Endogenous inverse demand functions
aa r X i v : . [ q -f i n . M F ] D ec Endogenous inverse demand functions
Maxim Bichuch ∗ Zachary Feinstein † Wednesday 16 th December, 2020
Abstract
In this work we present an equilibrium formulation for price impacts. This is motivatedby the B¨uhlmann equilibrium in which assets are sold into a system of market participantsand can be viewed as a generalization of the Esscher premium. Existence and uniqueness ofclearing prices for the liquidation of a portfolio are studied. We also investigate other desiredportfolio properties including monotonicity and concavity. Price per portfolio unit sold is alsocalculated. In special cases, we study price impacts generated by market participants who followthe exponential utility and power utility.
Pricing is a fundamental question in Mathematical Finance. The classical pricing-by-replicationapproach answers this question by constructing a perfect hedge and therefore eliminating all riskfrom holding the position. Unfortunately, this approach cannot always be used outside of a completemarket setting, the existence of which is often violated in practice. Therefore a different approachto price the unhedgeable risk is required. One approach developed by [24] is the utility indifferenceprice. This price is the extra wealth that an individual needs to be compensated with in order tohold a given liability until maturity, so that he will be indifferent between that and remaining withhis initial position without the liability and the extra wealth. This approach is well-suited to pricethe risk that remains after the best hedge has been used. Therefore it can be shown (see, e.g., ∗ Department of Applied Mathematics and Statistics, Johns Hopkins University 3400 North Charles Street, Balti-more, MD 21218. [email protected] . Work is partially supported by NSF grant DMS-1736414. Research is partiallysupported by the Acheson J. Duncan Fund for the Advancement of Research in Statistics. † School of Business, Stevens Institute of Technology, Hoboken, NJ 07030, USA, [email protected] . exogenous inverse demand functions are prevalent in systemic risk whenstudying price-mediated contagion. In such studies, the inverse demand function is typically chosento follow a simple mathematical form for tractability rather than the financial meaning. Twoclassical inverse demand functions – the linear and the exponential – do indeed have simple financialinterpretation (constant absolute and relative price impacts respectively). As provided in Section 4,these classical inverse demand functions can be obtained in our equilibrium setting. Importantly, theequilibrium setting relates the form of the inverse demand function to the underlying assumptions2f the state of the market and returns of the traded asset(s). Additionally, we hypothesize thatnot every inverse demand function can be obtained from a given market setting, therefore, specialattention needs to be taken when exogenous forms are assumed for the inverse demand function.Furthermore, when considering fire sales of multiple illiquid assets, it is often assumed that theliquidation of one asset does not impact prices of the other assets (see, e.g., [23, 12, 13]). Herein,we find that the equilibrium price impacts may not generally satisfy such a condition.The organization of this paper is as follows. First, in Section 2, we will introduce the B¨uhlmannequilibrium problem and how we modify that problem in order to present the general financialsetting which we will utilize throughout this work. The main results are presented in Section 3; theseresults include necessary and sufficient conditions for the existence and uniqueness of clearing pricesfor asset liquidations. We also find sufficient conditions for, e.g., the monotonicity and concavityof the value obtained from liquidations. In Section 4, we demonstrate the form and properties ofour endogenous pricing functions under two special cases: when all market participants maximizethe exponential utility function and when they all maximize the power utility. The proofs for allresults are provided in an Online Appendix. We are motivated in our study by the notion of the B¨uhlmann equilibrium [9, 10] over a prob-ability space (Ω , F , P ) with some risk-neutral measure P . (We refer to Remark 1 below for theinterpretation of P as a risk-neutral measure.) Such an equilibrium endogenizes the price impactsof market behavior in a system of n market participants. Each participant 1 ≤ i ≤ n , with a twicedifferentiable, strictly increasing and concave utility function u i with dom u i = D ⊆ R and initialendowments X i ∈ L ∞ (such that X i ∈ D a.s.) of risky payoffs. Assumption 2.1.
Let X = P ni =1 X i , X i ∈ D a.s. We also assume that ess inf X > . Additionally,we will assume, by a possible change in num´eraire, that X ∈ R ++ . For simplicity of exposition, we also assume a zero risk-free rate r = 0. In this paper, we willconsider two classical settings, when the utility are defined in a half real line and on the entire realline, i.e., D = R ++ and D = R respectively. Each market participant is assumed to be a rationalagent insofar as each market participant wishes to maximize her expected utility. More specifically,3iven that each agent is endowed with (risky) endowment X i , ≤ i ≤ n , she may choose to tradequantities Y i to reduce her risk and maximize her utility. For an equilibrium, the market must clearand every agent must be “happy” with the trade. As such, the goal is to find the clearing pricesof these trades Y i , ≤ ≤ n . In other words, to find the pricing measure Q . This defines thesolution to the B¨uhlmann equilibrium problem as a pair ( Y, Q ) satisfying:1. Utility maximizing : Y i ∈ arg max ˆ Y i ∈ L ∞ n E P h u i (cid:16) X i + ˆ Y i − E Q [ ˆ Y i ] (cid:17)io for every i ∈ { , , ..., n } ;and2. Equilibrium transfers : P ni =1 Y i = 0.The measure Q is the endogenously defined probability measure which provides the price of theclaims, i.e., the value of Y i at time 0 is E Q [ Y i ]. Notably, the measure Q will (generically) differ fromrisk-neutral measure P . We wish to note that, in this static setting, the B¨uhlmann equilibriumcoincides with the Arrow-Debreu equilibrium [4] (see, e.g., [3]). Theorem 2.2.
There exists a unique B¨uhlmann equilibrium if:1. D = R and the absolute risk aversions z ρ i ( z ) := − u ′′ i ( z ) /u ′ i ( z ) > are Lipschitz continu-ous, i = 1 , ..., n ; or2. D = R ++ , the Inada conditions are satisfied (i.e., lim z → u ′ i ( z ) = ∞ and lim z →∞ u ′ i ( z ) = 0 ),and z zu ′ i ( z ) i = 1 , ..., n are nondecreasing.Proof. This is proven in [10] if D = R and [1] if D = R ++ .Consider the same market of n participants with utility functions u i and endowments X i , butnow with some external portfolio Z ∈ L ∞ being liquidated into the market. That is we considerthe modified B¨uhlmann equilibrium problem of determining the pair ( Y, Q ) satisfying:1. Utility maximizing : Y i ∈ arg max ˆ Y i E P h u i (cid:16) X i + ˆ Y i − E Q [ ˆ Y i ] (cid:17)i for every i ∈ { , , ..., n } ;and2. Equilibrium transfers : P ni =1 Y i = Z for externally sold position Z ∈ L ∞ .Note that if Z ≡ Z ∈ L ∞ so that the modified equilibrium ( Y, Q ) exists (see Section 3.1 below for somediscussion on this question). As presented in [10], the equilibrium probability measure Q mustsatisfy the fixed point problem: d Q d P ( ω ) = exp (cid:16) − n R X + Z ( ω ) − E Q [ Z ] X ρ ( γ ) dγ (cid:17) E P h exp (cid:16) − n R X + Z − E Q [ Z ] X ρ ( γ ) dγ (cid:17)i . (2.1)Within the construction of the pricing measure Q from (2.1) we, implicitly, consider ρ to be theharmonic average of risk aversions ρ i , ≤ i ≤ n , i.e., ρ ( γ ) = n n X i =1 − u ′ i ( Y i ( γ )) u ′′ i ( Y i ( γ )) ! − = n n X i =1 ρ i ( Y i ( γ )) ! − , where Y solves a differential system with equilibrium initial conditions. We refer to [10, 1, 25]for details of these constructs as well as the individual risk transfers Y i . We will refer to ρ as therisk-aversion of the harmonic representative agent .As will be investigated in greater detail below, we are interested in pricing these liquidatedcontingent claims Z ∈ L ∞ through the B¨uhlmann market mechanism. By construction, the valueof this contingent claim is given by E Q [ Z ] where Q is the B¨uhlmann pricing measure. Utilizing (2.1),this price can be seen to satisfy the fixed point condition E Q [ Z ] = E P h Z exp (cid:16) − n R X + Z − E Q [ Z ] X ρ ( γ ) dγ (cid:17)i E P h exp (cid:16) − n R X + Z − E Q [ Z ] X ρ ( γ ) dγ (cid:17)i . (2.2)It is this fixed point problem, and variations of it, which are the primary focus of this work. Remark 1.
We introduced P as a risk-neutral measure. This terminology is a consequence ofthe (modified) B¨uhlmann setup in which all market participants are themselves risk-neutral, i.e., u i ( x ) = x for every i , because Q = P in that setting. Thus the distortion of Q from P , which comesfrom the risk aversion and limited resources of the market participants, provides a measure of theincompleteness of the financial market.Therefore, we have that the price on the one hand is v = E Q [ Z ] , and on the other hand, bydefinition of risk-neutral pricing, is given by v = E P [ δ T ( Z ) Z ] , where δ T ( Z ) = d Q d P is the equilibrium iscount factor, when considering pricing of a claim Z with maturity T > . Note that we haveassumed that the risk-free interest rate is zero, and this is not the same as the discount factor. Thediscount factor is literally the factor the equates the risk-neutral and the true price. It turns outthis discount factor depends on the position Z . Notably, the risk-free rate r = 0 , provides the exactdiscount factor δ T ( Z ) = 1 only if Z is deterministic. Motivated by the B¨uhlmann equilibrium setup, let R : D → R be a nondecreasing function. Fromthe B¨uhlmann equilibrium setup of the prior section, we can view R as the integral of the absoluterisk aversion of the harmonic representative agent, i.e., R ( z ) = 1 n Z z X ρ ( γ ) dγ. In fact, the function R is bijective with the set of strictly increasing and concave utility functions u for the harmonic representative agent (with equivalence class defined up to multiplicative andadditive constants): u ( z ) = Z z exp( − R ( X + y )) dy. We also note that, using the fact that harmonic mean is bounded from below by its minimum, R isconcave if the harmonic representative agent has a nonincreasing absolute risk aversion (in wealth). Assumption 3.1.
