. It can be inter-preted as an annuity which pays a dividend at the rate 1 during [0 , T ], as well as aunit lump sum payment at time T .Let Γ, the coefficient space , denote the set of all pairs γ = ( µ, σ ), where µ is ascalar-valued process and σ is an R × d -valued bmo process. For simplicity, we oftenidentify the market A γ with its coefficient pair γ = ( µ, σ ), and talk simply about the market γ . The set of all markets given by (2.2) is not bijectively parametrizedby Γ as not every γ ∈ Γ defines a market. Indeed, the terminal condition A T = 1imposes a nontrivial relationship between µ and σ ; for example, if µ is deterministic, σ either has to vanish or one of its components has to be truly stochastic. The setof those γ ∈ Γ that do define a market is denoted by Γ f and its elements are saidto be feasible . If we need to stress that it comes with feasible coefficients γ ∈ Γ f ,we write A γ for the process given by (2.2) .2.1.3. Agents.
There are a finite number I ∈ N of economic agents, each of whichis characterized by the following four elements:(1) the risk-aversion coefficient α i >
0. It fully characterizes the agent’sutility function U i which is of exponential form U i ( c ) = − α i e − α i c for c ∈ R . (2) the random-endowment (stochastic income) rate . Each agent re-ceives an endowment of the consumption good at the rate e it = e i ( t, ξ t ) anda lump sum e iT = e i ( T, ξ T ) at time T , for some function e i : [0 , T ] × R d → R .(3) the initial holding π i ∈ R is the initial number of shares of the annuity A held by the agent.With the cumulative endowment rate defined by e = P Ii =1 e i , we impose the following regularity conditions: TOCHASTIC ANNUITY IN EQUILIBRIUM 5
Assumption 2.3 (Regularity of the endowment rates) . (1) Each e i is bounded and continuous, while its terminal section e i ( T, · ) is α -Hölder continuous for some α ∈ (0 , e t = e ( t, ξ t ) is a semimartingale withthe decomposition e ( t, ξ t ) = e (0 , x ) + Z t µ e ( s, ξ s ) ds + Z t σ e ( s, ξ s ) dB s , where the drift function µ e : [0 , T ] × R d → R is bounded and continuousand σ e ( s, ξ s ) is a bmo process.We will often abuse notation and write e i both for the function e i : [0 , T ] × R d → R and the stochastic process e it = e i ( t, ξ t ). The same applies to other functions appliedto ( t, ξ t ) - such as e or µ e . Remark . It is worth stopping here to give a few examples of state processes ξ and the functions e i which satisfy all the regularity conditions imposed so far.Once the coefficients Λ and Σ for ξ are picked so as to satisfy Assumption 2.1, thenAssumption 2.3 is easy to check for a sufficiently smooth e i by a simple applicationof Itô’s formula.The more interesting observation is that there is room for improvement. It mayseem that the boundedness imposed in Assumption 2.1 rules out some of the mostimportant classes of state processes such as the classical mean-reverting (Ornstein-Uhlenbeck) processes. This is not the case, as we have the freedom to choose boththe state process ξ , and the deterministic function e i applied to it, while only caringabout the resulting composition. We illustrate what we mean by that with a simpleexample. The reader will easily add the required bells and whistles to it, and adaptit to other similar frameworks.We assume that d = 1 and that we are interested in the random endowment rate e i ( t, η t ) where e i is a bounded and appropriately smooth function, and η t is anOrnstein-Uhlenbeck process with the dynamics dη t = θ (¯ η − η t ) dt + σ η dB t , and parameters θ, σ η > η , ¯ η ∈ R . Since the drift function x θ (¯ η − x ) isnot bounded, the process η does not satisfy the conditions of Assumption 2.1. Theprocess η admits, however, an explicit expression in terms of a stochastic integralof a deterministic process with respect to the underlying Brownian motion: η t = ¯ η + ( η − ¯ η ) e − θt + σ η e − θt Z t e θs dB s (2.3)If we define the state process ξ by dξ t = e − θt dB t , ξ = 0 , i.e., if we set Λ( t, x ) = 0 and Σ( t, x ) = e − θt , the boundedness of the time horizon[0 , T ] allows use to conclude that Λ and Σ satisfy Assumption 2.1. Moreover, by TOCHASTIC ANNUITY IN EQUILIBRIUM 6 (2.3), the choice f i ( t, x ) = e i ( t, ¯ η + ( η − ¯ η ) e − θt + σ η x ) yields f i ( t, ξ t ) = e i ( t, η t ) . This way, we can represent a function of an interesting, but not entirely compliantstate process η as a (modified) function of a regular state process ξ . The bounded-ness (and other regularity properties) of the function e i are inherited by f i , thanksto the boundedness from above and away from zero of the function t e − θt .2.2. Admissibility and equilibrium.Definition 2.5.
