Modelling multi-period carbon markets using singular forward backward SDEs
aa r X i v : . [ q -f i n . M F ] A ug MODELLING MULTI-PERIOD CARBON MARKETS USINGSINGULAR FORWARD BACKWARD SDES
JEAN-FRANÇOIS CHASSAGNEUX ˚ , HINESH CHOTAI : AND DAN CRISAN ; Abstract.
We introduce a model for the evolution of emissions and the price of emis-sions allowances in a carbon market such as the European Union Emissions TradingSystem (EU ETS). The model accounts for multiple trading periods, or phases, withmultiple times at which compliance can occur. At the end of each trading period, theparticipating firms must surrender allowances for their emissions made during that pe-riod, and additional allowances can be used for compliance in the following periods. Weshow that the multi-period allowance pricing problem is well-posed for various mecha-nisms (such as banking, borrowing and withdrawal of allowances) linking the tradingperiods. The results are based on the analysis of a forward-backward stochastic dif-ferential equation with coupled forward and backward components, a discontinuousterminal condition and a forward component that is degenerate. We also introduce aninfinite period model, for a carbon market with a sequence of compliance times andwith no end date. We show that, under appropriate conditions, the value function forthe multi-period pricing problem converges, as the number of periods increases, to avalue function for this infinite period model, and that such functions are unique.
Keywords: forward-backward systems, decoupling field, carbon markets, emission trad-ing systems, market stability reserve, multiple compliance periods
MSC Classification (2020):
Primary 60H30; secondary 91G80.1.
Introduction
Carbon emission markets have been implemented in several regions worldwide as ameasure to mitigate against climate change. These are cap and trade schemes, where aregulator sets a cap on the total amount of emissions of all the participants in a particularmarket. The regulator releases a number of allowances, the number being equal to thisaggregate cap. For each unit of emissions made, a regulated firm must surrender oneallowance. The allowances can be traded amongst firms in the market, typically calleda carbon market. Within a cap and trade scheme, firms that can reduce their emissionscheaply will do so, and these firms will sell excess allowances. On the other hand, firmsthat can not reduce their emissions cheaply will buy allowances to cover their emissions.If the aggregate cap is set appropriately, emissions reduction can be achieved.The European Union (EU) has had its own emissions trading system (ETS) since2005. So far, there have been three different phases, with the fourth phase, phase 4, setto start in 2021. Every year, any operator that falls under the remit of the EU ETSmust submit an emissions report outlining its level of emissions for the year and thensurrender EUAs (European Union Allowances) for all of its emissions by April 30 of the ∗ Laboratoire de Probabilités, Statistique et Modélisation, Université de Paris. [email protected] † Department of Mathematics, Imperial College London. [email protected] ‡ Citigroup, London. [email protected] following year. Each EUA is worth 1 tonne of CO e, (equivalent tonnes of CO ). Foreach verified tonne of CO e made by an operator and not accompanied by a EUA in theappropriate year, an operator must pay a penalty. Since 2013, the penalty has been set at e
100 per tonne of CO e, rising with the EU inflation rate [20]. Phase 1 was a trial phaserunning from 2005 to 2008. During phase 1, only power generators and energy-intensiveindustries were covered. Almost all EUAs were given to firms for free. Towards the endof phase 1, it became clear that the total number of allowances issued would exceed thelevel of emissions. Since the EUAs from phase 1 could not be carried forward to thefollowing phase, phase 2, the EUA price decreased to 0 towards the end of phase 1 [23].Phase 2 ran from 2008 to 2012 and covered more sectors and companies than phase 1,and featured a lower cap on emissions. The proportion of EUAs allocated freely fell, andsome EUAs were initially allocated through auctions. At the end of phase 2, firms couldcarry over, or ‘bank’ any unused EUAs to the following phase, phase 3. This bankingmechanism is expected to continue for transitions between all future phases and it iscovered by the models in this paper.As of 2020, phase 3 (2013-2020) is in operation. It covers more sectors and moregreenhouse gases (GHGs) than previous phases. In addition, auctioning is now thedefault method for allocating allowances [21]. Since 2019, the EU ETS has also featureda market stability reserve (MSR). The aim of the MSR is to address the large surplus ofallowances currently in the market. Each year, the European Commission will considerthe total number of allowances in circulation. This number will, according to pre-definedrules, be used to decide whether a proportion of allowances that would have been releasedin the following year will be placed into the reserve, or whether allowances in the reservewill be released into the market. The reserve began with 900 million allowances thatwere deducted from auctioning volumes in the years 2014-2016 [22]. Full details of theMSR can be found in [26]. The multi-period model presented in this paper is appropriatefor modelling a MSR in a setting with two compliance periods and can be appropriatelyextended to model an MSR with an arbitrary number of compliance periods; see Example2 in the following section.In 2013, 8715 million tonnes of CO e worth of EU emissions allowances were traded.This was higher than in any year of phase 2. The majority of allowances, approximately6000 million tonnes of CO e worth in 2013, are traded on an exchange [24]. There isevidence that the EU ETS has led to significant emissions abatement without a significantreduction in competitiveness. In addition, the EU ETS may have also led some operatorsto consider innovation activities that would lead to emissions reduction; see [34] for areview of various economic studies.Besides the EU ETS, there are many other carbon markets in operation worldwide,and the results and analysis in this paper should be equally applicable to these othermarkets. Notable examples include the California cap-and-trade program and SouthKorea’s emissions trading scheme.China is planning to introduce a national emissions trading scheme for greenhouse gasemissions in 2020. In preparation for this, in 2013, seven pilots were introduced in sevendifferent regions in China. Since their inception up until 31 July 2015, over 57 milliontons of carbon had been traded under the pilots, and this quantity was valued at US$308million [33, 38]. By 2018, pilot schemes had begun in the regions of Shenzhen, Shanghai,Beijing and Guangdong. It is set to be fully functional in 2020 and, when it is, China’semissions trading scheme will be the largest carbon market in the world, covering moreCO e of GHGs than any other such market. In 2019, the Energy Transitions Commission INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 3 (ETC) released a report explaining that China can achieve net zero carbon emissionswhile becoming a fully developed economy by 2050 [18].In 2019, the World Bank stated that, in 2018, governments raised approximatelyUS$44 billion in carbon pricing revenues over 2018, constituting an increase of approxi-mately US$11 billion compared with the corresponding amount for 2017 [36]. The High-Level Commission on Carbon Prices, in [37], concluded that achievement of the targetwithin the Paris Agreement would be consistent with a carbon price level of betweenUS$40 and US$80 by 2020 and between US$50 and US$100 by 2030. According to theWorld Bank’s report, less than five percent of global emissions covered by carbon pricinginitiatives are priced at this level; the majority are priced below this level. In 2019 areport convened by the Science Advisory Group to UN Climate Action Summit 2019 [19]concluded that countries’ intended nationally determined contributions (NDCs) wouldroughly need to be tripled to be consistent with the target of at most a 2 ˝ C rise in meanglobal temperature from pre-industrial levels.For the EU ETS, between 2017 and 2018, total emissions from stationary installa-tions declined by 4.1%. Overall, total ETS emissions from stationary installations havedeclined by around 29% between 2005 and 2018. It is expected that, stationary emis-sions are set to decrease by 36% compared with 2005 levels by 2030 and that, even withadditional measures, this reduction would be 41%, which is still lower than the targetvalue of 43% [25].Carbon price formation is complex and can be approached in many different ways.This paper presents one particular approach. Broadly speaking, models for pricing emis-sions allowances in carbon markets can be separated into three categories: full equi-librium, risk-neutral and reduced form models; see [29] for further details. Consider amarket consisting of firms producing goods which cause emissions which is connected toan emissions trading system comprising a liquid market