For the rest of this paper we will assume that R : D → R is strictly increasing,differentiable, and concave. For the remainder of this work we will focus on and utilize this generalized function R to encodethe financial market and the harmonic representative agent’s utility u .The specific study of this work, rather than the modified B¨uhlmann equilibrium itself, is todetermine the price and value of a contingent claim Z ∈ L ∞ : V ( Z ) = FIX v (cid:26) H Z ( v ) := E P [ Z exp ( − R ( X + Z − v ))] E P [exp ( − R ( X + Z − v ))] (cid:27) . (3.1)This pricing function V can be seen as satisfying the fixed point condition of (2.2). We will also6tudy two special cases ( R ( x ) = α ( x − X ) and R ( x ) = η log( x/ X )) in Section 4 below whichcorrespond directly with the modified B¨uhlmann setup under specific choices of utility functions.In addition to the pricing function V , we also wish to consider the inverse demand functionsgenerated by this market. That is, given a portfolio q being liquidated in the market, we wish tofind the marginal price f q ( s ) for the s th unit sold and the average price for those same units ¯ f q ( s ).Such inverse demand functions satisfy the relation: V ( sq ) = Z s f q ( γ ) dγ = s ¯ f q ( s ) . (3.2)It is these inverse demand functions that are often introduced and presented in the literature with V derived through the relations of (3.2). For instance, we refer to [2] as an important work on fire salesin systemic risk which provides sufficient results on the uniqueness of (external) system liquidationsthrough the application of monotonicity conditions on the volume weighted average price ¯ f q and s s ¯ f q ( s ). However, in this equilibrium setup of market impacts, we find the construction ofthe pricing function V from (3.1) to be more natural; in Section 3.3, we study the inverse demandfunctions f q and ¯ f q derived from V .We wish to remind the reader that the proof of all results are presented in the Online Appendix. Consider a generalized structure from the B¨uhlmann setup in (3.1). We first investigate the exis-tence of the unique solution to the fixed point problem (3.1). That is, we study the conditions sothat the market is capable of providing a well-defined price for a contingent claim. This is presentedin Theorem 3.2 and expanded in Corollary 3.3. The equilibrium pricing problem (3.1) endogenizesthe market impacts due to the limited liquidity and preferences of the market participants; this isin contrast to the exogenous valuation taken in the aforementioned works (via assumed forms ofthe inverse demand function).
Theorem 3.2.
Assume R satisfies Assumption 3.1. Then:1. if D = R , then for any Z ∈ L ∞ , there exists a unique solution V ( Z ) = H Z ( V ( Z )) ;2. if D = R ++ , then for any Z ∈ L ∞ + , there exists at most one solution V ( Z ) = H Z ( V ( Z )) ; urthermore, V ( Z ) exists if and only if H Z ( X + ess inf Z ) ≤ X + ess inf Z . Theorem 3.2 gives a convenient condition for the existence and uniqueness of the fixed pointproblem (3.1) (and therefore the pricing measure Q ). However, in the case when D = R ++ thiscondition is far from intuitive. Therefore we have the following corollary: Corollary 3.3.
Assume R satisfies Assumption 3.1. Let D = R ++ , and Z ∈ L ∞ + .1. If E P [exp ( − R ( Z − ess inf Z ))] < ∞ and E P [ Z ] ≤ X + ess inf Z , then Z ∈ dom V .2. If E P [exp ( − R ( Z − ess inf Z ))] = ∞ then Z ∈ dom V . These conditions are much more intuitive financially. Indeed, when E P [ Z − ess inf Z ] ≤ X ,i.e., the external market is selling risky positions (in risk-neutral value) for less than the marketcapitalization X , the clearing value is well defined in any case. Whereas, when E P [ Z − ess inf Z ] islarge, then the clearing price is guaranteed to exist if E P [exp ( − R ( Z − ess inf Z ))] = ∞ .Theorem 3.2 and Corollary 3.3 give us an understanding about the domain of V , i.e., whichcontingent claims can be cleared by the market. We now wish to consider some intuitive propertiesof V . Namely, we prove that V is bounded, law invariant, cash translative, monotonic in the marketliquidity X , and continuous. Further properties of the pricing function V , through an extensionintroduced in Section 3.2, are provided in Lemma 3.8. Proposition 3.4.
Assume R satisfies Assumption 3.1.1. V ( Z ) ∈ [ess inf Z, ess sup Z ] for any Z ∈ dom V .2. V is law invariant.3. V ( Z + z ) = V ( Z ) + z for any Z ∈ dom V with z ∈ R if D = R or z ≥ − ess inf Z if D = R ++ .4. If V ( Z ; X ) exists then V ( Z ; X ) exists and is at least V ( Z ; X ) for any X ≥ X .5. V is continuous in the strong topology. V We have now successfully defined V on L ∞ in case D = R , but have more severe restrictions if D = R ++ ; specifically, we cannot define V outside of L ∞ + . Since our ultimate goal is to define an8nverse demand pricing function using V , we want to define V on a larger set than L ∞ + , as otherwise,we limit ourselves to pricing only positions with nonnegative payoffs, which is a severe restriction.Therefore, in this section we look for a way to extend the definition of V to the entire space L ∞ even in the case when D = R ++ . Of course, this definition must coincide with the current definitionof V in all the cases in which we have already successfully defined it.For Z ∈ L ∞ consider the following definition:¯ V ( Z ) := sup { v ∈ R + | H Z − ess inf Z ( v ) ≥ v, X − v ∈ D } + ess inf Z. (3.3)This definition intuitively provides the maximum price the market participants are willing to payfor the portfolio Z which can be supported by those market participants. Though this may not bea “fair” price in the B¨uhlmann sense, we assume, as in the fire sale literature [11, 2, 16], that theexternal seller of Z is forced to complete the liquidation and thus must accept the value provided bythe market. The market is willing to support the value ¯ V ( Z ) of Z because, due to the monotonicityof H Z − ess inf Z ( · ), at this level the market views Z as being undervalued and thus a good deal.First, we show that this is a well-defined extension in the following theorem: Theorem 3.5.
Under Assumption 3.1, the definition of ¯ V in (3.3) is a well defined extension of V . Specifically,1. ¯ V ( Z ) = V ( Z ) for every Z ∈ dom V ; and2. ¯ V ( Z ) ∈ R for every Z ∈ L ∞ , i.e., ¯ V exists for every Z ∈ L ∞ . In fact, ¯ V has an easy to calculate form even when V is not defined. As hinted at in ourconstruction of the extension ¯ V , whenever Z − ess inf Z dom V , the market finds that it does nothave sufficient liquidity to pay a fair (clearing) value for the claim Z − ess inf Z . Since the marketis only limited by the cash available to it, this means the market will, instead, offer X for the claim Z − ess inf Z (which will be accepted due to the assumption that the external liquidation of theclaim is forced on the market). This argument is formalized in the following corollary. Corollary 3.6.
Assume R satisfies Assumption 3.1. The extension ¯ V , defined in (3.3) , can equiv- lently be formulated as: ¯ V ( Z ) = V ( Z − ess inf Z ) + ess inf Z if Z − ess inf Z ∈ dom V X + ess inf Z if Z − ess inf Z dom V. (3.4)It turns out the same extension ¯ V can also be achieved using a wholly different notion provided D = R ++ and lim z → R ( z ) = −∞ . Let B [ p ] ∼ Bern ( p ) be a Bernoulli random variable representingthe ruin (with probability 1 − p ) of the banking system. It follows that the position Z ∈ L ∞ + becomes B [ p ] Z as it is only payable if the system has not defaulted. As p tends towards 1, i.e., the probabilityof systemic ruin tends towards 0, the claim Z is again recovered. Armed with this observation, weformulate an alternative definition for ¯ V :ˆ V ( Z ) = lim p ր V ( B [ p ]( Z − ess inf Z )) + ess inf Z. (3.5)That is, up to modification via the essential infimum, the (extended) price ˆ V ( Z ) of Z is providedby the limiting behavior of the price of B [ p ] Z as the probability of systemic ruin tends towards 0.The following result shows that ˆ V = ¯ V . Thus, this new extension ˆ V provides the interpretationthat the market prices Z as if the probability of systemic ruin is negligibly small rather than theexplicitly setting the probability to 0. Corollary 3.7.
Assume R satisfies Assumption 3.1. Assume also that D = R ++ and lim z → R ( z ) = −∞ . Let Z ∈ L ∞ and assume that that B [ p ] in the construction (3.5) are independent of Z . Then ˆ V ( Z ) = ¯ V ( Z ) . We now revisit the properties of V from Proposition 3.4 and show that the extension ¯ V enjoysthose same properties. We also formulate additional properties for ¯ V under certain conditions on R . Specifically, we provide conditions for the monotonicity and concavity of ¯ V ; that is, respectively,greater liquidations provide a larger value and the marginal increase in value is decreasing. We notethat, taken together, these properties of ¯ V construct a monetary risk measure (with modificationup to negative signs) which may be of interest for future study. Lemma 3.8.