Given a feasible set of coefficients γ = ( µ, σ ) ∈ Γ f , a pair ( π, c ) ofscalar processes is said to be a γ -admissible strategy for agent i if(1) | c | + | π ( A γ µ − | ∈ P and πA γ σ ∈ bmo.(2) the gains process X = X π,γ = πA γ is a semimartingale which satisfiesthe self-financing condition dX = π dA γ + ( e i − c + π ) dt. The set of all γ -admissible strategies for agent i is denoted by A iγ , and the subsetof A iγ consisting of the strategies with π (0) = π is given by A iγ ( π ). Definition 2.6.
We say that γ ∗ ∈ Γ f is the set of equilibrium market coeffi-cients (and A γ ∗ an equilibrium market ) if there exist γ ∗ -admissible strategies(ˆ π i , ˆ c i ) ∈ A iγ ∗ ( π i ), i = 1 , . . . , I , such that the following two conditions hold:(1) Single-agent optimality:
For each i and all ( π, c ) ∈ A iγ ∗ ( π i ) we have E [ R T U i (ˆ c it ) dt ] + E [ U i ( X ˆ π i , ˆ c i T + e iT )] ≥ E [ R T U i ( c t ) dt ] + E [ U i ( X π,cT + e iT )](2) Market clearing: P Ii =1 ˆ π i = 1 and P Ii =1 ˆ c i = e + 1 on [0 , T ) , and P Ii =1 X ˆ π i , ˆ c i T = 1 , a.s.3. Results
A BSDE characterization.
Our first result is a characterization of equilibriain terms of a system of backward stochastic differential equations (BSDE). Thesesystems consist of 1 + I equations, with the first component generally playing adifferent role from the other I . For that reason, it pays to depart slightly from theclassical notation ( Y, Z ), where Y has as many components as there are equations,and the driver Z is a matrix process whose additional dimension reflects the numberof driving Brownian motions. Instead, we use the notation (cid:0) ( a, Y ) , ( σ, Z ) (cid:1) where a is a scalar and Y is R I × -valued. Similarly, σ and Z are R × d - and R I × d -valuedprocesses, respectively. As usual, we say that (( a, Y ) , ( σ, Z )) is an ( S ∞ × bmo)-solution if all the components of a and Y are in S ∞ , and all components of ( σ, Z )are in bmo. To simplify the notation, we also introduce the following, derived,quantities: α := P Ii =1 1 α i , and κ i := ¯ αα i > P Ii =1 κ i = 1 . (3.1) TOCHASTIC ANNUITY IN EQUILIBRIUM 7
Theorem 3.1 (A BSDE Characterization) . Suppose that P Ii =1 π i = 1 , that As-sumption 2.3 holds, and that (cid:0) ( a, Y ) , ( σ, Z ) (cid:1) is an S ∞ × bmo -solution to da = σ dB + (cid:16) ¯ αµ e − P Il =1 κ l (cid:12)(cid:12) Z l (cid:12)(cid:12) − exp( − a ) (cid:17) dt, a T = 0 dY i = Z i dB + (cid:16) (cid:12)(cid:12) Z i (cid:12)(cid:12) + exp( − a )(1 + a + Y i − α i e i ) (cid:17) dt, Y iT = α i e iT , ≤ i ≤ I. (3.2) Then A = exp( a ) is an equilibrium annuity price with market coefficients ( µ, σ ) ∈ Γ f , where µ is given by µ = ¯ αµ e + | σ | − P Ii =1 κ i (cid:12)(cid:12) Z i (cid:12)(cid:12) . (3.3) Remark . We note that the validity of Theorem 3.1 above does not depend onAssumption 2.1. In fact, no Markovian assumption is needed for it, at all. Moreover,the full force of Assumption 2.3 is not needed, either. It would be enough toassume that each e i is in bmo and that the cumulative endowment process e is asemimartingale of the form de = µ e dt + σ e dB , where µ e and σ e are general bmoprocesses and not necessarily bounded functions of a state process. Proof.