for emissions allowances. In a fullequilibrium model, one considers the interaction of individual firms in the market. Suchmodels often lead to an optimization problem in which firms optimize their productionof goods and number of emissions allowances. Some examples of full equilibrium modelscan be found in [2, 3, 8, 9, 10]. In a reduced form model, the coupling or interdependencebetween the allowance price and the level of emissions is not modelled explicitly. For ex-ample, we may assume that the level of cumulative emissions follows a standard process,such as geometric Brownian motion, independently of the allowance price. Such modelsare more tractable numerically compared with full equilibrium models. Some examplesof reduced form models can be found in [11, 14, 28, 35]. Finally, in a risk-neutral model,the price of an emissions allowance and the level of emissions are both modelled directly,without reference to individual firms. For these models, results are derived by using thetools and methods of risk-neutral pricing. Examples of risk-neutral models include themodel described in [12] and all models which use FBSDEs to model a carbon market,which are described below. The model studied in this paper fits into the same risk-neutralframework. Other models that do not necessarily fit into only one of the aforementionedcategories can be found in [1, 12, 16, 27].The model studied here is based on a class of singular forward-backward stochasticdifferential equations (FBSDEs). These equations have three distinct features: theirforward and backward component are coupled, the terminal condition for the backwardcomponent is a discontinuous function of the terminal value of the forward component,and the forward dynamics are degenerate. In particular, the equation satisfied by the cu-mulative emissions process has no volatility term. Such equations have already been used
JEAN-FRANÇOIS CHASSAGNEUX, HINESH CHOTAI AND DAN CRISAN to model the evolution of the cumulative emissions and price of an emissions allowancein a carbon market such as the EU ETS; see [5, 7, 13, 29].The novelty in this work is that we introduce a model with multiple trading periods.More precisely we consider a market consisting of participating firms whose activitiescause emissions during the time interval r , T s . For a positive integer q , representingthe number of periods, the overall time interval r , T s is divided into q trading periods: r T : “ , T s , r T , T s ,..., r T q ´ , T q : “ T s . During any trading period r T k ´ , T k s , for ď k ď q , emissions regulation is in effect. Let p E qt q t Pr ,T s be a real valued continuousprocess representing, at time t , the cumulative emissions made in the market up to time t . For every integer ď k ď q , at time T k , the regulator records the level of cumulativeemissions E qT k , and, for each ď k ď q ´ , a cap on the level of emissions at T k ` isdefined. At each time T k ` , the regulator checks whether the emissions made during the r T k , T k ` s have exceeded the time T k ` cap and market participants must pay a penaltyfor each unit of emissions above the cap. We will denote by p Y qt q t Pr ,T s the spot price ofan allowance certificate at time t . At each time T k ` the spot price will depend on thevalue of the cumulative emissions E T k ` but also on the cap imposed by the regulator,which in turn, will depend on E T k .This model is more realistic than the single-period model introduced in previous work[5, 7, 13]. In the single trading period model, any unused allowances become worthless atthe end of the trading period. The multi-period model allows for the caps on the level ofemissions E qT k ` at time T k ` to depend on the level of emissions E qT k accumulated fromtime 0 up to time T k , the beginning of the k -th trading period. In this framework, unusedallowances can be ported to the next period (except for the final period) depending onthe mechanism being modelled. This permits the modelling of mechanisms includingthe banking, borrowing or withdrawal of allowances between compliance periods. SeeExample 1 below for further details.The main result of the paper is to give a characterization of the pair of process p E qt , Y qt q t Pr ,T s as the unique solution of a set of q FBSDEs that are linked through theirtransition values at times T k , k “ , . . . , q ´ and terminal conditions at times T k , k “ , . . . , q . In addition, the consecutive FBSDEs can be linked so as to model banking,borrowing and withdrawal of allowances, as described above. The linking of the FBSDEsin the multi-period model means that it is not possible to directly consider each FBSDEas a separate single-period model. This is the main technical difficulty of studying themulti period model. As usual, the study of this FBSDE is closely linked to the study ofthe associated value function (known as decoupling field in the FBSDE literature). Thisdecoupling field can be considered to be an entropy solution to a degenerate quasilinearelliptic PDE. Even though we rely on some important results given in [6], we also establishnew estimates concerning this value function, see e.g. Lemma 3.6, Lemma 3.7 and Lemma3.8 below. They constitute a step forward in the study of singular FBSDEs and theirassociated decoupling fields.As described above, a multi-period model is more applicable than a single periodmodel because it allows one to model multiple compliance periods. One disadvantageof the multi-period model studied here, however, is that, for a q period model, onemust specify the end date T q . This is important because, at T q , the spot price Y qT q ofan allowance certificate is different from the one specified at every prior time T k , for ď k ď q ´ . The time T q is the time at which all emissions regulation ceases and thisis why Y qT q “ if at the time T q cumulative emissions are below the time T q cap. For amore realistic model, one can consider a model for a carbon market with no specified end INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 5 date. In the setting of the EU ETS, for example, there is currently no time at which onecan say with certainty that emissions regulation will cease or the banking of allowancesto next period will be prohibited. In the second part of the paper, we introduce a modelfor a carbon market in operation over the time period r , with no end date and showthat it is well posed under certain conditions. More precisely, we give a characterizationof a pair of processes, p E t , Y t q t Pr , , say, E representing the cumulative emissionsand Y the allowance price, as the unique solution of an infinite sequence of FBSDEsthat are linked through their transition values at times T k , k ě . Moreover we showthat, under reasonable conditions, the spot price Y qt of an allowance certificate for the q -period model converges, as the number of periods q increases, to the spot price Y t ofan allowance certificate for the infinite period model. Again, the results are obtained bya careful study of the associated decoupling field.Carbon markets face many criticisms. Some critics claim that they reduce industries’competitiveness, while others believe that the average carbon price today is not highenough to motivate a substantial reduction in greenhouse gas emissions. Proponentsof emissions trading systems claim that they lead to real emissions reductions whenregulators operate them in an appropriate way. In any case, it is clear that emissionstrading systems are becoming increasingly important and prevalent. Scientific, partic-ularly mathematical, studies of them are needed in order to expose more about theiradvantages, their shortcomings, and their efficient implementation. It is hoped that theresults of this paper will improve the understanding of carbon markets and help regu-lators to implement them in a way that brings the greatest social benefit in the actionagainst the effects of climate change.The rest of this paper is organised as follows. In Section 2, we introduce the mainassumptions and key notions used in using singular FBSDEs to model carbon markets ina multi-period setting, and give generic statements of the main results. In particular, westate the wellposedness of a class of singular FBSDEs with the terminal condition for thebackward equation that is a discontinuous function of the terminal value of the forwardequation, and the forward dynamics may be degenerate (having no volatility term). ThisFBSDE is autonomous in the sense that it does not depend on other processes. Thisproperty holds for all of the q FBSDEs comprising a q period model. In Section 3 westate and prove the wellposedness of the multi-period model. We also present several newresults concerning singular FBSDEs that are needed to study the infinite period model.In Section 4, we state and prove the wellposedness of the infinite period model as wellas the convergence of the value function for a q -period multi-period model to a valuefunction for the infinite period model, as q tends to infinity. The paper is concluded withan appendix in which some results for single period models that were used in this paperare presented. Notation.