Assume R satisfies Assumption 3.1. . ¯ V ( Z ) ∈ [ess inf Z, ess sup Z ] for any Z ∈ L ∞ .2. ¯ V is law invariant.3. ¯ V ( Z + z ) = ¯ V ( Z ) + z for any Z ∈ L ∞ and z ∈ R .4. ¯ V ( Z ; X ) ≥ ¯ V ( Z ; X ) for X ≥ X for any Z ∈ L ∞ .5. ¯ V is continuous in the strong topology.6. ¯ V ( Z ) ≥ ¯ V ( Z ) for Z ≥ Z a.s. if z z exp( − R ( X + z )) is nondecreasing. Additionally, ¯ V is Lipschitz continuous with Lipschitz constant if z z exp( − R ( X + z )) is nondecreasing.7. ¯ V is concave if z z exp( − R ( X + z )) is nondecreasing and concave. Additionally, ¯ V is uppersemicontinuous in the weak* topology if z z exp( − R ( X + z )) is nondecreasing and concave. Remark 2.
Recall the R can be viewed as the integral of the absolute risk aversion of the harmonicrepresentative agent. The condition for monotonicity can be viewed with respect to the risk aversionof the agent. That is, z z exp( − R ( X + z )) is nondecreasing if and only if zR ′ ( X + z ) ≤ . Inparticular, by concavity, this is true if zR ′ ( z ) ≤ for every z > . Therefore, for monotonicityof ¯ V , it is sufficient to bound the relative risk aversion of the harmonic representative agent fromabove by . Now that we have a good definition of ¯ V that is valid over the entire space L ∞ , we are finally able torigorously define the inverse demand functions. There is no unique way to do so, and we choose todemonstrate how this can be done following the example set in [6, 7]: using the order book densityand the volume weighted average price (VWAP) function. For the former, we set f q : R + → R , sothat ¯ V ( sq ) = R s f q ( t ) dt , i.e., f q ( s ) = ∂∂s ¯ V ( sq ). For the latter, we define ¯ f q : R + → R by¯ f q ( s ) = E P [ q ] if s = 0¯ V ( sq ) /s if s > . (3.6)The order book density function f q provides the price of the next marginal unit of the portfolio q given the total number of units already sold; in this way we can encode a dynamic notion of pricing11n a single period framework. In contrast, the VWAP function ¯ f q provides the average price perunit of liquidated portfolio; this construction is implicitly utilized in much of the fire sale literature,see, e.g., [11, 2, 16].As discussed previously, these inverse demand functions ( f q or ¯ f q ) are often the objects intro-duced exogenously. Such an approach, though valid, does not necessarily follow from a financialequilibrium. By first studying the value of arbitrary portfolios, we are able to consequently talkabout the price per unit of any asset or portfolio. Though only presented in the special cases ofmarkets generated by the exponential or power utility functions (as detailed in Section 4 below),we can consider the cross-impacts that liquidating one asset can have on the price of another; thisis in contrast to the typical, simplifying, assumption that there are no cross-impacts as taken in,e.g., [23].We first show that order book density function f q is well defined. That is, we can meaningfullydiscuss the price of the next marginal unit of the portfolio q . Lemma 3.9.
Assume R satisfies Assumption 3.1. Consider the setting in which a single portfoliois being liquidated proportionally, i.e., f q : R + → R defined by f q ( s ) = E P [ q (1 − [ sq − ¯ V ( sq )] R ′ ( X + sq − ¯ V ( sq ))) exp( − R ( X + sq − ¯ V ( sq )))] E P [(1 − [ sq − ¯ V ( sq )] R ′ ( X + sq − ¯ V ( sq ))) exp( − R ( X + sq − ¯ V ( sq )))] if sq ∈ dom V ess inf q else . ¯ V ( sq ) = R s f q ( t ) dt for any s ∈ R + .2. If z z exp( − R ( X + z )) is nondecreasing then f q ( s ) ≥ ess inf q for every s ∈ R + andif, additionally, z z exp( − R ( X + z )) is strictly increasing and P ( q > ess inf q ) > then f q ( s ) > ess inf q for every s ∈ R + such that sq ∈ dom V .3. f q is continuous on int { s ∈ R + | sq ∈ dom V } if R is continuously differentiable.4. f q is nonincreasing if z z exp( − R ( X + z )) is nondecreasing and concave. Now we want to consider the volume weighted average price ¯ f q . As shown below, this pricingfunction satisfies the expected conditions automatically in contrast to the order book density func-tion f q . This VWAP function provides exactly the average price obtained by the seller per unit ofthe portfolio q , i.e., ¯ f q ( s ) = s ¯ V ( sq ) for s >
0. 12 emma 3.10.
Assume R satisfies Assumption 3.1. Consider the setting in which a single portfolio q is being liquidated proportionally, i.e., ¯ f q : R + → R is given by (3.6) .1. ¯ f q ( s ) ≥ ess inf q for every s ∈ R + and if, additionally, P ( q > ess inf q ) > then ¯ f q ( s ) > ess inf q for every s ∈ R + .2. ¯ f q is continuous.3. ¯ f q is nonincreasing. Remark 3.
The order book density function f q and VWAP function ¯ f q are related through: ¯ V ( sq ) = Z s f q ( t ) dt = s ¯ f q ( s ) . Remark 4.
Often (see, e.g., [17, 6]), the liquidity of the inverse demand function near 0 is of greatimportance. Notably, by construction, the inverse demand functions coincide with the risk-neutralprice of q when no assets are being sold, i.e., f q (0) = ¯ f q (0) = E P [ q ] . The liquidity of the marketcan then be described by the velocity that prices are impacted by a small, additional, liquidation.The order book density function provides an initial liquidity value of ( f q ) ′ (0) = − R ′ ( X ) Var( q ) ;the VWAP provides an initial liquidity value of ( ¯ f q ) ′ (0) = − R ′ ( X ) Var( q ) . That is, the marketclearing price drops no matter how it is measured when the first (marginal) unit is sold (unless q isdeterministic) and is proportional to both the absolute risk aversion of the harmonic representativeagent at the market wealth X and to the variance of portfolio q . Remark 5.
It is logical to ask if the inverse demand functions f q and ¯ f q are convex. First, wenote that ¯ f q is convex if f q is convex as utilized in [7]; therefore if ¯ f q is nonconvex in general thesame must be true of f q as well. Now consider ¯ f q . If skew( q ) < then ( ¯ f q ) ′′ (0) < and, thus, theVWAP inverse demand function is concave at 0. However, skew( q ) > is not sufficient to provethe convexity of ¯ f q as will be shown in both special cases of R in the following section. We now specialize the generic framework of the previous section to consider specific examples ofthe pricing and inverse demand functions. We highlight that these functions are the result of the13odified B¨uhlmann equilibrium setting presented in Section 2 with specific choices for the util-ity functions of the market participants. Specifically, we consider two settings in the modifiedB¨uhlmann framework: exponential utility maximizers and power (or logarithmic) utility maximiz-ers. With these utility maximizing settings, we illustrate how asset and portfolio prices can beobtained as a result of the equilibrium problem. In this way, in Example 4.3 we are able to recover,e.g., two classical inverse demand functions – the linear and exponential inverse demand functions –which are commonly used. This is significant as it makes explicit a number of assumptions that con-tribute to these (and other) specific pricing functions, though other combinations of utility functionsand underlying distributions of q may recover these same inverse demand functions. Furthermore,we hypothesize that not every inverse demand function can be achieved as an equilibrium from aspecific set of market participants (as encoded in their utility functions and aggregate holdings X ).As such, extra care needs to be taken when using a specific, exogenous, inverse demand function,as it may not be achievable with the set of market participants or, potentially, from the specificdistribution of the returns. Consider the B¨uhlmann equilibrium construction from Section 2 in which every market participanthas exponential utility function u i ( x ) := 1 − exp( − α i x ) with risk aversion α i >
0. Let α := (cid:16)P ni =1 1 α i (cid:17) − . Then α , up to a multiplication by n , is the harmonic average of the risk aversions.As the absolute risk aversion in this case is constant, it immediately follows that we can constructa payment system with the function: R ( x ) = α ( x − X ) with D = R , (4.1) V ( Z ; α ) = E P [ Z exp( − αZ )] E P [exp( − αZ )] . (4.2)As D = R , it immediately follows that ¯ V ≡ V . We further wish to note that V is defined as theEsscher premium [20]. Proposition 4.1.
Let the pricing function V : L ∞ → R be defined as in (4.2) . As more marketparticipants enter the system, the market becomes more liquid (liquidation value goes up), i.e., V ( Z ; α ) ≤ V ( Z ; α ) for α ≥ α > . q = ( q , ..., q m ) ⊤ , the order book density function and VWAP inversedemand function are consequently defined as f q ( s ) = E P [ q exp( − α s ⊤ q )] E P [exp( − α s ⊤ q )] + (4.3) α E P [( s ⊤ q ) exp( − α s ⊤ q )] E P [ q exp( − α s ⊤ q )] − E P [exp( − α s ⊤ q )] E P [( s ⊤ q ) q exp( − α s ⊤ q )] E P [exp( − α s ⊤ q )] , ¯ f q ( s ) = E P [ q exp( − α s ⊤ q )] E P [exp( − α s ⊤ q )] . (4.4)As highlighted in the below proposition, these inverse demand functions exhibit no cross-impactson the prices as is often assumed (see, e.g., [23, 12, 13]) so long as the components of q are pairwiseindependent. Proposition 4.2.