Having fixed an ( S ∞ , bmo)-solution (cid:0) ( a, Y ) , ( σ, Z ) (cid:1) , we set A = exp( a ) anddefine µ as in (3.3), so that A satisfies (2.2). With the market coefficients γ = ( µ, σ )fixed, we pick an agent i ∈ { , . . . , I } and a pair ( π, c ) ∈ A iγ ( π i ), and defineprocesses X i , ˜ V i and V i by X i = πA, ˜ V i = − exp( − α i X i /A − Y i ) and V i = ˜ V i + Z · − exp( − α i c t ) dt. The self-financing property of ( π, c ) implies that the semimartingale decompositionof V i is given by dV i = µ V dt + σ V dB , where µ V = − exp( − α i c ) + − ˜ V i A (cid:0) − log( − ˜ V i A ) (cid:1) − (cid:0) α i c (cid:1) − ˜ V i A and σ V = − ˜ V i Z i . Young’s inequality implies that µ V ≤ µ V and σ V areregular enough to conclude that V i is a supermartingale for all admissible ( π, c ).Therefore, E [ Z T U i ( c s ) ds ] + E [ U i ( X iT + e iT )] == α i E [ Z T − exp( − α i c s ) ds ] + α i E [exp( − α i ( X iT + e iT ))]= α i (cid:16) E [ Z T − exp( − α i c s ) ds ] + E [exp( − α i X iT /A T − Y iT )] (cid:17) = α i E [ V iT ] ≤ α i V i = − α i exp( − α i π i − Y i ) . Next, in order to characterize the optimizer, we construct a consumption processfor which µ V = 0. More precisely, we let the process ˆ X i be the unique solution ofthe following linear SDE:ˆ X i = π i A , d ˆ X i = (cid:16) µ ˆ X i + ( e i − α i ( a + Y i ) − ˆ X i A ) (cid:17) dt + ˆ X i σ dB, (3.4) TOCHASTIC ANNUITY IN EQUILIBRIUM 8 and set ˆ c i = α i ( a + Y i ) + ˆ X i A , ˆ π i = ˆ X i A . It follows immediately that (ˆ π i , ˆ c i ) ∈ A iγ ( π i ) and that the process ˆ X is the associ-ated gains process. The choice of ˆ c i , through ˆ X i , makes the process V i a martingaleand the pair (ˆ π i , ˆ c i ) optimal for agent i .Turning to market clearing, we consider the process F = a + P Ii =1 κ i Y i − ¯ αe , whosedynamics are given by dF = ( σ + P Ii =1 κ i Z i − ¯ ασ e ) dB + exp( − a ) F dt, F T = 0 . (3.5)In other words, the pair ( Y, ζ ) = (
F, σ + P Ii =1 κ i Z i − ¯ ασ e ) is an S ∞ × bmo-solutionto the linear BSDE dY = ζ dB + exp( − a ) Y dt, Y T = 0 . Since a is bounded, the coefficients of this BSDE are globally Lipschitz, and, there-fore, by the uniqueness theorem (see [Zha17, Theorem 4.3.1, p. 84]), we can concludethat F = 0. That implies that a + P Ii =1 κ i Y i = ¯ αe on [0 , T ] , and, so, P Ii =1 ˆ c i = e + A P Ii =1 ˆ X i . The form of the dynamics (3.4) of each ˆ X i leads to the following dynamics forˆ X = P Ii =1 ˆ X i : d ˆ X = ( µ ˆ X − A ˆ X ) dt + ˆ Xσ dt. (3.6)The assumption that P π i = 1 implies that ˆ X = A , which, in turn, implies thatthe process A is also a solution to (3.6). By uniqueness, we must have ˆ X = A andconclude that the clearing conditions are satisfied. (cid:3) Existence of an equilibrium.
Next, we show that under additional assump-tions on the problem ingredients - most notably that of a Markovian structure -the characterization of Theorem 3.1 can be used to establish the existence of anequilibrium market.
Theorem 3.3.
Under Assumptions 2.1 and 2.3, the system (3.2) admits an S ∞ × bmo -solution. The BSDE characterization of Theorem 3.1 immediately implies the main result ofthe paper:
Corollary 3.4.
Under Assumptions 2.1 and 2.3, there exists a set γ ∗ = ( µ ∗ , σ ∗ ) of feasible market coefficients such that A γ ∗ is an equilibrium market.Proof of Theorem 3.3. In certain situations it will be convenient to standardize thenotation, so we also write Y for a , Z for σ , and set g i ( x ) = ( , i = 0 ,α i e i ( T, x ) , ≤ i ≤ I. TOCHASTIC ANNUITY IN EQUILIBRIUM 9
The dt -terms in (3.2) define the driver f : [0 , T ] × R d × R I +1 × R ( I +1) × d → R I +1 in the usual way: f ( t, x, y, z ) = ¯ αµ e ( t, x ) − P Il =1 κ l (cid:12)(cid:12) z l (cid:12)(cid:12) − exp( − y ) ,f i ( t, x, y, z ) = (cid:12)(cid:12) z i (cid:12)(cid:12) + exp( − y ) (cid:16) y + y i − α i e i ( t, x ) (cid:17) , for i = 1 , . . . , I. The system (3.2), written in the new notation, becomes dY it = f i ( t, ξ t , Y t , Z t ) dt + Z it dB t Y iT = g i ( ξ T ) , i = 0 , . . . , I. (3.7) Step 1 (truncation).