In the following we will use the following spaces ‚ For fixed ď a ă b ă `8 and I “ r a, b s or I “ r a, b q , S ,k p I q is the set of R k -valued càdlàg F t -adapted processes Y , s.t. } Y } S : “ E „ sup t P I | Y t | ă 8 . Note that we may omit the dimension and the terminal date in the norm notationas this will be clear from the context. S ,k c p I q is the subspace of process withcontinuous sample paths. French acronym for right continuous with left limits.
JEAN-FRANÇOIS CHASSAGNEUX, HINESH CHOTAI AND DAN CRISAN
We also consider S ,k pr , the vector space of càdlàg adapted processes Y , withvalues in R k , and such that E “ sup ď t ď b | Y t | ‰ ă 8 for every b ą . S ,k c pr , denotes the subspace of such processes having continuous paths. ‚ For fixed ď a ă b ă `8 , and I “ r a, b s or I “ r a, b q again, we denote by H ,k p I q the set of R k -valued progressively measurable processes Z , such that } Z } H : “ E „ż I | Z t | dt ă 8 . H ,k pr , is the set of R k -valued progressively measurable processes Z , suchthat E ”ş b | Z t | dt ı ă 8 , for all b ą .For any process X and time t , we denote by X t ´ and X t ` respectively, the left and rightlimit of the process X at t , i.e X t ´ “ lim s Ò t X s , X t ` “ lim s Ó t X s . For φ : R d ˆ R Ñ R , measurable and non-decreasing in its second variable, the functions φ ´ and φ ` are the left and right continuous versions, respectively defined, for p p, e q P R d ˆ R , by, φ ´ p p, e q “ sup e ă e φ p p, e q φ ` p p, e q “ inf e ą e φ p p, e q . (1.1)Moreover, we denote by } ¨ } the essential supremum: } φ } “ esssup p p,e qP R d ˆ R | φ p p, e q| . Framework and main result
Framework for the multi-period model.
Let p Ω , F , P q be a complete probabil-ity space. We denote by W a d dimensional Brownian motion defined on p Ω , F , P q startedat t “ , and t F t u ď t ď T the complete filtration generated by the Brownian motion W .In the following, we consider a market with q ě trading periods, denoted r T , T s , r T , T s ,..., r T q ´ , T q s with T : “ ă ¨ ¨ ¨ ă T k ă ¨ ¨ ¨ ă T q “ : T for T ą . The market isgoverned by three processes p P, E, Y q all defined on p Ω , F , P q : ‚ The process p Y t q ď t ď T represent the spot price of a carbon emissions allowance.The constant r will denote the instantaneous risk-free interest rate, which willbe assumed to be fixed and deterministic throughout this paper. r is such thatinvestment of x at time yields e rt x at time t , for any t P r , T s . We will as-sume that the allowances are traded assets, and that the discounted price process p e ´ rt Y t q t Pr T k ,T k ` q is an F t -adapted martingale (see also Remark 1). Equivalently,the dynamics of Y can be written Y t “ Y t ` ż t t rY s ds ` ż t t Z s dW s , (2.1) Although European Union Allowance (EAU) futures are most commonly traded in the EU ETS, wemodel the spot price rather than the futures price to simplify the presentation in the multi-period model.For a one period model, futures prices can be directly modelled. Note that, in our setting of a constant,deterministic interest rate r , the spot and futures prices only differ by a multiplicative deterministicdiscount factor. INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 7 for every T k ď t ď t ă T k ` , where p Z t q t Pr T k ,T k ` q is a progressively measurableprocess such that E „ ş T k ` T k | Z s | ds ă 8 , k ď q . ‚ The process p P t q ď t ď T represents factors in the market that will also drive emis-sions. We assume that this process is purely autonomous and independent ofthe level of cumulative emissions and the allowance price. For example, in thepresentation of [5] for an electricity market with emissions regulation, p P t q ď t ď T could be a vector consisting of fuel prices and an inelastic demand curve for elec-tricity. Setting P “ p , a deterministic constant, we assume that its dynamicsare given by d P t “ b p P t q d t ` σ p P t q d W t , t P r , T s , (2.2)for functions b and σ such that strong existence and uniqueness holds for the SDE(2.2). (The conditions on all coefficient functions will be made precise below andin the following section). ‚ The process p E t q ď t ď T represents the cumulative emissions in the market; itresults from integrating the quantity µ p P t , Y t q , where µ : R d ˆ R Ñ R is afunction representing the market emissions rate. In other words, we assume thatthe dynamics of the cumulative its dynamics are given by d E t “ µ p P t , Y t q d t, t P r , T s , (2.3) Remark 1.
We are implicitly assuming here that we work under a risk neutral probabilitymeasure P . In this setting, the discounted future cash flow of any tradable asset is amartingale. To summarise, we shall assume that the 4-tuple p P t , E t , Y t , Z t q t ď t ď T satisfies the follow-ing forward-backward stochastic differential equation on each period r T k , T k ` q , k ď q : d P t “ b p P t q d t ` σ p P t q d W t , d E t “ µ p P t , Y t q d t, d Y t “ rY t d t ` Z t d W t . (2.4)This description is obviously incomplete as one must understand what happens at theend of each period in other to link them together and hopefully obtain a market processon the whole time interval r , T s . We now describe the mechanisms that are put in placein the ETS market and that we will take into account in our model.For every integer ď k ď q ´ , the number of allowances in circulation at T k ´ , the startof the r T k ´ , T k s period, will be assumed to be a deterministic function of E T k ´ , namely Γ k p E T k ´ q where R Q e Ñ Γ k p e q P R . This is the cap on emissions during that period;compliance occurs at time T k , if and only if E T k ´ E T k ´ ă Γ k p E T k ´ q or equivalentlyif E T k ă Λ k p E T k ´ q , where Λ k p e q : “ Γ k p e q ` e . The quantity Λ k p E T k ´ q represents thenthe cap on cumulative emissions at T k , namely a cap for all emissions from time totime T k . For every k ě , at time T k there is a penalty for non-compliance ρ k , which isusually set to be equal to 1, incurred if the cumulative emissions up to that time E T k have exceeded the cap Λ k . In the sequel, we will work with penalty functions Λ whichbelong to the following class. Definition 2.1.
Let Θ be the class of functions Λ : R Ñ R such that e ÞÑ Γ p e q : “ Λ p e q´ e is monotone decreasing and satisfies lim e Ñ`8 Γ p e q “ ´8 . We now give examples of cap functions that can be encountered in practice.
JEAN-FRANÇOIS CHASSAGNEUX, HINESH CHOTAI AND DAN CRISAN
Example 1.