Let q be a m -dimensional random vector. Let the order book density function f q : R m + → R m and VWAP inverse demand function ¯ f q : R m + → R m be defined as in (4.3) and (4.4) respectively. If the components of q are pairwise independent then the inverse demand functionsexhibit no price cross-impacts, i.e., f q k ( s ) = f q k k ( s k ) and ¯ f q k ( s ) = ¯ f q k k ( s k ) for every s ∈ R m + and k = 1 , ..., m .Proof. This follows directly by independence and properties of the exponential function.Though the Esscher principle provides a clear, analytical, structure for the pricing function V and the inverse demand functions, it only satisfies those properties that hold generally for suchfunctions. For instance, it is not true that V is monotonic or concave in general (see, e.g., thediscussion for the inverse demand function under a Poisson distribution). Similarly, the orderbook density function f q does not provide any clear structure as it can be negative (see, e.g.,the discussion for the inverse demand function under either the normal or Poisson distributions).However, those properties proven above that hold generally (e.g., continuity and monotonicity ofthe VWAP inverse demand function ¯ f q ) will hold herein.The exponential utility setting provides an added benefit; so long as q has distribution with amoment generating function, we can easily define the inverse demand functions f q and ¯ f q . Notably,this allows us to consider a larger domain than ( L ∞ ) m , including, e.g., the multivariate normaldistribution. In those cases, motivated by the previous sections, we will define V using (3.1), which15n this setting (with exponential utility and linear R in (4.1)) simplifies to (4.2). We then compute f q and ¯ f q , without going through the proofs and properties in the previous sections.In Example 4.3 below, we will consider a few well known distributions to provide the structure ofthe inverse demand functions. We also wish to highlight a final, discrete, distribution that providesa counterexample for the convexity of these inverse demand functions in general. Example 4.3.
Let f q and ¯ f q be constructed from the exponential utility function. Under the fol-lowing distributions of q we can determine the structure of the inverse demand functions explicitly. • Multivariate normal : If q ∼ N ( µ, C ) then f q ( s ) = µ − αC s and ¯ f q ( s ) = µ − αC s . That is,we recover the linear inverse demand function common in the literature (see, e.g., [23, 8, 13]).This makes this example one of the most important examples presented here. While we arenot claiming any kind of uniqueness, this is likely to be the only tractable case in which thelinear inverse demand function can be recovered. • Poisson : If q ∼ Pois( λ ) then f q ( s ) = (1 − αs ) λ exp( − αs ) and ¯ f q ( s ) = λ exp( − αs ) . Thatis, we recover the exponential inverse demand function common in the literature (see, e.g.,[32, 17]). This is the second most important example in the paper. We, not only, recovered theexponential inverse demand function, but this example also shows that the inverses demandfunction of the order book density can become negative, and that it need not be convex for all s ≥ . • Bernoulli : If q ∼ Bern( p ) then f q ( s ) = p +(1 − αs ) p (1 − p ) exp( αs )( p +(1 − p ) exp( αs )) and ¯ f q ( s ) = pp +(1 − p ) exp( αs ) . • Gamma : If q ∼ Γ( k, θ ) then f q ( s ) = kθ (1+ αθs ) and ¯ f q ( s ) = kθ αθs . • Discrete distribution : We wish to conclude with a simple distribution that results in non-convex inverse demand functions f q and ¯ f q . Let q ∈ { , , } with P ( q = 0) = 0 . , P ( q =1) = 0 . , P ( q = 16) = 0 . . Then skew( q ) = . . / = 0 . > which impliesboth f q and ¯ f q are convex near . Consider now s = α , ¯ f q ( s ) ≈ . but ( ¯ f q ) ′′ ( s ) = α E Q [( q − ¯ f q ( s )) ] ≈ − . α < . Similarly it can be shown that the order book densityfunction f q is nonconvex as well. .2 Power utility Consider the B¨uhlmann equilibrium construction from Section 2 in which every market participanthas power utility function u i ( x ) := x − η − − η if η = 1 and u i ( x ) = log( x ) if η = 1 for constant relativerisk aversion η ≥
0. As the relative risk aversion in this case is constant, it immediately follows thatwe can construct a payment function with the function R ( x ) = η (log( x ) − log( X )) (with D = R ++ ),i.e., V ( Z ) = FIX v E P [ Z ( X + Z − v ) − η ] E P [( X + Z − v ) − η ] . We wish to note that, generally, there is no closed form for this construction. As discussed pre-viously, due to D = R ++ , the domain of V is not the entire space L ∞ . In fact, as proven inProposition 4.4 below, dom V = { Z ∈ L ∞ + | E P [( Z − ess inf Z ) − η ] ≤ X E P [( Z − ess inf Z ) − η ] } .In addition to the properties listed below, the extension ¯ V also satisfies all properties that holdgenerally (e.g., translativity). Proposition 4.4.
Let the pricing function V be defined as in (4.2) and let ¯ V be its extension asdefined in (3.3) .1. dom V = { Z ∈ L ∞ + | E P [( Z − ess inf Z ) − η ] ≤ X E P [( Z − ess inf Z ) − η ] } .2. If η ∈ [0 , then ¯ V is nondecreasing and concave. It is, additionally, Lipschitz continuous inthe strong topology and weak* upper semicontinuous.3. If η ∈ (0 , then ¯ V is strictly increasing and strictly concave on dom V . With the construction for ¯ V , we can consider the inverse demand functions; for simplicity we willfirst present the setting with a single portfolio liquidated proportionally. However, these functionsdo not provide any analytical expression except one w.r.t. ¯ V , i.e., f q ( s ) = E P [ q (1 − η sq − ¯ V ( sq ) X + sq − ¯ V ( sq ) )( X + sq − ¯ V ( sq )) − η ] E P [(1 − η sq − ¯ V ( sq ) X + sq − ¯ V ( sq ) )( X + sq − ¯ V ( sq )) − η ] if s ≤ X E P [( q − ess inf q ) − η ] E P [( q − ess inf q ) − η ] ess inf q else , ¯ f q ( s ) = E P [ q ( X + sq − ¯ V ( sq )) − η ] E P [( X + sq − ¯ V ( sq )) − η ] if s ≤ X E P [( q − ess inf q ) − η ] E P [( q − ess inf q ) − η ] X s + ess inf q else . R , the properties for the order book density function f q (for asingle portfolio being liquidated proportionally) hold for specific choices of η . The general propertiesof both inverse demand functions (as summarized in Lemmas 3.9 and 3.10) hold for every η ≥ Corollary 4.5.
Consider the order book density function f q with a single portfolio being liquidatedproportionally. Assume P ( q > ess inf q ) > . If η ∈ [0 , then f q is nonincreasing and bounded frombelow by ess inf q . If, additionally, η > then f q ( s ) > ess inf q for every s ∈ [0 , X E P [( q − ess inf q ) − η ] E P [( q − ess inf q ) − η ] ) .Proof. This is a direct consequence of Lemma 3.9.We wish to highlight that, as opposed to the exponential utility case above, even if the assetsare independent it is not guaranteed that f q k ( s ) can be separated into a function f q k k ( s k ). This isclear from the construction of the inverse demand functions for a m -dimensional random vector q ,i.e., for s ∈ R m + such that E P [( s ⊤ q − ess inf s ⊤ q ) − η ] E P [( s ⊤ q − ess inf s ⊤ q ) − η ] ≤ X : f q ( s ) = E P [ q (1 − η s ⊤ q − ¯ V ( s ⊤ q ) X + s ⊤ q − ¯ V ( s ⊤ q ) )( X + s ⊤ q − ¯ V ( s ⊤ q )) − η ] E P [(1 − η s ⊤ q − ¯ V ( s ⊤ q ) X + s ⊤ q − ¯ V ( s ⊤ q ) )( X + s ⊤ q − ¯ V ( s ⊤ q )) − η ] , ¯ f q ( s ) = E P [ q ( X + s ⊤ q − ¯ V ( s ⊤ q )) − η ] E P [( X + s ⊤ q − ¯ V ( s ⊤ q )) − η ] . The induced price cross-impacts implies that there may exist complicated dependencies betweenprices of (statistically) independent assets.In addition to the properties satisfied by these inverse demand functions, we also wish to notethat, e.g., s s ⊤ ¯ f q ( s ) is nondecreasing and concave for η ∈ [0 ,
1] due to Proposition 4.4. Further,as with the exponential utility function, neither inverse demand function is convex in general; hereinthis can be seen with a single asset q at s ∗ = X E P [( q − ess inf q ) − η ] E P [( q − ess inf q ) − η ] for q > ess inf q a.s. In particular,we leave it as an exercise to the reader to plot f q and ¯ f q for q ∼ LogN( − σ , σ ) to investigatethe convexity at s ∗ . Furthermore, motivated by Corollary 3.7, f q and ¯ f q are not convex even if P ( q = ess inf q ) > q = B [ p ]ˆ q for B [ p ] ∼ Bern( p ) with p ≈ q ∼ LogN( − σ , σ ).We wish to conclude this special case with a few quick comments on the difficulties inherentin finding analytical forms for the pricing function ¯ V . Though numerical computation of ¯ V ( Z )is straightforward through Monte Carlo simulation, analytical construction is hampered by the18eed for non-integer moments of a constant plus Z . The combination of these requirements on thedistribution generally make an explicit representation of the fixed point of V intractable. In this work we have introduced an equilibrium model for pricing externally liquidated assets.This builds upon the seminal work by B¨uhlmann [9, 10] to endogenize the price impacts to finda clearing price in a financial market. This is in contrast to the typical approach in the fire saleliterature in which the form of inverse demand functions are exogenously given. In order to studythese endogenous inverse demand functions we prove the existence and uniqueness of the pricingmeasure. We additionally find that the resulting pricing functions satisfy the axioms of monetaryrisk measures; further study of this class of risk measures is left for future research. Utilizingthese results, we analyze two special cases – all banks are exponential or power utility maximizers– to study analytical structures. The exponential utility setup provides a direct connection withthe Esscher transform and provides analytical structure to the inverse demand functions whereasthe power utility setup satisfies useful mathematical properties for any claim being liquidated.Importantly, we find an example – the power utility setting – in which these inverse demandfunctions generate price cross-impacts even for statistically independent assets.