We start by truncating the driver f to obtain a sequence ofwell-behaved, Lipschitz problems. More precisely, given N > ι N ( x ) = max(min( x, N ) , − N ) for x ∈ R and q N ( z ) = | z | ι N ( | z | ) , for z ∈ R × d , so that ι N and q N are Lipschitz functions with Lipschitz constants 1 and N , respec-tively. Moreover, | ι N ( x ) | ≤ N and | q N ( z ) | ≤ N | z | . Using the functions defined above, for each N ∈ N we pose a truncated version of(3.2): da = σ dB + (cid:16) ¯ αµ e − P Il =1 κ l q N ( Z l ) − exp( − ι N ( a )) (cid:17) dt,dY i = Z i dB + (cid:16) q N ( Z i ) + exp( − ι N ( a ))(1 + ι N ( a ) + ι N ( Y i ) − α i e i ) (cid:17) dt (BSDE N )with the terminal conditions Y iT = ι N ( g iT ) and a T = 0. We define the driver( t, x, y, z ) f ( N ) ( t, x, y, z ) from the dt -terms in the standard way.For each N ∈ N , f ( N ) is continuous in all of its variables, uniformly Lipschitz in both z and y , and f ( N ) ( t, x, ,
0) is bounded. Assumption 2.1 guarantees that the sameis true for the function F ( N ) ( t, x, y, z ) = − f ( N ) ( t, x, y, z Σ − ( t, x )). Therefore, wecan apply Proposition 4.1 in the Appendix to conclude that there exists a solution( Y ( N ) , Z ( N ) ) to (BSDE N ) of the form Y ( N ) t = v ( N ) ( t, ξ t ) , Z ( N ) t = w ( N ) ( t, ξ t ) , with v ( N ) : [0 , T ] × R d → R I +1 bounded and w ( N ) : [0 , T ] × R d → R ( I +1) × d suchthat Z ( N ) is a bmo process. We note that existence for (BSDE N ) is also guaranteedby the classical result [PP90, Theorem 3.1, p. 58], but only in the class S × H ,which is too big for our purposes. Step 2 (uniform estimates).
The bounds guaranteed by Proposition 4.1 all dependon the truncation constant N , so our next task is to explore the special structureof our system and establish bounds in terms of universal quantities. A universalconstant, in this proof, will be a quantity that depends on the constants α i , thetime-horizon T and the S ∞ -bounds on e i and µ e , but not on N . We denote sucha constant by C , and allow it to change from line to line. TOCHASTIC ANNUITY IN EQUILIBRIUM 10
Let (cid:0) ( a ( N ) , Y i, ( N ) ) , ( σ ( N ) , Z ( N ) ) (cid:1) be the solution to the truncated system from Step1 above. It follows from the dynamics of a ( N ) and the fact that q N ( z ) ≥ z ∈ R × d that a ( N ) − R · ¯ αµ e dt is a supermartingale, so that for all t ∈ [0 , T ], a ( N ) t ≥ E [ a ( N ) T − Z Tt ¯ αµ e dt |F t ] ≥ − ( T − t ) || ¯ αµ e || S ∞ , i.e., a ( N ) t ≥ − C. Next, we turn to Y ( N ) and use the fact that the components of Y are coupled onlythrough a . This way, we can get uniform bounds on Y i, ( N ) if we manage to producea uniform bound on the function of a appearing on the right-hand side. We startby using the following easy-to-check inequalityexp( − x )(1 + | x | ) ≤ exp(2 x − ) , for all x ∈ R , and the fact that ( ι N ( x )) − ≤ ( x ) − for all x , to obtain that for all t ∈ [0 , T ],exp( − ι N ( a ( N ) t )) (cid:16) (cid:12)(cid:12) ι N ( a ( N ) t ) (cid:12)(cid:12)(cid:17) ≤ C. It is readily checked that there exist a bounded measurable function δ ( N ) : R × d → R d , such that q N ( z ) = zδ ( N ) ( z ) = P dj =1 z j δ j, ( N ) ( z ) , for z = ( z , . . . , z j ) ∈ R × d . Therefore, for each i = 1 , . . . , I , there exists a probability measure P i (= P i,N ) ∼ P under which the process ˜ B i = B + R · δ ( N ) ( Z i, ( N ) ) dt is a Brownian motion on [0 , T ].Since Z i, ( N ) is guaranteed to be in bmo, it remains in bmo under the measure P i (see [Kaz94, Theorem 3.3, p. 57]). Therefore, the process R Z i, ( N ) dB i is a P i -martingale and we can take the