We give examples to show how the cap functions p Λ k q ď k ď q can be chosento model different mechanisms that are in force in the EU ETS market, namely ‚ Banking: allowances that are not used in one period can be carried forward forcompliance in the next period. ‚ Withdrawal: for any ď i ď q ´ , if the cap on emissions is exceeded at T i ,then the regulator removes a quantity of allowances from the r T i , T i ` s marketallocation. The quantity of allowances removed is equal to the level of excessemissions at T i . ‚ Borrowing: for any ď i ď q ´ , firms may trade some of the allowancesto be released at T i during r T i ´ , T i s . If each trading period represents a year,this means that firms can, in a particular year that is not the final year, use thefollowing year’s allowance allocation for compliance.Suppose that the regulator releases c k ` ě allowances into circulation at each time T k for k “ , , ..., q ´ .To take into account banking, borrowing and withdrawal, we can set Γ k p e q “ p k ` q^ q ÿ i “ c i ´ e , (2.5) For banking and withdrawal only, we can set Γ k p e q “ k ÿ i “ c i ´ e , (2.6) for every ď k ď q . We can also present a very simple example in which the functions Λ k are not all constantfunctions. Example 2. [Simple market stability reserve for a two period model] Let q “ for atwo period model. Suppose that, similarly to Example 1, for each k “ , , the regulatorhas a quantity of allowances c k ` ě to be released into the market at T k and to be usedfor compliance at any time after T k To specify the mechanism that links the two periods,we simply need to specify the two functions Γ and Γ or, equivalently, Λ and Λ . Let Λ “ c , thus e ÞÑ Γ p e q “ c ´ e . Assuming E “ , the cap at T is simply a constant,equal to c .Suppose that, at T , the regulator considers the number of allowances in circulationinstantaneously after all allowances for emissions up to T have taken place and checkswhether it is within the interval r κ L , κ U s where κ L ă κ U and κ L and are κ U are, respec-tively, lower and upper thresholds. The regulator then adjusts the number of allowancesin circulation in the following way. The regulator adds a fixed quantity c if the number ofallowances in circulation would be below the threshold, reduces the number of allowancesin circulation by a proportion p ´ α q of the total if the number of allowances in circu-lation would be above the threshold, and does not make an adjustment if the number ofallowances would be within the threshold. Here, c ą and ă α ă are constants.Mathematically, this is expressed as Γ p E T q “ $’’&’’% ˜Γ p E T q ` c, if ˜Γ p E T q ă κ L , ˜Γ p E T q , if κ L ď ˜Γ p E T q ď κ U ,α ˜Γ p E T q , if ˜Γ p E T q ą κ U , (2.7) INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 9 where ˜Γ p E T q represents the number of allowances in circulation at T instantaneouslybefore adjustment with ˜Γ given by e ÞÑ ˜Γ p e q “ c ` c ´ e . (2.8) The above setting is a simple version of a market stability reserve. Such a reserve wasestablished and began operation in January 2019 in the EU ETS ( [26] ). This reserve isdesigned to reduce the large surplus of allowances that has built up in the EU ETS sincephase 2. In the EU ETS, if the total number of allowances in circulation is too high,then 12% of these allowances will be removed and placed in the reserve. Similarly, if thenumber of allowances in circulation is too low (less than 400 million) in a given year,then a quantity of allowances up to 100 million will be released from the reserve and addedto the volume of allowances set to be released through auction. See [26] for full details.To model a market stability reserve more realistically for a q period model, where q ą ,we would need to study a model in which, for each ă k ď q , the cap on emissions at T k depends not only on E T k ´ , but on all quantities E T , E T ,..., E T k ´ . Although we do notconsider such a model in this paper, the model presented here can be extended to such asetting and, under additional conditions, the results regarding existence and uniquenesscan be proved. Remark 2.
The modelling assumption of the emissions process being continuous at alltimes including the compliance times can be justified in the following way: Assumingcontinuity at each compliance time is required to specify the initial and terminal conditionsfor those FBSDEs, and it implies that the solutions of the different FBSDEs defining themulti period model are not independent. This is realistic and means that a multi periodmodel is different to several separate copies of a single period model. In a realistic marketsetting, an increase or decrease in emissions requires an adjustment in the factors ofproduction and we argue that this can not be carried out instantaneously in such a waythat the emissions process would develop a point of discontinuity at any time. Anotherargument is the following. For each ď k ď q , the cap at time T k , which will affect themarket dynamics over r T k , T k ` s is a deterministic function of E T k ´ ; it is already knownat time T k ´ . Based on this, it is reasonable to assume that E will not have a jump orany kind of discontinuity at T k for any ď k ď q . We now describe the link between periods for the Y -process, it follows from thefollowing heuristics : If the cumulative emissions at time T k is less than the cap at thattime, i.e., E T k ă Λ k p E T k ´ q , then the spot price of the allowance will be ported to the nextperiod, in other words, lim t Ñ T k Y t “ Y T k . However, if E T k ą Λ k p E T k ´ q , then the spotprice should converge to the penalty for non-compliance, in other words, lim t Ñ T k Y t “ .For the last period, as the market ends at T q , we set by convention Y T q “ .Before stating our main result concerning the existence and uniqueness of an equilib-rium for the market described above in this multi-period setting, we give the assumptionson the coefficient functions parameters that will be used throughout this paper. Assumption 1.
The functions b : R d Ñ R d , σ : R d Ñ R d ˆ d and µ : R d ˆ R Ñ R aresuch that there exist three constants L ě , l , l ą , { L ď l ď l ď L satisfying(1) b and σ are L -Lipschitz continuous: | b p p q ´ b p p q| ` | σ p p q ´ σ p p q| ď L | p ´ p | , p, p P R d . (2.9) See e.g. [10] for an equilibrium argument in the one period setting. (2) µ is L -Lipschitz continuous, satisfying | µ p p, y q ´ µ p p , y q| ď L ` | p ´ p | ` | y ´ y | ˘ , p, p P R d , y, y P R . (2.10) Moreover, for any p P R d , the real function y ÞÑ µ p p, y q is strictly decreasing and µ satisfies the following monotonicity condition l | y ´ y | ď p y ´ y q ` µ p p, y q ´ µ p p, y q ˘ ď l | y ´ y | , p P R d , y, y P R . (2.11) Remark 2.2.
The strict monotonicity of µ (2.11) is required to prove some bounds usedin the analysis of singular FBSDEs such as (2.4) . In practical applications, µ can beinterpreted as an emissions rate function. In such settings, at a time t , given an allowanceprice Y t and a vector of factors that drive emissions, P t , µ p P t , Y t q is the rate of emissionsper unit time. So the strict monotonicity amounts to assuming that the emissions rate isstrictly decreasing in the allowance price. This is economically reasonable because higherallowance prices can be expected to promote lower level of emissions. In the literature,when (2.4) has been studied for a carbon market operating within an electricity market, µ is explicitly constructed, using functions that arise in the modelling of the electricitymarket, in such a way that guarantees that the strict monotonicity of µ holds [5, 13, 29] .Also, as described in [6] , we can interpret the strict monotonicity of µ as convexity ofthe anti-derivative of ´ µ , which is standard when studying scalar conservation laws. TheFBSDE (2.4) has strong links to the theory of scalar conservation laws: as shown in [6] , Section 3, the value function constructed for (2.4) can be related to a solution of theinviscid Burgers equation. Our first main result is the well-posedness for the multi-period pricing problem.
Theorem 2.3.