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A.1 Proof of Theorem 3.2
Proof.
First, consider the derivative of H Z for a fixed external liquidation Z . For simplicity of notation,define R := R ( X + Z − v ) and R ′ := R ′ ( X + Z − v ). H ′ Z ( v ) = E P [exp( − R )] E P [ ZR ′ exp( − R )] − E P [ Z exp( − R )] E P [ R ′ exp( − R )] E P [exp( − R )] = E P [exp( − R )] E P [( X + Z − v ) R ′ exp( − R )] − E P [( X + Z − v ) exp( − R )] E P [ R ′ exp( − R )] E P [exp( − R )] Recall, by construction, R ′ >
0. Therefore, H ′ Z ( v ) ≤ v such that ˆ Z = X + Z − v ∈ D almost surely if E P [ ˆ Z exp( − R ( ˆ Z ))] E P [ R ′ ( ˆ Z ) exp( − R ( ˆ Z ))] − E P [exp( − R ( ˆ Z ))] E P [ ˆ ZR ′ ( ˆ Z ) exp( − R ( ˆ Z ))] ≥ . Indeed, for any ˆ Z ∈ L ∞ such that ˆ Z ∈ D almost surely, let A ( ω ) := { ¯ ω ∈ Ω | ˆ Z (¯ ω ) ≤ ˆ Z ( ω ) } . Then E P [ ˆ Z exp( − R ( ˆ Z ))] E P [ R ′ ( ˆ Z ) exp( − R ( ˆ Z ))] − E P [exp( − R ( ˆ Z ))] E P [ ˆ ZR ′ ( ˆ Z ) exp( − R ( ˆ Z ))]= Z Ω ˆ Z ( ω ) exp( − R ( ˆ Z ( ω ))) P ( dω ) Z Ω R ′ ( ˆ Z (¯ ω )) exp( − R ( ˆ Z (¯ ω ))) P ( d ¯ ω ) − Z Ω exp( − R ( ˆ Z ( ω ))) P ( dω ) Z Ω ˆ Z (¯ ω ) R ′ ( ˆ Z (¯ ω )) exp( − R ( ˆ Z (¯ ω ))) P ( d ¯ ω )= Z Ω Z Ω exp( − R ( ˆ Z ( ω ))) exp( − R ( ˆ Z (¯ ω ))) h ˆ Z ( ω ) − ˆ Z (¯ ω ) i R ′ ( ˆ Z (¯ ω )) P ( d ¯ ω ) P ( dω )= Z Ω Z A ( ω ) exp( − R ( ˆ Z ( ω ))) exp( − R ( ˆ Z (¯ ω ))) h ˆ Z ( ω ) − ˆ Z (¯ ω ) i R ′ ( ˆ Z (¯ ω )) P ( d ¯ ω ) P ( dω )+ Z Ω Z A ( ω ) c exp( − R ( ˆ Z ( ω ))) exp( − R ( ˆ Z (¯ ω ))) h ˆ Z ( ω ) − ˆ Z (¯ ω ) i R ′ ( ˆ Z (¯ ω )) P ( d ¯ ω ) P ( dω )= Z Ω Z A ( ω ) exp( − R ( ˆ Z ( ω ))) exp( − R ( ˆ Z (¯ ω ))) h ˆ Z ( ω ) − ˆ Z (¯ ω ) i R ′ ( ˆ Z (¯ ω )) P ( d ¯ ω ) P ( dω )+ Z Ω Z A ( ω ) exp( − R ( ˆ Z ( ω ))) exp( − R ( ˆ Z (¯ ω ))) h ˆ Z (¯ ω ) − ˆ Z ( ω ) i R ′ ( ˆ Z ( ω )) P ( d ¯ ω ) P ( dω )= Z Ω Z A ( ω ) exp( − R ( ˆ Z ( ω ))) exp( − R ( ˆ Z (¯ ω ))) | {z } > h ˆ Z ( ω ) − ˆ Z (¯ ω ) i| {z } ≥ h R ′ ( ˆ Z (¯ ω )) − R ′ ( ˆ Z ( ω )) i| {z } ≥ P ( d ¯ ω ) P ( dω ) ≥ . This is nonnegative due to concavity of R implying the monotonic nonincreasing property for R ′ .Since V ( Z ) is a scalar, by continuity and nonincreasing-ness of H Z , there exists at most one solution ofthe fixed point problem defining V ( Z ).1. If D = R , then:(a) If H Z (0) = 0 then, trivially, there exists a fixed point V ( Z ) = 0. b) If H Z (0) > V ( Z ) ∈ (0 , H Z (0)].(c) If H Z (0) < V ( Z ) ∈ [ H Z (0) , D = R ++ , then by the assumptions, H Z (0) ≥
0, and we have that:(a) If H Z ( X + ess inf Z ) ≤ X + ess inf Z then there exists a fixed point V ( Z ) ∈ [0 , X + ess inf Z ].(b) If H Z ( X + ess inf Z ) > X + ess inf Z then it would need to hold that V ( Z ) > X + ess inf Z .However, R ( X + Z − v ) is undefined for v > X + ess inf Z ; therefore there cannot exist any fixedpoints. A.2 Proof of Corollary 3.3
Proof.
1. Indeed, we need to verify that E P [ Z exp ( − R ( Z − ess inf Z ))] ≤ E P [( X + ess inf Z ) exp ( − R ( Z − ess inf Z ))] . (A.1)Note that from the assumption E P [exp ( − R ( Z − ess inf Z ))] < ∞ , it follows that the denominator of H Z ( X + ess inf Z ) is a real number and (A.1) is indeed equivalent to H Z ( X + ess inf Z ) ≤ X + ess inf Z .Assuming for convenience that ess inf Z = 0, the above is true if and only if E P [( Z − X ) exp ( − ( R ( Z ) − R ( X )))] ≤ , where we have also used that X is deterministic.Now assume that E P [ Z ] ≤ X . We then have that ( Z − X ) exp ( − ( R ( Z ) − R ( X ))) ≤ Z − X as R is anincreasing function. Therefore, E P [( Z − X ) exp ( − ( R ( Z ) − R ( X )))] ≤ E P [ Z − X ] ≤ .
2. For any ε >
0, without loss of generality assume that E P (cid:2) exp ( − R ( Z − ess inf Z )) I { Z − ess inf Z ≥ ε } (cid:3) < ∞ .Therefore E P (cid:2) exp ( − R ( Z − ess inf Z )) I { ≤ Z − ess inf Z<ε } (cid:3) = ∞ , as follows from the assumption of thecorollary. Then, H Z ( X + ess inf Z ) = E P [ Z exp( − R ( Z − ess inf Z ))] E P [exp( − R ( Z − ess inf Z ))]= E P [( Z − ess inf Z ) exp( − R ( Z − ess inf Z ))] E P [exp( − R ( Z − ess inf Z ))] + ess inf Z = ess inf Z < X + ess inf Z. o deduce the equality H Z ( X + ess inf Z ) = ess inf Z , consider H Z − ess inf Z ( X ) = E P [( Z − ess inf Z ) exp( − R ( Z − ess inf Z ))] E P [exp( − R ( Z − ess inf Z ))]= E P (cid:2) ( Z − ess inf Z ) exp ( − R ( Z − ess inf Z )) (cid:0) I { ≤ Z − ess inf Z<ε } + I { Z − ess inf Z ≥ ε } (cid:1)(cid:3) E P (cid:2) exp ( − R ( Z − ess inf Z )) (cid:0) I { ≤ Z − ess inf Z<ε } + I { Z − ess inf Z ≥ ε } (cid:1)(cid:3) = E P (cid:2) ( Z − ess inf Z ) exp ( − R ( Z − ess inf Z )) I { ≤ Z − ess inf Z<ε } (cid:3) E P (cid:2) exp ( − R ( Z − ess inf Z )) I { ≤ Z − ess inf Z<ε } (cid:3) ≤ ε, where the third equality holds because E P (cid:2) exp ( − R ( Z − ess inf Z )) I { ≤ Z − ess inf Z<ε } (cid:3) = ∞ . Since ε > A.3 Proof of Proposition 3.4
Proof.
1. This is trivial as V ( Z ) = E P [ Z exp( − R ( X + Z − V ( Z )))] / E P [exp( − R ( X + Z − V ( Z )))] ≥ (ess inf Z ) E P [exp( − R ( X + Z − V ( Z )))] / E P [exp( − R ( X + Z − V ( Z )))] = ess inf Z and similarly for theupper bound V ( Z ) ≤ ess sup Z .2. This follows immediately by the law invariance of Z H Z ( v ).3. By construction H Z + z ( v ) = H Z ( v − z )+ z , therefore V ( Z + z ) = H Z + z ( V ( Z + z )) = H Z ( V ( Z + z ) − z )+ z .Take the ansatz that V ( Z + z ) = V ( Z ) + z , then V ( Z + z ) = H Z ( V ( Z )) + z = V ( Z ) + z . As thissatisfies the fixed point problem, and by Theorem 3.2 there exists a unique solution, this structuremust be satisfied for every Z, z .4. Monotonicity of V w.r.t. X follows from the monotonicity of H Z ( v ; X ) by noting the symmetry of v and X and applying the same logic as in the proof of Theorem 3.2. It remains to show that if V ( Z ; X )exists then so does V ( Z ; X ) in the case that D = R ++ (as V ( Z ; X ) exists for all Z, X if D = R ). Thisfollows trivially by the monotonicity of H Z ( v ; X ), i.e., for X ≥ X with V ( Z ; X ) existing H Z ( X + ess inf Z ; X ) = H Z ( X + ess inf Z ; X ) ≤ X + ess inf Z ≤ X + ess inf Z and thus, by Theorem 3.2, V ( Z ; X ) exists.5. Define D k := { Z ∈ L ∞ | k Z k ∞ ≤ k } and let H k : D k × [ − k, k ] → [ − k, k ] be the mapping H k ( Z, v ) := H Z ( v ) for any Z ∈ D k and v ∈ [ − k, k ]. By Property (1), V ( Z ) = H k ( Z, V ( Z )) for any Z ∈ D k . Since H k is jointly continuous (by construction), [18, Lemma C.1] implies { ( Z, V ( Z )) | Z ∈ D k } is closed. Bythe closed graph theorem, V restricted to D k is continuous. Consider now the sequence ( Z n ) n ∈ N → Z n dom V . Let k ∗ > k Z k ∞ . By construction, there exists some N ∈ N such that Z n ∈ D k ∗ for every n ≥ N . Thus by continuity over D k ∗ , it must follow that V ( Z ) = lim n →∞ V ( Z n ). B Proofs from Section 3.2
B.1 Proof of Theorem 3.5
Proof.