expectation of the i -th equation with respect to P i to obtain (cid:12)(cid:12) Y i, ( N ) t (cid:12)(cid:12) ≤ (cid:12)(cid:12) E i [ ι N ( α i e i ( T )) |F t ] (cid:12)(cid:12) + Z Tt E i h exp( − ι N ( a ( N ) s )) (cid:16) (cid:12)(cid:12) ι N ( a ( N ) s ) (cid:12)(cid:12)(cid:17) |F t i ds + Z Tt E i h exp( − ι N ( a ( N ) s )) (cid:12)(cid:12) ι N ( Y i, ( N ) s ) − α i e is (cid:12)(cid:12) |F t i ds ≤ C (cid:16) Z Tt E i [ (cid:12)(cid:12) Y i, ( N ) s (cid:12)(cid:12) |F t ] ds (cid:17) ≤ C (cid:16) Z Tt y i ( s ) ds (cid:17) , where y i ( t ) = || Y i, ( N ) t || L ∞ . Thus, y i satisfies y i ( t ) ≤ C (1 + Z Tt y i ( s ) ds ) , for all t ∈ [0 , T ] , for some universal constant C . Gronwall’s inequality implies that y i (0) = || Y i, ( N ) || S ∞ is bounded by another universal constant, so we conclude that there exists a uni-versal S ∞ -bound on all Y i, ( N ) .Our next goal is to produce universal bmo bounds on the processes Z i, ( N ) . Thiswill follow by using the universal boundedness of the Z -free terms in the driver of Y i, ( N ) obtained above. Since the i -th component of the driver f ( N ) depends on Z ( N )TOCHASTIC ANNUITY IN EQUILIBRIUM 11 only through Z i, ( N ) , for 1 ≤ i ≤ I , we can apply standard exponential-transformestimates. We adapt the argument in [EB13, Proposition 2.1, p. 2925] and define φ ( x ) := exp(2 | x | ) − − | x | x ∈ R , noting that both φ and φ ′ are nonnegative and increasing, while φ ∈ C ( R ) with φ ′′ − | φ ′ | = 1. Thus, for any stopping time τ in [0 , T ], Itô’s Lemma gives us that0 ≤ φ ( Y i, ( N ) τ ) ≤ E [ φ ( Y i, ( N ) T ) |F τ ] + E "Z Tτ (cid:12)(cid:12) φ ′ ( Y i, ( N ) s ) (cid:12)(cid:12) C (cid:16) k Y i, ( N ) k S ∞ (cid:17) |F τ + E "Z Tτ (cid:18)(cid:12)(cid:12) φ ′ ( Y i, ( N ) s ) (cid:12)(cid:12)(cid:12)(cid:12) Z i, ( N ) s (cid:12)(cid:12) − φ ′′ ( Y i, ( N ) s ) (cid:12)(cid:12) Z i, ( N ) s (cid:12)(cid:12) (cid:19) ds |F τ ≤ φ ( k Y i, ( N ) k S ∞ )++ C Z T φ ′ ( k Y i, ( N ) k S ∞ )(1 + k Y i, ( N ) k S ∞ ) ds − E "Z Tτ (cid:12)(cid:12) Z i, ( N ) s (cid:12)(cid:12) ds |F τ . Rearranging terms yields E (cid:2) Z Tτ (cid:12)(cid:12) Z i, ( N ) s (cid:12)(cid:12) ds (cid:12)(cid:12)(cid:12) F τ i ≤ φ ( k Y i, ( N ) k S ∞ )++ C Z T φ ′ ( k Y i, ( N ) k S ∞ )(1 + k Y i, ( N ) k S ∞ ) ds. The right-hand side admits a universal bound (independent of N and τ ), and, hence,so does the left-hand side.Finally, we go back to the equation satisfied by a ( N ) and note that the termexp( − ι N ( a ( N ) )) is bounded because ( a ( N ) ) − is. We can bound a ( N ) from abovein an N -independent manner, by a combination of the bmo-bounds on Z ( N ) andthe sup norm of µ e . By taking expectations and using universal boundedness/bmo-property of all the other terms, we conclude that σ ( N ) also admits a universalbmo-bound.Having the universal bounds on Y ( N ) and a ( N ) , we can remove some of the trun-cations introduced in (BSDE N ). Indeed, for N larger than the largest of the S ∞ -bounds on Y ( N ) and a ( N ) , we have ι N ( Y i, ( N ) ) = Y i, ( N ) and ι N ( a ( N ) ) = a ( N ) . Therefore, there exists a constant N such that for N ≥ N the processes ( Y ( N ) , a ( N ) )together with ( Z ( N ) , σ ( N ) ) solve the intermediate system da = σ dB + (cid:16) ¯ αµ e − P j κ j q N ( Z j ) − exp( − ι N ( a )) (cid:17) dt,dY i = Z i dB + (cid:16) q N ( Z i ) + exp( − ι N ( a ))( ι N ( Y i ) + ι N ( a ) − α i e i + 1) (cid:17) dt (BSDE ′ N )with the same terminal conditions as (3.2). TOCHASTIC ANNUITY IN EQUILIBRIUM 12
Step 3 (Bensoussan-Frehse conditions and the existence of a Lyapunov function).