Let q ě be an integer representing the number of trading periods andlet Λ k P Θ , ď k ď q be the cap functions, recall Definition 2.1. We consider a q period multi-period model over r T , T s , ...., r T q ´ , T q s with T : “ and T q : “ T . UnderAssumption 1, there exists a unique càdlàg process p Y t q ď t ď T P S , pr , T sq , a continuousprocess p E t q ď t ď T P S , c pr , T sq , a continuous process p P t q ď t ď T P S ,dc pr , T sq and aprocess p Z t q ď t ď T P H ,d pr , T sq satisfying the dynamics (2.4) on each period r T k ´ , T k q , ď k ď q . The process Y is continuous on r T k ´ , T k q ; it can have a jump at T k . Thereit satisfies, for every ď k ď q , almost surely, Y T k ´ “ Y T k , if E T k ă Λ k p E T k ´ q ,Y T k ´ “ , if E T k ą Λ k p E T k ´ q ,Y T k ď Y T k ´ ď , if E T k “ Λ k p E T k ´ q , (2.12) with, by convention, Y T q “ .Moreover, there exists a unique function v r , T s ˆ R d ˆ R ˆ R Q p t, p, e, e q ÞÑ v p t, p, e, e q P R such that, for each e P R , p t, p, e q ÞÑ v p t, p, e, e q is continuous on r T k ´ , T k q ˆ R d ˆ R , and Y t “ v p t, P t , E t , E T k ´ q , T k ´ ď t ă T k , ď k ď q . By convention, v p T, p, e, e q “ , for p p, e, e q P R d ˆ R ˆ R . Remark 2.4. i) The function v is a key object to obtain existence and uniquenessto the FBSDEs and is known as the decoupling field for equation (2.4) . In oursetting, it appears also as a pricing function for the allowance contract, with respectto the underlying process p P, E q . Further properties of the function v are given inProposition 3.15 below. INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 11 ii) Here, we consider a càdlàg Y. This differs slightly from what is done in the one periodsetting [6] where the càglàd version is considered. This is merely a convention thatsimplifies our presentation in the multi-period setting.iii) For a single period model comprising a single copy of equation (2.4) on a time in-terval r , T s say, it is shown in [6] that in general, given a monotone increasing realvalued function φ taking values in r , s (note that φ is not assumed to be Lipschitzcontinuous), one can not construct a process Y satisfying the dynamics (2.4) andsuch that Y T “ φ p E T q almost surely. It is shown there that existence and uniquenessof solutions to the FBSDE does hold when, instead, the relaxed terminal condition, P r φ ´ p E T q ď Y T ´ ď φ ` p E T qs “ , (2.13) is imposed. To model a single period carbon market in which the penalty for overemission is 1 and the cap on emissions is λ , one would typically take φ p x q “ r λ, `8q p x q for every x P R . The terminal condition, (2.12) , presented in Theo-rem 2.3, is the multi period version of this terminal condition. Notice that in themulti-period model, the cap on emissions at time T k is Λ k p E T k ´ q . Infinite period model.
The multi period model introduced in the previous sectionis more realistic and applicable than a single period model because it allows one to modelmultiple times at which compliance occurs and a new allowance allocation is released intoa carbon market. One disadvantage of this, however, is that, for a q period model, onemust specify the end date T q . The time T q is the time at which all emissions regulationceases and this is why the terminal condition at this time specifies that allowances at T q will have price Y T q “ if the time T q cumulative emissions are below the time T q cap.For an even more realistic model, one might prefer to consider a model for a carbonmarket with no specified end date. Phase 2 of the EU ETS was followed by Phase 3,with no interruptions. Although new rules came into effect at the start of Phase 3,any Phase 2 allowances could be carried forward (banked) into Phase 3 and used forcompliance later and, according to the EU ETS handbook [20], this can be expected tocontinue for Phase 4 and all future phases. Therefore, in the setting of the EU ETS,there is currently no time at which one can say that emissions regulation will cease orthe banking of allowances will be prohibited.We thus introduce and study a model with an infinite number of period to addressthe aforementioned limitations. This can be thought of as a model for a carbon marketin operation over the time period r , with no end date. The cap and trade periodsare still connected through the banking, withdrawal and borrowing rules. For this infiniteperiod model, we shall work under the following setting: Assumption 2.
We set T k “ kτ , for every k ě , where τ ą . The cap functions p Λ k q k ě are constants and given by, Λ k “ kλ , for k ě and λ ą . (By a slight abuseof notation, we make no difference between the function and its constant value.) We shall also require one of the two following assumptions:
Assumption 3.
The coefficient b and σ are such that for p P R d , b i p p q and σ i ¨ p p q onlydepend on p i . Moreover, there exists β P R such that, denoting L b the Lipschitz constantof b , r ´ L b ě β ą . Assumption 4. i) The matrix σ is uniformly elliptic, namely, there exists β ą suchthat υ J σ p p q σ p p q J υ ě β | υ | , @p p, υ q P R d ˆ R d . ii) The interest rate r is strictly positive. Remark 2.5. i) The constant τ ą represents the length of each period, which istypically one year. In this case, the constant λ is the yearly cap.In light of Example 1, assuming the Λ k constant is not much of a restriction inapplications. Assumption 3 (resp. Assumption 4(i)) is a technical assumption usedto guarantee that some Lipschitz constant does not explode when considering aninfinite number of periods, see Lemma 3.7 (resp. Lemma 3.8) below.ii) Assumption 3 implies, in particular, that the rate r is strictly positive. Our main result in this setting is the following.
Theorem 2.6.
Let Assumptions 1 - 2 - 3 or 1 - 2 - 4 hold. Then there exist processes P P S ,dc pr , , E P S , c pr , , Y P S , pr , and Z P H ,d pr , satisfying oneach period r T k , T k ` q : d P t “ b p P t q d t ` σ p P t q d W t d E t “ µ p P t , Y t q d t d Y t “ rY t d t ` Z t d W t . (2.14) The process Y is continuous on r T k ´ , T k q . It can have a jump at T k , ď k , where itsatisfies, almost surely Y T k ´ “ Y T k , if E T k ă Λ k ,Y T k ´ “ , if E T k ą Λ k ,Y T k ď Y T k ´ ď , if E T k “ Λ k . (2.15) Moreover, there exists a continuous function w : r , τ q ˆ R d ˆ R Ñ R such that Y t “ w p t ´ T k ´ , P t , E t ´ Λ k ´ q , t P r T k ´ , T k q , k ě . Setting, for e P R , Φ p p, e q “ " w p , p, e ´ λ q , if e ă λ, , otherwise. (2.16) the function w satisfies, Φ ´ p p, e q ď lim inf t Ò τ w p t, p t , e t q ď lim sup t Ò τ w p t, p t , e t q ď Φ ` p p, e q , (2.17) for any family p p t , e t q ď t ă τ converging to p p, e q as t tends to τ . Remark 2.7.
The pricing function w has the remarkable property of a link betweenits terminal condition and its value at time . Precisely, this link is shown in (2.18) and (2.19) below. We explain heuristically below why this coupling emerges. It is alsointeresting to note that we are able to prove uniqueness of this value function in anappropriate setting; see Proposition 4.2 for further details. Let us now explain the structure of the decoupling field, w , in this infinite periodsetting. Assume that we are given a process E P S , pr , such that, for any t ą , E t represents the cumulative emissions up to time t in the market setting introducedhere. Further, assume that there is a unique price for emissions allowances in the marketsuch that at any time t with t ‰ T k for any k ě , the allowance price Y t is equal toa deterministic function, y , of the time t , the time t cumulative emissions E t and the INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 13 time t value of the random factors in the market P t i.e. assume that there exists adeterministic function y such that Y t “ y p t, P t , E t q whenever t P r , T , T , .... u . Byright continuity of Y , we also consider that y p T k , P T k , E T k q “ Y T k ` and moreover lim t Ò T k y p t, P t , E t q “ " y p T k , P T k , E T k q , if E T k ă Λ k , , if E T k ą Λ k , (2.18)for every integer k ě . Now, we argue that the dynamics over the period r T k ´ , T k q areidentical to the dynamics on the time period r T k , T k ` q , except that, in the latter timeperiod, the cap is higher by a quantity λ . The cap being λ units larger is equivalent tothe cumulative emissions being λ units lower. Therefore, we impose that the function y should satisfy y p T k , p, e q “ y p T k ´ , p, e ´ λ q , (2.19)for every integer k ě and every p p, e q P R d ˆ R . On the first period, we see that y satisfies the condition (2.16) and is in fact equal to w for this period. We demonstratethat in fact y (and thus Y ) can be constructed from a slight modification of the valuefunction on the first period. At the level of the pricing function, this expresses thestationarity of our setting. Let us note also that the different periods are decoupled inthe sense that, for each integer k ě , the values of the function w over the time interval r T k ´ , T k q do not depend directly on its values over the time interval r T k , T k ` q , thanksto the time periodicity. This is in contrast with the finite multi-period setting, but atthe price of the strong coupling between initial and terminal condition seen in (2.16).3. The Multi - Period Model
To prove the above results, we first need to obtain results for one-period FBSDEs ofthe form (2.4).3.1.