1. Fix Z ∈ dom V (in particular, if D = R ++ then we assume Z ∈ L ∞ + ). By Proposition 3.4(3), V ( Z ) = V ( Z − ess inf Z ) + ess inf Z . Therefore it is necessary and sufficient to prove V ( Z − ess inf Z ) =sup { v ∈ R + | H Z − ess inf Z ( v ) ≥ v, X − v ∈ D } . By Proposition 3.4(1), V ( Z − ess inf Z ) ≥ V ( Z − ess inf Z ) is feasible for this problem and v > V ( Z − ess inf Z ) is not feasible.By construction of the equilibrium payment V ( Z − ess inf Z ) and the monotonicity of H Z − ess inf Z asproven in Theorem 3.2, X − V ( Z − ess inf Z ) ∈ D , H Z − ess inf Z ( V ( Z − ess inf Z )) = V ( Z − ess inf Z ),and H Z − ess inf Z ( v ) < v for any v > V ( Z − ess inf Z ). This proves the result.2. Fix Z ∈ L ∞ . The result follows if we can bound ¯ V ( Z ) from above and below. First, v = 0 is feasiblefor the optimization problem within the construction of ¯ V by H Z − ess inf Z (0) ≥ H Z − ess inf Z and X ∈ D . Therefore ¯ V ( Z ) ≥ ess inf Z . Second, we need to consider our two domains for R :(a) If D = R then sup { v ∈ R + | H Z − ess inf Z ( v ) ≥ v, X − v ∈ D } ≤ ess sup Z − ess inf Z and, thus,¯ V ( Z ) ≤ ess sup Z .(b) If D = R ++ then sup { v ∈ R + | H Z − ess inf Z ( v ) ≥ v, X − v ∈ D } ≤ X and, thus, ¯ V ( Z ) ≤X + ess inf Z . B.2 Proof of Corollary 3.6
Proof.
Fix Z ∈ L ∞ . First assume Z − ess inf Z ∈ dom V . By construction of the extension ¯ V and Theo-rem 3.5, ¯ V ( Z ) = ¯ V ( Z − ess inf Z ) + ess inf Z = V ( Z − ess inf Z ) + ess inf Z . Second assume Z − ess inf Z dom V . This implies, as per Theorem 3.2, D = R ++ and H Z − ess inf Z ( X ) > X . By monotonicity of H Z − ess inf Z it therefore follows that sup { v ∈ R + | H Z − ess inf Z ( v ) ≥ v, X − v ∈ D } = X and the proof is completed. .3 Proof of Corollary 3.7 Proof.
Note that B [ p ]( Z − ess inf Z ) ∈ dom V for any Z ∈ L ∞ and p ∈ (0 ,
1) by Corollary 3.3(2). First,we will prove lim p ր V ( B [ p ]( Z − ess inf Z )) + ess inf Z exists for arbitrary Z ∈ L ∞ . Then we will utilizerepresentation (3.4) of ¯ V to prove that ˆ V ( Z ) = ¯ V ( Z ) for any Z ∈ L ∞ .We will prove lim p ր V ( B [ p ]( Z − ess inf Z )) + ess inf Z exists by showing that p ∈ (0 , V ( B [ p ]( Z − ess inf Z )) + ess inf Z is monotonic in p . Fix p ∈ (0 ,
1) and B [ p ] independent from Z . To simplify notation,let ˆ Z := Z − ess inf Z , V := V ( B [ p ] ˆ Z ) and V ′ := ∂∂p V ( B [ p ] ˆ Z ). Noting ess inf ˆ Z = 0: V = E P [ ˆ Z exp( − R ( X + ˆ Z − V ))] p E P [exp( − R ( X + ˆ Z − V ))] p + exp( − R ( X − V ))(1 − p ) . Therefore, assuming the derivative V ′ exists, it must satisfy: V ′ (cid:16) E P [exp( − R ( X + ˆ Z − V ))] p + exp( − R ( X − V ))(1 − p ) (cid:17) + V (cid:16) V ′ E P [ R ′ ( X + ˆ Z − V ) exp( − R ( X + ˆ Z − V ))] p (cid:17) + V (cid:16) E P [exp( − R ( X + ˆ Z − V ))] + V ′ R ′ ( X − V ) exp( − R ( X − V ))(1 − p ) − exp( − R ( X − V )) (cid:17) = V ′ E P [ ˆ ZR ′ ( X + ˆ Z − V ) exp( − R ( X + ˆ Z − V ))] p + E P [ ˆ Z exp( − R ( X + ˆ Z − V ))] ⇒ V ′ (cid:16) E P [(1 − ( ˆ Z − V ) R ′ ( X + ˆ Z − V )) exp( − R ( X + ˆ Z − V ))] p + (1 + V R ′ ( X − V )) exp( − R ( X − V ))(1 − p ) (cid:17) = E P [( ˆ Z − V ) exp( − R ( X + ˆ Z − V ))] + V exp( − R ( X − V )) ⇒ V ′ E P [(1 − ( B [ p ] ˆ Z − V ) R ′ ( X + B [ p ] ˆ Z − V )) exp( − R ( X + B [ p ] ˆ Z − V ))]= E P [ ˆ Z exp( − R ( X + ˆ Z − V ))] + V (exp( − R ( X − V )) − E P [exp( − R ( X + ˆ Z − V ))]) . Therefore V ′ exists if E P [(1 − ( B [ p ] ˆ Z − V ) R ′ ( X + B [ p ] ˆ Z − V )) exp( − R ( X + B [ p ] ˆ Z − V ))] = 0. By constructionof V : E P [(1 − ( B [ p ] ˆ Z − V ) R ′ ( X + B [ p ] ˆ Z − V )) exp( − R ( X + B [ p ] ˆ Z − V ))]= E P [exp( − R ( X + B [ p ] ˆ Z − V ))] E P [exp( − R ( X + B [ p ] ˆ Z − V ))]+ E P [ B [ p ] ˆ Z exp( − R ( X + B [ p ] ˆ Z − V ))] E P [ R ′ ( X + B [ p ] ˆ Z − V ) exp( − R ( X + B [ p ] ˆ Z − V ))] E P [exp( − R ( X + B [ p ] ˆ Z − V ))] − E P [exp( − R ( X + B [ p ] ˆ Z − V ))] E P [ B [ p ] ˆ ZR ′ ( X + B [ p ] ˆ Z − V ) exp( − R ( X + B [ p ] ˆ Z − V ))] E P [exp( − R ( X + B [ p ] ˆ Z − V ))]which is strictly positive by construction in the proof of Theorem 3.2. Therefore, V ′ ≥ E P [ ˆ Z exp( − R ( X + ˆ Z − V ))] + V (exp( − R ( X − V )) − E P [exp( − R ( X + ˆ Z − V ))]) ≥
0. By ˆ Z ≥ P [ ˆ Z exp( − R ( X + ˆ Z − V ))] ≥ V ≥ − R ( X − V )) ≥ exp( − R ( X +ˆ Z − V )) by monotonicity of R . Therefore V ′ ≥
0. In particular, this implies that p ∈ (0 , → V ( B [ p ]( Z − ess inf Z ))+ess inf Z is nondecreasing. Additionally, V ( B [ p ]( Z − ess inf Z ))+ess inf Z ≤ X +ess inf Z for every p ∈ (0 ,
1) by the existence criteria of Theorem 3.2. Therefore, by application of the monotone convergencetheorem, we can guarantee the existence of ˆ V ( Z ).We will complete this proof by proving that this limit is equivalent to the form (3.4). Fix Z ∈ L ∞ . First,we will show ˆ V ( Z ) = ¯ V ( Z ) if Z − ess inf Z ∈ dom V . Second, we will consider Z − ess inf Z dom V .1. First, fix Z ∈ L ∞ such that V ( Z ) exists (in particular, we assume Z ∈ L ∞ + ). We will prove V ( Z ) =lim p ր V ( B [ p ]( Z − ess inf Z )) + ess inf Z by showing that p ∈ [0 , V ( B [ p ]( Z − ess inf Z ) + ess inf Z )is continuous through an application of Proposition 3.4(3). By Corollary 3.3(2) and choice of Z , V ( B [ p ]( Z − ess inf Z ) + ess inf Z ) exists for any p ∈ [0 , V ( B [ p ]( Z − ess inf Z ) + ess inf Z ) ∈ [ess inf Z, ess sup Z ] for any p ∈ [0 , p ∈ [0 , V ( B [ p ]( Z − ess inf Z ) + ess inf Z )= E P [( B [ p ]( Z − ess inf Z ) + ess inf Z ) exp( − R ( X + B [ p ]( Z − ess inf Z ) + ess inf Z − V ))] E P [exp( − R ( X + B [ p ]( Z − ess inf Z ) + ess inf Z − V ))]= E P [ Z exp( − R ( X + Z − V ))] p + ess inf Z exp( − R ( X + ess inf Z − V ))(1 − p ) E P [exp( − R ( X + Z − V ))] p + exp( − R ( X + ess inf Z − V ))(1 − p ) . That is, V ( B [ p ]( Z − ess inf Z ) + ess inf Z )= FIX v (cid:26) ¯ H ( p, v ) := E P [ Z exp( − R ( X + Z − v ))] p + ess inf Z exp( − R ( X + ess inf Z − v )) E P [exp( − R ( X + Z − v )] p + exp( − R ( X + ess inf Z − v ))(1 − p ) (cid:27) . As ¯ H : [0 , × [ess inf Z, ess sup Z ] → [ess inf Z, ess sup Z ] is jointly continuous and V ( B [ p ]( Z − ess inf Z ) + ess inf Z ) is unique for every p ∈ [0 , p ∈ [0 , → V ( B [ p ]( Z − ess inf Z ) + ess inf Z ) is continuous. Finally the resultfollows by Proposition 3.4(3) as V ( B [ p ]( Z − ess inf Z ) + ess inf Z ) = V ( B [ p ]( Z − ess inf Z )) + ess inf Z .Now, let Z ∈ L ∞ such that Z − ess inf Z ∈ dom V then ˆ V ( Z ) = ˆ V ( Z − ess inf Z ) + ess inf Z = V ( Z − ess inf Z ) + ess inf Z = ¯ V ( Z ).2. Fix Z ∈ L ∞ such that Z − ess inf Z dom V (and in particular, by Corollary 3.3(2), Z > ess inf Z a.s.)then, by Theorem 3.2, H ˆ Z ( X ) > X with ˆ Z := Z − ess inf Z . By definition, this is equivalent to E P [( ˆ Z −X ) exp( − R ( ˆ Z ))] >