Mere boundedness in S ∞ × bmo is not sufficient to guarantee subsequential conver-gence of the solution ( Y ( N ) , a ( N ) ) of the truncated system to a limit which solves(3.2) or (BSDE ′ N ). It has been shown, however, in [XŽ18, Theorem 2.8, p. 501],that an additional property - namely the existence of a uniform Lyapunov function -will guarantee such a convergence. The existence of such a function can be deducedfrom another result of the same paper, [XŽ18, Proposition 2.11, p. 503], once itsconditions are checked. This proposition states that a uniformly bounded sequenceof solutions of a sequence of BSDE such as (BSDE ′ N ) admits a common Lyapunovfunction if the structure of its drivers satisfies the so called Bensoussan-Frehse con-ditions uniformly in N (see [XŽ18, Definition 2.10, p 502] for the definition). Itapplies here because our system is of upper-triangular form when it comes to itsquadratic dependence on z . More precisely, the driver of the system (BSDE ′ N ) canbe represented as a sum of two functions f ( N )1 and f ( N )2 given by( f ( N )1 ) i ( t, x, y, z ) = ¯ αµ e ( t, x ) − exp( − ι N ( y )) i = 0exp( − ι N ( y ))( ι N ( y i ) + ι N ( y ) − α i e i ( t, x ) + 1) (cid:17) , ≤ i ≤ I ( f ( N )2 ) i ( t, x, y, z ) = ( − P Il =1 κ l q N ( z l ) i = 0 q N ( z i ) 1 ≤ i ≤ I where the convention that a = Y and σ = Z is used. Therefore, there exists auniversal constant C such that, for all 1 ≤ i ≤ I + 1, we have (cid:12)(cid:12)(cid:12) ( f ( N )1 ) i ( t, x, y, z ) (cid:12)(cid:12)(cid:12) ≤ C as well as (cid:12)(cid:12)(cid:12) ( f ( N )2 ) i ( t, x, y, z ) (cid:12)(cid:12)(cid:12) ≤ C (1 + P ij =1 (cid:12)(cid:12) q N ( z j ) (cid:12)(cid:12) ) ≤ C (1 + P ij =1 (cid:12)(cid:12) z j (cid:12)(cid:12) ) . Therefore, f ( N ) can be split into a subquadratic (in fact bounded) and an uppertriangular component, allowing us to conclude that a uniform Lyapunov functionfor ( f ( N ) ) N ≥ N can be constructed. Step 4 (Passage to a limit).
It remains to use [XŽ18, Theorem 2.8, p. 501] toconclude that a subsequence of v ( N ) converges towards a continuous function v :[0 , T ] × R d → R I +1 such that Y t = v ( t, ξ t ) and Z t = Dv ( t, ξ t ) solves the limitingsystem da = σ dB + (cid:16) ¯ αµ e − P l κ l (cid:12)(cid:12) Z l (cid:12)(cid:12) − exp( − ι N ( a )) (cid:17) dt.dY i = Z i dB + (cid:16) (cid:12)(cid:12) Z i (cid:12)(cid:12) + exp( − ι N ( a ))( ι N ( Y i ) + ι N ( a ) − α i e i + 1) (cid:17) dt, (BSDE’)for i = 1 , . . . , I , where, as above a = Y and σ = Z . As far as the conditionsof Theorem 2.8 in [XŽ18] are concerned, the most difficult one, the existence of aLyapunov function, has been settled in Step 3. above. The other conditions - theuniform Hölder boundedness of the terminal conditions, and a-priori boundedness- are easily seen to be implied by our standing assumptions. Finally, since Y is a TOCHASTIC ANNUITY IN EQUILIBRIUM 13 pointwise limit of a sequence of functions bounded by N , the same processes ( Y, Z )also solve the original BSDE (3.2) (without truncation at N ). (cid:3) Bounded solutions of Lipschitz quasilinear systems
The main result of this section, Proposition 4.1, collects some results on systemsof heat equations with Lipschitz nonlinearities on derivatives up to the first order.We suspect that these results may be well-known to PDE specialists, but we wereunable to find a precise reference under the same set of assumptions in the literature,and, therefore, decided to include a fairly self-contained proof.In the sequel, D denotes the derivative operator with respect to all spatial variables,i.e., all variables except t . For d, J ∈ N and β ≥
0, we define the following threeBanach spaces:(1) L ∞ = L ∞ ( R d , R J ) or L ∞ = L ∞ ( R d ; R J × d ), depending on the context,(2) W , ∞ = W , ∞ ( R d ; R J ), with the norm || U || W , ∞ = || U || L ∞ + || DU || L ∞ .(3) L β = L β ([0 , T ); W , ∞ ) - the Banach space of measurable functions u :[0 , T ] → W , ∞ , endowed with the exponentially weighted norm || u || L β = Z T e − β ( T − t ) || u ( t, · ) || W , ∞ dt. The infinitesimal generator of the state process ξ is given by A u ( t, x ) = Du ( t, x )Λ( t, x ) + Tr (cid:16) D u ( t, x ) Σ( t, x )Σ T ( t, x ) (cid:17) for ( t, x ) ∈ [0 , T ] × R d . Proposition 4.1.