Well-posedness of one-period singular FBSDEs.
In this section, we will presentresults for one-period model, that will be useful in the sequel. They are often direct ex-tensions of the results obtained in [6].A first ingredient of our study is to be able to consider terminal condition that dependon both E and P . We now introduce the class of terminal conditions we will work within the sequel. Definition 3.1.
Let K be the class of functions φ : R d ˆ R Ñ r , s such that φ is L φ -Lipschitz in the first variable for some L φ ą and non-decreasing in its second variable,namely | φ p p, e q ´ φ p p , e q| ď L φ | p ´ p | for all p p, p , e q P R d ˆ R d ˆ R , (3.1) φ p p, e q ě φ p p, e q if e ě e , (3.2) and moreover satisfying, sup e φ p p, e q “ and inf e φ p p, e q “ for all p P R d . (3.3)We now state an existence and uniqueness result for the one-period model with termi-nal condition depending on P and E . This result is a restatement of some of the resultsin Chapter 2 of the thesis [15]. For completeness, the results have been stated in moredetail in the appendix; see Proposition A.1 and Theorem A.2. Proposition 3.2.
Let Assumption 1 hold and let φ belong to K . Given any initial con-dition p t , p, e q P r , τ q ˆ R d ˆ R , there exists a unique progressively measurable 4-tuple ofprocesses p P t ,p,et , E t ,p,et , Y t ,p,et , Z t ,p,et q t ď t ď τ P S ,d c pr t , τ sqˆ S , pr t , τ sqˆ S , pr t , τ qqˆ H ,d pr t , τ sq satisfying the dynamics d P t ,p,et “ b p P t ,p,et q d t ` σ p P t ,p,et q d W t , P t ,p,et “ p P R d , d E t ,p,et “ µ p P t ,p,et , Y t ,p,et q d t, E t ,p,et “ e P R , d Y t ,p,et “ rY t ,p,et d t ` Z t ,p,et d W t , (3.4) and such that P „ φ ´ p P t ,p,eτ , E t ,p,eτ q ď lim t Ò τ Y t ,p,et ď φ ` p P t ,p,eτ , E t ,p,eτ q “ . (3.5) The function defined by r , τ q ˆ R d ˆ R Q p t , p, e q Ñ v p t , p, e q “ Y t ,p,et P R is continuous and satisfies(1) For any t P r , τ q , the function v p t, ¨ , ¨q is {p l p τ ´ t qq -Lipschitz continuous withrespect to e ,(2) For any t P r , τ q , the function v p t, ¨ , ¨q is C -Lipschitz continuous with respect to p , where C is a constant depending on L , τ and L φ only.(3) Given p p, e q P R d ˆ R , for any family p p t , e t q ď t ă τ converging to p p, e q as t Ò τ ,we have φ ´ p p, e q ď lim inf t Ñ τ v p t, p t , e t q ď lim sup t Ñ τ v p t, p t , e t q ď φ ` p p, e q . (3.6)The following result arises from the proof of Proposition 3.2; see Remark 3.4 below. Corollary 3.3 (Approximation result) . Let Assumption 1 hold. Let p φ n q n ě be a se-quence of smooth functions belonging to K and converging pointwise towards φ as n goesto `8 . For ǫ ą , consider then v ǫ,n the solution to: B t u ` µ p p, u qB e u ` L p u ` ǫ pB ee u ` ∆ pp u q “ ru and u p τ , ¨q “ φ n (3.7) where ∆ pp is the Laplacian with respect to p , and L p is the operator L p p ϕ qp t, p, e q “ B p ϕ p t, p, e q b p p q `
12 Tr “ a p p qB pp ‰ p ϕ qp t, p, e q , (3.8) with B p denotes the Jacobian with respect to p , and a “ σσ J , where J is the transposeand B pp is the matrix of second derivative operators.Then the functions v ǫ,n are C , (continuously differentiable in t and twice continu-ously differentiable in both p and e ) and lim n Ñ8 lim ǫ Ñ v ǫ,n “ v where the convergenceis locally uniform in r , τ q ˆ R d ˆ R . Remark 3.4. i) The proof of Proposition 3.2 is not given in this paper. As describedin the Appendix, this result is almost identical to Theorem 2.2 and Proposition 2.10from [6] with a similar proof. The difference between Proposition 3.2 here and theaforementioned results from [6] is that, here the terminal condition belongs to K ,while in [6] , the authors considered terminal conditions φ : R Ñ r , s which aremonotone increasing, having limit at ´8 and ` at `8 . Moreover, INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 15 (a) In that paper, (and also for the proof of Proposition 3.2), the value function v is constructed in the following way. First, one assumes that the coefficientfunctions b , σ and µ , and the terminal condition φ are Lipschitz smooth withbounded derivatives of all order. Then one adds to the system (2.4) mollifyingnoise of variance ǫ for a small ǫ ą . In this setting, results from [17] allow oneto show that the corresponding version of FBSDE (2.4) has a unique solutionand a smooth value function satisfying a PDE of the form (3.7) . Then, usingsome a priori estimates, a value function, v , for the FBSDE in the originalsetting can be obtained by taking limits, as the noise converges to and theterminal condition converges to the true, discontinuous terminal condition, ofvalue functions for FBSDE (3.4) with additional mollifying noise and a smoothapproximation of the true terminal condition, namely the functions p v ǫ,n q inCorollary 3.3.(b) Uniqueness is obtained in [6] by a direct duality argument; see Section 2.2 thereinfor details. We note however that the novel estimate given in Lemma 3.6 yieldsthe uniqueness result too.ii) Corollary 3.3 presents the key approximation procedure used to construct v by intro-ducing a vanishing viscosity (and smoothing of the terminal condition). Also, notethat if one heuristically considers a solution to (3.4) of the form Y t “ v p t, P t , E t q with v being C , then an application of Itô’s formula yields the PDE (3.7) with ǫ “ and terminal condition replaced by φ , namely B t u ` µ p p, u qB e u ` L p u “ ru and u p τ , ¨q “ φ. (3.9) Proposition 3.2 tells us then also that v can be considered as the unique “entropysolution” to (3.9) and that it can be represented using the “random characteristics” p P, E, Y Z q [4, 31] .iii) By the results in [15] , Corollary 3.3 holds when the sequence p φ n q n ě converges toany function ˆ φ satisfying φ ´ ď ˆ φ ď φ ` . In the thesis [15] , for a function φ P K , anexplicit construction of sequences of functions p φ n, ´ q n ě and p φ n, ` q n ě , belonging to K , converging to φ ´ and φ ` , respectively, and with each function having the sameLipschitz constant as φ , is given. We now collect some properties of the function v defined above that will be useful in thesequel. Proposition 3.5.
Let Assumption 1 hold. Let v be the function defined in Proposition3.2 associated to φ P K . Then the following hold.i) Limits for v : lim e Ñ`8 v p t, p, e q “ e ´ r p τ ´ t q and lim e Ñ´8 v p t, p, e q “ (3.10) ii) L -integrability property: there exists a constant C ą depending only on L and τ such that sup p P R d ż ´8 v p t, p, e q d e ď e ´ r p τ ´ t q ˜ sup p P R d ż ´8 φ p p, e q d e ` C ¸ . (3.11) iii) Comparison property: Let ˜ φ P K such that ˜ φ p p, e q ě φ p p, e q for every p p, e q P R d ˆ R .Then ˜ v p t, p, e q ě v p t, p, e q for every p t, p, e q P r , τ q ˆ R d ˆ R , where ˜ v is defined inProposition 3.2 with terminal condition ˜ φ . Proof.