0. Let ǫ ∗ := min (cid:16) X , inf { ǫ ∈ R + | E P [( ˆ Z − ( X − ǫ )) exp( − R ( ˆ Z + ǫ ))] ≥ } (cid:17) ; ǫ ∗ > ǫ ∈ R + E P [( ˆ Z − ( X − ǫ )) exp( − R ( ˆ Z + ǫ ))]. For any ǫ ∈ (0 , ǫ ∗ ) define p ǫ := X − ǫ ) exp( − R ( ǫ ))( X − ǫ ) exp( − R ( ǫ ))+ E P [( ˆ Z − ( X − ǫ )) exp( − R ( ˆ Z + ǫ ))] ∈ (0 , H B [ p ǫ ] ˆ Z ( X − ǫ ) = X − ǫ (i.e., V ( B [ p ǫ ] ˆ Z ) = X − ǫ ). Additionally, by construction, H B [ p ] ˆ Z ( X − ǫ ) ≥ X − ǫ for every p ∈ [ p ǫ ,
1) which implies p ǫ > p ǫ for ǫ < ǫ . Finally, it is trivial to observe lim ǫ ց p ǫ = 1. Weconclude:lim p ր V ( B [ p ] ˆ Z ) + ess inf Z = lim ǫ ց V ( B [ p ǫ ] ˆ Z ) + ess inf Z = lim ǫ ց ( X − ǫ ) + ess inf Z = X + ess inf Z. B.4 Proof of Lemma 3.8
Proof.
Before considering the properties themselves, we will often use the following reformulation of ¯ V :¯ V ( Z ) = sup { v ∈ R + | E P [( Z − ess inf Z − v ) exp( − R ( X + Z − ess inf Z − v ))] ≥ , X − v ∈ D } + ess inf Z which holds because of the definition of H Z − ess inf Z ( v ).1. First consider the lower bound; as utilized in the proof of Theorem 3.5, H Z − ess inf Z (0) ≥ X ∈ D . Therefore ¯ V ( Z ) ≥ ess inf Z by construction. Now consider the upper bound; H Z − ess inf Z ( v ) = E P [( Z − ess inf Z ) exp( − R ( X + Z − ess inf Z − v ))] / E P [exp( − R ( X + Z − ess inf Z − v ))] ≤ (ess sup Z − ess inf Z ) E P [exp( − R ( X + Z − ess inf Z − v ))] / E P [exp( − R + Z − ess inf Z − v ))] = ess sup Z − ess inf Z .Therefore, by the monotonicity of H Z − ess inf Z , v > ¯ V ( Z − ess inf Z ) for any v > ess sup Z − ess inf Z ,i.e., ¯ V ( Z ) ≤ ess sup Z .2. This follows immediately by the law invariance of Z ess inf Z and Z H Z − ess inf Z ( v ).3. By construction ¯ V ( Z + z ) = sup { v ∈ R + | H Z + z − ess inf[ Z + z ] ( v ) ≥ v, X − v ∈ D } + ess inf[ Z + z ] =sup { v ∈ R + | H Z − ess inf Z ( v ) ≥ v, X − v ∈ D } + ess inf Z + z = ¯ V ( Z ) + z .4. Let X ≥ X , then H Z − ess inf Z ( v ; X ) ≥ H Z − ess inf Z ( v ; X ) (as X 7→ H Z − ess inf Z ( v ; X ) is monotonic asdetailed in the proof of Proposition 3.4(4)) and X − v ≥ X − v . Therefore, by construction of ¯ V , theresult follows.5. First, if D = R then by dom V = L ∞ , ¯ V = V (by Theorem 3.5), and strong continuity of V (byProposition 3.4(5)) the result follows. Second, consider D = R ++ . By the Berge maximum theorem,if Z D ( Z ) := { v ∈ [0 , X ] | E P [( Z − ess inf Z − v ) exp( − R ( X + Z − ess inf Z − v ))] ≥ } is a set-valued continuous mapping (in the Vietoris topology) then the result holds because Z ss inf Z is continuous in the strong topology. By the closed graph theorem and continuity of z z exp( − R ( X + z )), D is an upper continuous mapping. Now, let V ⊆ [0 , X ] be open in the subspacetopology and define D − [ V ] := { Z ∈ L ∞ | D ( Z ) ∩ V 6 = ∅} ; if D − [ V ] is open then D is lower continuous.Let Z ∈ D − [ V ] and, in particular, let v ∈ V such that v ∈ D ( Z ).(a) If H Z − ess inf Z ( v ) − v > N Z around Z such that N Z ⊆ D − [ V ]by continuity of Z H Z − ess inf Z ( v ) − v .(b) If H Z − ess inf Z ( v ) − v = 0 then:i. If v > V open, take ǫ > v − ǫ ∈ V . By ˆ v H Z − ess inf Z (ˆ v ) − ˆ v strictlydecreasing, H Z − ess inf Z ( v − ǫ ) − ( v − ǫ ) > v = 0 then, by construction, D − [ V ] = L ∞ and the result follows.6. (a) Let Z ≥ Z . We will, initially, assume Z ≥ V ( Z ) ≥ ¯ V ( Z ) if, and only if, { v ∈ R + | E P [( Z − v ) exp( − R ( X + Z − v ))] ≥ , X + ess inf Z − v ∈ D }⊇ { v ∈ R + | E P [( Z − v ) exp( − R ( X + Z − v ))] ≥ , X + ess inf Z − v ∈ D } . By Theorem 3.5, and because Z ≥ E P [( Z − v ) exp( − R ( X + Z − v ))] ≥ E P [( Z − v ) exp( − R ( X + Z − v ))] and X + ess inf Z − v ≥ X +ess inf Z − v . Therefore the constraints are more restrictive w.r.t. Z than Z and the resultfollows.Now, let us relax the nonnegativity assumption on Z , i.e., let ess inf Z ∈ R arbitrary. Byconstruction Z − ess inf Z ≥ Z − ess inf Z ≥ V ( Z − ess inf Z ) ≥ ¯ V ( Z − ess inf Z ). Finally, by Property (3), this provides the desired monotonicity ¯ V ( Z ) ≥ ¯ V ( Z ).(b) By translativity and monotonicity, for any Z , Z ∈ L ∞ , ¯ V ( Z ) ≤ ¯ V ( Z + k Z − Z k ∞ ) = ¯ V ( Z )+ k Z − Z k ∞ . Taking the same inequality but switching Z and Z proves | ¯ V ( Z ) − ¯ V ( Z ) | ≤k Z − Z k ∞ .7. (a) ¯ V is concave if D := { ( Z, v ) ∈ L ∞ × R + | E P [( Z − ess inf Z − v ) exp( − R ( X + Z − ess inf Z − v ))] ≥ , X + ess inf Z − v ∈ D } is convex since ( Z, v ) v and Z ess inf Z are concave functions.Let ( Z , v ) , ( Z , v ) ∈ D and λ ∈ [0 , λZ + (1 − λ ) Z ] − [ λv + (1 − λ ) v ] ≥ λ [ess inf Z − v ] + (1 − λ )[ess inf Z − v ]. Additionally ( Z, v ) E P [( Z − ess inf Z − ) exp( − R ( X + Z − ess inf Z − v ))] is concave (on its domain) due to the monotonicity andconcavity assumptions on z z exp( − R ( X + z ). From these properties it is trivial to conclude λ ( Z , v ) + (1 − λ )( Z , v ) ∈ D and the proof is concluded.(b) ¯ V is weak* upper semicontinuous if and only if { Z ∈ L ∞ | ¯ V ( Z ) ≥ v } is weak* closed forevery v ∈ R . By [26, Proposition 5.5.1] and the concavity of ¯ V , this is true if and only if { Z ∈ L ∞ | ¯ V ( Z ) ≥ v, k Z k ∞ ≤ k } is closed in probability for every v ∈ R and k ∈ R ++ . Let Z n → Z in probability so that Z n ∈ { Z ∈ L ∞ | ¯ V ( Z ) ≥ v, k Z k ∞ ≤ k } . First, k Z k ∞ ≤ k trivially. Now we wish to show that ¯ V ( Z ) ≥ v . If v ≤ ess inf Z then this is true trivially byProperty (1). Let v > ess inf Z and define D k := { Z ∈ L ∞ | k Z k ∞ ≤ k } . By construction,¯ V ( Z ) ≥ v if and only if E P [( Z − v ) exp( − R ( X + Z − v ))] ≥ X + ess inf Z − v ∈ D . As Z ∈ D k E P [( Z − v ) exp( − R ( X + Z − v ))] and Z ∈ D k
7→ X + ess inf Z − v are continuous w.r.t.convergence in probability, the result follows. C Proofs for Section 3.3
C.1 Proof of Lemma 3.9
Proof.