Suppose that g : R d → R J and F : [0 , T ] × R × d × R J × R J × d → R J are measurable functions such that • | g ( x ) | ≤ M , • | F ( t, x, , | ≤ M , and • | F ( t, x, y , z ) − F ( t, x, y , z ) | ≤ M ( | y − y | + | z − z | ) ,for some M and all t, x, y , y , z , z , and that the functions Λ and Σ satisfy theconditions of Assumption 2.1 (with the constant K ). Then the following statementshold:(1) The PDE system u t + A u + F ( · , · , u, Du ) = 0 , u ( T, · ) = g (4.1) admits a weak solution u on [0 , T ] . Moreover u ( t, · ) ∈ W , ∞ for all t ∈ [0 , T ) and there exists a constant C = C ( J, d, M, T, K ) ∈ [0 , ∞ ) such that || u ( t, · ) || L ∞ ≤ C for all t ∈ [0 , T ] and Z T || Du ( t, · ) || L ∞ dt ≤ C (2) Let u denote a solution of (4.1) as in (1) above and let { ξ t } t ∈ [0 ,T ] be astrong solution of the SDE dξ t = Λ( t, ξ t ) dt + Σ( t, ξ t ) dB t . TOCHASTIC ANNUITY IN EQUILIBRIUM 14
The pair ( Y t , Z t ) , where Y t = u ( t, ξ t ) and Z t = Du ( t, ξ t )Σ( t, ξ t ) is an S ∞ × bmo -solution to the system dY it = − F i (cid:16) t, ξ t , Y t , Z t Σ − ( t, ξ t ) (cid:17) dt + Z it dB t , Y iT = g i ( ξ T ) , i = 1 , . . . , I (4.2) Proof.
Throughout the proof, C will denote a constant which may depend on J, d, M, T or K , but not on β , t, s or x , and can change from line to line; wewill call such a constant universal. The assumptions on F imply that uniformly in t , and for all U, V ∈ W , ∞ , we have || F ( t, · , U, DU ) − F ( t, · , V, DV ) || L ∞ ≤ C || U − V || W , ∞ , and (4.3) || F ( t, · , U, DU ) || L ∞ ≤ C (1 + || U || W , ∞ ) . (4.4)Let p ( t, x ; s, x ′ ) denote a fundamental solution associated to the operator u t + A u ,i.e., ( t, x ) p ( t, x, s, x ′ ) solves p t + A p = 0 for t, x ∈ [0 , s ) × R d classically and satisfies the boundary condition lim t ր s R R d ψ ( x ) p ( t, x, s, x ′ ) dx = ψ ( x ′ ) for each bounded and continuous ψ . We refer the reader to [Fri64, Theorem10, p. 23] and the discussion preceding it for existence of a positive fundamentalsolution under the conditions of Assumption 2.1. Moreover, the equations (6.12)and (6.13) on p. 24 of [Fri64] state that there exist universal constants C, λ > t < s and all x, x ′ we have | p ( t, x, s, x ′ ) | ≤ Cϕ λ ( t, x, s, x ′ ) and | ∂ x k p ( t, x, s, x ′ ) | ≤ C √ s − t ϕ λ ( t, x, s, x ′ ) , (4.5)for all k = 1 , . . . , d , where ϕ λ ( t, x ; s, x ′ ) = πλ ( s − t )) d/ exp (cid:16) − λ ( s − t ) | x ′ − x | (cid:17) , is the scaled heat kernel (which is, itself, a fundamental solution associated to u t + λ ∆ u .).These properties, in particular, allow us to define the function Φ[ u ] : [0 , T ] × R d → R J byΦ[ u ]( t, x ) = Z R d Z Tt F (cid:16) s, x ′ , u ( s, x ′ ) , Du ( s, x ′ ) (cid:17) p ( t, x ; s, x ′ ) ds dx ′ , (4.6)for each u ∈ L β . The equation (4.4) guarantees that Φ[ u ] is well-defined withΦ[ u ]( t, · ) ∈ L ∞ .The Gaussian bounds in (4.5) imply that one can pass the derivative under theintegral sign to obtain ∂ x k Φ[ u ]( t, x ) = Z R d Z Tt F (cid:16) s, x ′ , u ( s, x ′ ) , Du ( s, x ′ ) (cid:17) ∂ x k p ( t, x ; s, x ′ ) ds dx ′ , (4.7)Consequently t Φ[ u ]( t, · ) is an a.e-defined measurable map [0 , T ] → W , ∞ , foreach u ∈ L β . To bound the norm of Φ[ u ] we start