Parts i) and iii) follow directly from Proposition A.1 which is presented in theappendix.For part ii), we use the estimate in part (5) of Proposition A.1. The following holds e ă Λ ùñ v p t, p, e q ď e ´ r p τ ´ t q ˜ E r φ p P t,pτ , Λ qs ` ^ C p ` | p | a q ˆ Λ ´ eL p τ ´ t q ˙ ´ a +¸ , for a ě where C is a constant that depends only on L and τ ; it may change from lineto line in this proof. Assume Λ ă and set a “ , e “ p ` a ` | p | q Λ , in the previousinequality, then @ Λ ă , v p t, p, p ` a ` | p | q Λ q ď e ´ r p τ ´ t q ˆ E “ φ p P t,pτ , Λ q ‰ ` ^ C | Λ | ˙ . Integrating on Λ , we obtain ż ´8 v p t, p, e q d e ď e ´ r p τ ´ t q ˆ E „ż ´8 φ p P t,pτ , e q d e ` C ˙ . The proof is concluded by taking the supremum in p accordingly. l The following lemma is key to obtain uniqueness for the infinite period model.
Lemma 3.6.
Let Assumption 1 hold. For i P t , u , i φ P K , we denote by i v the functiondefined in Proposition 3.2 and associated to i φ . Set ∆ v : “ v ´ v , ∆ φ “ φ ´ φ .Then, for any p t , p q P r , τ q ˆ R d , it holds E „ż | ∆ v |p t, P t ,pt , e q d e ď e ´ r p τ ´ t q E „ż | ∆ φ |p P t ,pτ , e q d e , (3.12) for any t P r t , τ q , where P t ,p is solution to (2.2) started at t with P t “ p . Proof.
The proof is carried out in several steps. We first make use of Corollary 3.3and prove a form of (3.12) which is valid when the functions v and v are replaced byapproximating sequences and the integrals in (3.12) are over a compact set. Then, wetake limits to complete the proof.1.a For i P t , u , let i v ǫ,n be the C , function defined in Corollary 3.3 and associated toa Lipschitz smooth approximating sequence of i φ . Denote i v n “ lim ǫ Ó i v ǫ,n and let usintroduce r , τ s ˆ R d Q p t, p q ÞÑ ̺ ǫ,n p t, p q : “ ż p v ǫ,n ´ v ǫ,n qp t, p, e q η p e q d e P R , r , τ s ˆ R d Q p t, p q ÞÑ ̺ n p t, p q : “ ż p v n ´ v n qp t, p, e q η p e q d e P R and r , τ s ˆ R d Q p t, p q ÞÑ ̺ p t, p q : “ ż ∆ v p t, p, e q η p e q d e P R , where η is a smooth, bounded function with compact support whose form will be chosenlater. By a direct application of the dominated convergence theorem, we observe that lim ǫ Ó ̺ ǫ,n “ ̺ n and lim n Ñ8 ̺ n “ ̺ . Moreover, ̺ ǫ,n is a C , solution to B t u ` L p u ` ǫ B pp u “ ru ` h ǫ,n p t, p q ` g ǫ,n p t, p q and u p τ , ¨q “ φ n ´ φ n (3.13) INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 17 where r , τ s ˆ R d Q p t, p q ÞÑ h ǫ,n p t, p q : “ ż t M p p, v ǫ,n q ´ M p p, v ǫ,n up t, p, e q η p e q d e P R , (3.14) r , τ s ˆ R d Q p t, p q ÞÑ g ǫ,n p t, p q “ ´ ǫ ż t v ǫ,n ´ v ǫ,n up t, p, e q η p e q d e P R (3.15)with M p p, y q “ ş y µ p p, υ q d υ , y P R . For later use, we remark that g ǫ,n Ñ as ǫ Ñ andthat lim n Ñ8 lim ǫ Ó h ǫ,n p t, p q “ h η p t, p q : “ ż t M p¨ , v q ´ M p¨ , v up t, p, e q η p e q d e. Let p P ǫt q t Pr t ,T s be the solution to P ǫt “ p ` ż tt b p P ǫs q d s ` ż tt σ p P ǫs q d W s ` ǫ p W s ´ W t q , , where W is a Brownian motion independent from W , and observe that, by classicalarguments, lim ǫ Ó E ” sup t Pr t ,τ s | P ǫt ´ P t ,pt | ı “ . Since p ̺ ǫ,n q ǫ ą are uniformly Lipschitz,we straightforwardly deduce that lim ǫ Ó E ” | ̺ ǫ,n p t, P ǫt q ´ ̺ n p t, P t ,pt q| ı “ . Now, we apply Itô’s Formula to p e ´ rt ̺ ǫ,n p t, P ǫt qq ď t ă τ . Using the PDE (3.13), we get E “ e ´ rt ̺ ǫ,n p t, P ǫt q ‰ “ e ´ rt ̺ ǫ,n p t , p q ` E „ż tt e ´ rs p h ǫ,n ` g ǫ,n qp s, P ǫs q d s . Taking the limit in ǫ first and then n (recall uniform linear growth in p ), we obtain E ” e ´ rt ̺ p t, P t ,pt q ı “ e ´ rt ̺ p t , p q ` E „ż tt e ´ rs h η p s, P t ,ps q d s To conclude this step, we observe that using (3.6), it follows from the dominated conver-gence theorem lim t Ñ τ ż v p t, p, e q η p e q d e “ ż φ p p, e q η p e q d e. Combining the above observation with the dominated convergence theorem again, we get lim t Ñ τ E r ̺ p t, P t qs “ E „ż ∆ φ p P τ , e q η p e q d e . We thus have E „ż ∆ v p t, P t , e q η p e q d e “ E „ e ´ r p τ ´ t q ż ∆ φ p P τ , e q η p e q d e ` ż τt e ´ r p s ´ t q h η p s, P s q d s (3.16)1.b Note that the previous reasoning can be applied to ˆ φ “ φ ^ φ P K and ˇ φ “ φ _ φ P K and the terminal condition Ă ∆ φ : “ ˇ φ ´ ˆ φ : “ | φ ´ φ | . Denoting ˆ v and ˇ v theassociated functions, we have from the comparison result given in Proposition 3.5 (iii),that ˆ v ď v ^ v and ˇ v ě v _ v and then ˇ v ´ ˆ v ě | ∆ v | . Combined with (3.16), thisleads to E „ż | ∆ v p t, P t , e q| η p e q d e ď E „ e ´ r p τ ´ t q ż | ∆ φ |p P τ , e q η p e q d e ` ż τt | h η |p s, P s q d s . (3.17)
2. We recall that for φ P K , for all p t, p q P r , τ s ˆ R d , we have lim e Ñ`8 v p t, p, e q “ e ´ r p τ ´ t q and lim e Ñ´8 v p t, p, e q “ (3.18)see Proposition 3.5 (i).For this step, we let η be a smooth approximation of r´ R,R s for R ą with first andsecond derivatives bounded (uniformly in R ), equal to on r´ R, R s and with support in r´p R ` q , p R ` qs (in other words a smooth truncation function).Observe that, for all p t, p q P r , τ s ˆ R d , lim R Ñ`8 r R,R ` s p e q| M p p, v qp t, p, e q ´ M p p, v qp t, p, e q| “| lim e Ñ`8 M p p, v qp t, p, e q ´ lim e Ñ`8 M p p, v qp t, p, e q| “ , where we used (3.18). Similarly, lim R Ñ`8 r´ R ´ , ´ R s p e q| M p p, v qp t, p, e q ´ M p p, v qp t, p, e q| “ . Observing that | M p p, v qp t, p, e q| ď L p `| p |q , we deduce from the dominated convergencetheorem and the previous observations that lim R Ñ`8 E „ż τt | h η |p s, P t ,ps q d s “ . (3.19)Moreover, the monotone convergence theorem implies that lim R Ñ`8 E „ż | ∆ v p t, P t ,pt , e q| η p e q d e “ E „ż | ∆ v p t, P t ,pt , e q| d e , and lim R Ñ`8 E „ż | ∆ φ |p P t ,pτ , e q η p e q d e “ E „ż | ∆ φ |p P t ,pτ , e q d e . The proof is concluded by combining the above equalities with (3.19) and (3.17). l Lemma 3.7.