1. First we will show that f q ( s ) ∈ R for every s ∈ R + . This is trivially true if sq dom V .Assume sq ∈ dom V ; f q ( s ) exists if and only if the denominator ( E P [(1 − [ sq − ¯ V ( sq )] R ′ ( X + sq − ¯ V ( sq ))) exp( − R ( X + sq − ¯ V ( sq )))]) is nonzero. In fact, we will demonstrate that this denominatoris strictly positive. Note also that ¯ V ( sq ) = V ( sq ) by Theorem 3.5. For simplicity of notation, let V := V ( sq ), R := R ( X + sq − V ), and R ′ := R ′ ( X + sq − V ). E P [(1 − [ sq − V ] R ′ ) exp( − R )]= E P [exp( − R )] + (cid:0) E P [ sq exp( − R )] E P [ R ′ exp( − R )] − E P [exp( − R )] E P [ sqR ′ exp( − R )] (cid:1) E P [exp( − R )] , which is strictly positive by construction in the proof of Theorem 3.2.Second assume s ∈ int { s ∈ R + | sq ∈ dom V } . We now wish to consider ∂∂s V ( sq ). Using the samenotation as above, and in addition V ′ := ∂∂s V ( sq ): V ′ E P [exp( − R )] − V E P [( q − V ′ ) R ′ exp( − R )] = E P [ q exp( − R )] − E P [ sq ( q − V ′ ) R ′ exp( − R )] ⇒ V ′ E P [(1 − [ sq − V ] R ′ ) exp( − R )] = E P [ q (1 − [ sq − V ] R ′ ) exp( − R )] V ′ = E P [ q (1 − [ sq − V ] R ′ ) exp( − R )] E P [(1 − [ sq − V ] R ′ ) exp( − R )] . That is, V ′ = f q ( s ) in this case.Third assume s ∈ int { s ∈ R + | sq dom V } . By Corollary 3.6, ¯ V ( sq ) = X + s ess inf q . Immediatelythis implies ∂∂s ¯ V ( sq ) = f q ( s ) in this case.Finally, by continuity of s ¯ V ( sq ) (see Lemma 3.8(5)) and the fact that ¯ V (0) = 0 the result follows.2. The assumption implies, for sq ∈ dom V , (1 − [ sq − V ( sq )] R ′ ( X + sq − V ( sq ))) exp( − R ( X + sq − V ( sq ))) ≥ f q ( s ) ≥ ess inf q for every s ∈ R + trivially and if sq ∈ dom V with P ( q > ess inf q ) > f q ( s ) > ess inf q .3. The result follows trivially by continuity of R ′ and ¯ V .4. By the relation in Property (1), if s ¯ V ( sq ) is concave then f q is nonincreasing. Therefore therelation holds by Lemma 3.8(7). C.2 Proof of Lemma 3.10
Proof.
1. ¯ f q (0) = E P [ q ] ≥ ess inf q and, for s >
0, ¯ f q ( s ) = ¯ V ( sq ) s ≥ ess inf q by Lemma 1. Now consider P ( q > ess inf q ) > sq ∈ dom V then ¯ f q ( s ) = E P [ q exp( − R ( X + sq − V ( sq )))] E P [exp( − R ( X + sq − V ( sq )))] > ess inf q by Corollary 3.6.(b) If sq dom V then, by Theorem 3.2, it must be that D = R ++ and, in particular, X ∈ R ++ . ByCorollary 3.6, ¯ f q ( s ) = X + s ess inf qs > ess inf q .2. Recall, ¯ f q ( s ) = ¯ V ( sq ) /s for s > f q (0) = E P [ q ]. By continuity of ¯ V (see Lemma 3.8(5)),continuity of the inverse demand function holds so long as lim s → ¯ f q ( s ) = E P [ q ]. We will consider twocases: D = R and D = R ++ .(a) Let D = R . Then ¯ V = V by Theorem 3.5 as dom V = L ∞ . Note that, here, ¯ f q ( s ) = V ( sq ) /s = E P [ q exp( − R ( X + sq − V ( sq )))] / E P [exp( − R ( X + sq − V ( sq )))] for every s >
0. Noting V (0) = 0implies lim s ց ¯ f q ( s ) = E P [ q ] and the result is proven.(b) Let D = R ++ . First we we want to consider a small remark on the domain of V ; if Z ∈ L ∞ + suchthat k Z − ess inf Z k ∞ < X / Z ∈ dom V since H Z ( X + ess inf Z ) < X / < X + ess inf Z .If q is deterministic, then by Lemma 3.8.1 ¯ V ( sq ) = sq , and therefore ¯ f q ( s ) = q. Other-wise, if ess sup q − ess inf q >
0, we have that ¯ f q ( s ) = ¯ V ( sq ) s = V ( s ( q − ess inf q ))+ s ess inf qs for < X / [2(ess sup q − ess inf q )]. As with the prior case, for this small s ,¯ f q ( s ) = E P [( q − ess inf q ) exp( − R ( X + s [ q − ess inf q ] − V ( s [ q − ess inf q ])))] E P [exp( − R ( X + s [ q − ess inf q ] − V ( s [ q − ess inf q ])))] + ess inf q. Again, noting V (0) = 0 then trivially lim s ց ¯ f q ( s ) = E P [ q ].3. Note that ¯ f q ( s ) = sup { p ∈ R + | E P [( q − ess inf q − p ) exp( − R ( X + s ( q − ess inf q − p )))] ≥ , X − sp ∈ D } + ess inf q . By construction, ¯ f q ( s ) = ¯ f q − ess inf q ( s ) + ess inf q . Therefore, monotonicity holds ingeneral if it is true for every random variable q such that ess inf q = 0. Assume ess inf q = 0 andlet s ≥ s . Fix p ∈ [0 , ¯ f q ( s )]. Immediately X − s p ≤ X − s p ; by X − s p ∈ cl D the samemust be true for X − s p . Therefore monotonicity follows if E P [( q − p ) exp( − R ( X + s ( q − p )))] ≥ E P [( q − p ) exp( − R ( X + s ( q − p )))]. This holds because ∂∂s E P [( q − p ) exp( − R ( X + s ( q − p )))] = − E P [( q − p ) R ′ ( X + s ( q − p )) exp( − R ( X + s ( q − p )))] ≤ . D Proofs for Section 4
D.1 Proof of Proposition 4.1
Proof.
Recall that α = (cid:16)P ni =1 1 α i (cid:17) − . Let α ∗ := (cid:16)P n +1 i =1 1 α i (cid:17) − = (cid:16) α + α n +1 (cid:17) − < α . Consider now ∂∂α V ( Z ; α ). We will show that ∂∂α V ( Z ; α ) ≤ V ( Z ; α ) ≤ V ( Z ; α ∗ ) for every Z ∈ L ∞ . ∂∂α V ( Z ; α ) = E P [exp( − αZ )] ∂∂α E P [ Z exp( − αZ )] − E P [ Z exp( − αZ )] ∂∂α E P [exp( − αZ )] E P [exp( − αZ )] = E P [exp( − αZ )] E P [ − Z exp( − αZ )] − E P [ Z exp( − αZ )] E P [ − Z exp( − αZ )] E P [exp( − αZ )] . Let x = αZ . Then ∂∂α V ( Z ; α ) ≤ E P [exp( − x )] E P [ x exp( − x )] − E P [ x exp( − x )] ≥ E P [exp( − x )] E P [ x exp( − x )] − E P [ x exp( − x )] = E P [exp( − x )] E P [ x exp( − x )] − E P [exp( − x x exp( − x ≥ E P [exp( − x )] E P [ x exp( − x )] − E P [exp( − x ] E P [( x exp( − x ]= E P [exp( − x )] E P [ x exp( − x )] − E P [exp( − x )] E P [ x exp( − x )] = 0 , here the inequality above follows from the Cauchy-Schwartz inequality. D.2 Proof of Proposition 4.4
Proof.
1. By Theorem 3.2, Z ∈ dom V if and only if H Z ( X +ess inf Z ) ≤ X +ess inf Z , which by construc-tion is true if and only if E P [ Z ( Z − ess inf Z ) − η ] ≤ ( X + ess inf Z ) E P [( Z − ess inf Z ) − η ]. Rearrangingterms completes the proof.2. This follows immediately by Lemma 3.8 as z exp( − R ( X + z )) = z ( X + z ) − η .3. This follows by the same logic as Lemma 3.8 because z exp( − R ( X + z )) = z ( X + z ) − η ..