with the following estimate, TOCHASTIC ANNUITY IN EQUILIBRIUM 15 fueled by (4.5), || Φ[ u ]( t, · ) || W , ∞ ≤ C Z Tt (1 + || u ( s, · ) || W , ∞ ) Z R d (cid:16) p ( t, x ; s, x ′ ) + P Jk =1 | ∂ x k p ( t, x ; s, x ′ ) | (cid:17) dx ′ ds ≤ C Z Tt √ s − t (1 + || u ( s, · ) || W , ∞ ) ds. Furthermore, (4.6) and (4.7) imply that and, so, || Φ[ u ] || L β ≤ C Z T e β ( t − T ) Z Tt √ s − t (1 + || u ( s, · ) || W , ∞ ) ds dt = C Z T (1 + || u ( s, · ) || W , ∞ ) ds Z s √ s − t e β ( t − T ) dt ≤ C √ β (1 + || u || L β )A similar computation also yields || Φ[ u ] − Φ[ v ] || L β ≤ C √ β || u − v || L β . (4.8)Next, for g ∈ L ∞ , we defineΨ[ g ]( t, x ) = Z R d g ( x ′ ) p ( t, x ; T, x ′ ) dx ′ , so that, as above, || Ψ[ g ]( t, · ) || W , ∞ ≤ C √ T − t || g || L ∞ and || Ψ[ g ] || L β ≤ C √ β || g || L ∞ , and Ψ[ g ] ∈ L β for each g ∈ L ∞ . Therefore, the functionΓ[ u ] = Φ[ u ] + Ψ[ g ] , maps L β into L β and (4.8) implies that it is Lipschitz, with constant C/ √ β . Since C does not depend on β , we can turn Γ into a contraction by choosing a large-enough β , and conclude that Γ admits a unique fixed point u ∈ L β . The integralrepresentations in (4.6) and (4.7) allow us to conclude that u and Du are continuousfunctions on [0 , T ) × R d . Moreover, thanks to the Markov property of ξ , we have u ( t, ξ t ) = E [ g ( ξ T ) + Z Tt f ( s, ξ s ) ds |F t ] , a.s.where f ( s, x ) = F ( s, x, u ( s, x ) , Du ( s, x )) ∈ R J . (4.9)Since || f ( t, · ) || L ∞ ≤ C (1 + || u ( t, · ) || W , ∞ ) for all t , the map t
7→ || u ( t, · ) || W , ∞ belongs to L β , and (stripped of its norm) the space L β does not depend on thechoice of β . Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ( t, ξ t ) − E [ g ( ξ T ) |F t ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ≤ Z Tt || f ( s, · ) || L ∞ ds → t → T. TOCHASTIC ANNUITY IN EQUILIBRIUM 16
Since g is bounded, we have || u ( t, · ) || L ∞ ≤ C for all t . Moreover, the martingale E [ g ( ξ T ) |F t ] admits a continuous modification, so the process Y , defined by Y t = ( u ( t, ξ t ) , t < Tg ( ξ T ) , t = T, is a.s.-continuous. This allows us to conclude, furthermore, that Y t + R t f ( s, ξ s ) isa continuous modification of the martingale M t = E [ g ( ξ T ) + Z T f ( s, ξ s ) ds |F t ] , making Y a semimartingale. To show that ( Y, Z ) as in the statement indeedsolves (4.2), we need to argue that the martingale M t − M must be of the form R t Du ( s, ξ s )Σ( s, ξ s ) dB s . This can be proven by approximation as in the proof of[XŽ18, Lemma 4.4, p. 516]).The last step is to argue that ( Y, Z ) is an S ∞ × bmo-solution. The function u isuniformly bounded, so it suffices to establish the bmo-property of Z . This can bebootstrapped from the boundedness of Y by applying Itô’s formula to the boundedprocesses exp( cY i ), i = 1 , . . . , J , for large-enough constant c . A similar argument isalready presented on page 11, in the proof of Theorem 3.3, so we skip the details. (cid:3) References [Cal01] Laurent E. Calvet,
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E-mail address : [email protected] (Gordan Žitković) Department of Mathematics, The University of Texas at Austin, Austin,TX, USA
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