Consider φ P K and assume that Assumptions 1, 2 and 3 hold. Then,the corresponding function v , defined in Proposition 3.2, satisfies, for all p t, p, p , e q Pr , τ q ˆ R d ˆ R d ˆ R , | v p t, p, e q ´ v p t, p , e q| ď ˆ e ´ β p τ ´ t q L φ ` Ll ˙ | p ´ p | , (3.20) where L φ is such that | φ p p, e q ´ φ p p , e q| ď L φ | p ´ p | . Proof.
We split the proof into two steps. In the first step, we use the PDE (3.7) to prove(3.24) in a setting in which all coefficient functions are smooth with bounded derivativesof all orders, the terminal condition φ is smooth and the FBSDE in Proposition 3.2has additional mollifying noise. Then, we use a standard mollification argument andCorollary 3.3 to conclude.1. Using a mollification argument, the same as that used in Section 2 of [6], we start byassuming that the coefficient functions b , σ and µ are smooth with bounded derivativesof all orders. We also assume that the terminal condition φ is Lipschitz smooth. Inthis setting, following Section 2 of [6], given ǫ ą , we can associate with the FBSDEa function v ǫ which is a smooth solution to the PDE in (3.7) with terminal condition v ǫ p τ , ¨ , ¨q “ φ p¨ , ¨q . The function v ǫ has bounded and continuous derivatives of any orderon r , τ s ˆ R d ˆ R . INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 19
Without loss of generality, we can consider only the first component of the gradientof v ǫ : w ǫ : “ B p v ǫ . The proof will be similar for all other components. Using the PDE(3.7), we have that B t w ǫ ` ˜ b p p q ¨ B p w ǫ ` T r rt a p p q ` ǫ I d uB pp w ǫ s ` ǫ B ee w ǫ ` b p p q w ǫ ` µ p p, v ǫ qB e w ǫ ` rB p µ p p, v ǫ q ` B y µ p p, v ǫ q w ǫ sB e v ǫ “ rw ǫ (3.21)with, ˜ b p p q is a vector the following components: for ď i ď d , $&% ˜ b i p p q : “ b i p p q ` σ ¨ p p q J σ i ¨ p p q , if i “ , ˜ b i p p q : “ b i p p q ` σ ¨ p p q J σ i ¨ p p q , otherwise,and, for ď i ď d , we are denoting b i p p q “ B p i b i p p q and σ i ¨ p p q “ B p i σ i ¨ p p q ,where σ i ¨ p p q denotes row i of the matrix σ p p q . Now set, for t P r t , τ s , p U ǫt , V ǫt q “p w ǫ p t, ˜ P ǫt , ˜ E ǫt q , B p w ǫ p t, ˜ P ǫt , ˜ E ǫt qq with p ˜ P ǫ , ˜ E ǫ q strong solution to d ˜ P ǫt “ ˜ b p ˜ P ǫt q d t ` σ p ˜ P ǫt q d W t ` ǫ d W t , ˜ P ǫt “ p , (3.22) d ˜ E ǫt “ µ p ˜ P ǫt , v ǫ p t, ˜ P ǫt , ˜ E ǫt qq d t ` ǫ d B t , ˜ E ǫt “ e , (3.23)where p W , B q is two dimensional Brownian Motion independent from W . Applying Itô’sformula, we compute d U ǫt “ t r ´ b p ˜ P ǫt qu U ǫt d t ´ B p µ ´ ˜ P ǫt , v ǫ p t, ˜ P ǫt , ˜ E ǫt q ¯ B e v ǫ p t, ˜ P ǫt , ˜ E ǫt q d t ´ B y µ ´ ˜ P ǫt , v ǫ p t, ˜ P ǫt , ˜ E ǫt q ¯ B e v ǫ p t, ˜ P ǫt , ˜ E ǫt q U ǫt d t ` V ǫt σ p ˜ P ǫt q d W t ` ǫV ǫt d W t ` ǫ B e w p t, ˜ P ǫt , ˜ E ǫt q d B t . We now introduce the weights, I t “ exp ´ ż tt ` b p ˜ P ǫs q ´ r ˘ d s ¯ , J t “ exp ´ ż tt B y µ ` ˜ P ǫt , v ǫ p t, ˜ P ǫt , ˜ E ǫt q ˘ B e v p s, ˜ P ǫt , ˜ E ǫt q d s ¯ , and E t “ I t J t for t P r t , τ s . Setting p ¯ U ǫt , ¯ V ǫt q “ E t p U ǫt , V ǫt q for t P r t , τ s , we compute d ¯ U ǫt “ ´B p µ ´ ˜ P ǫt , v ǫ p t, ˜ P ǫt , ˜ E ǫt q ¯ B e v ǫ p t, ˜ P ǫt , ˜ E ǫt q E t d t ` ¯ V ǫt σ p ˜ P ǫt q d W t ` ǫ ¯ V ǫt d W t ` ǫ B e w p t, ˜ P ǫt , ˜ E ǫt q E t d B t . This leads to E “ ¯ U ǫt ‰ “ E “ ¯ U ǫτ ‰ ` E „ż τt B p µ ´ ˜ P ǫt , v ǫ p t, ˜ P ǫt , ˜ E ǫt q ¯ B e v ǫ p t, ˜ P ǫt , ˜ E ǫt q E t d t , “ E “ ¯ U ǫτ ‰ ` E »–ż τt I t B p µ ´ ˜ P ǫt , v ǫ p t, ˜ P ǫt , ˜ E ǫt q ¯ B y µ ´ ˜ P ǫt , v ǫ p t, ˜ P ǫt , ˜ E ǫt q ¯ J t d t fifl . By Assumptions 1 and 3, we have | B p µ B y µ | ď Lℓ and I t ď . Therefore |B p v ǫ p t , p, e q| ď E r E τ s |B p φ | ` Lℓ p ´ E r J τ sq . Using the fact that r ´ | b | ě β and J ¨ ď , we get |B p v ǫ | ď e ´ β p τ ´ t q |B p φ | ` Lℓ , which concludes the proof for this step.2. Using the mollification argument, we can approximate the true coefficient functions b , σ and µ by smooth functions with bounded derivatives of all orders and take limitsto show that (3.24) will hold for v ǫ without any additional smoothness or boundednessconditions on the coefficient functions.Finally, for a general terminal condition φ P K , we can approximate φ by a sequence ofLipschitz smooth functions in K and apply Corollary 3.3 to complete the proof. l Lemma 3.8.
Consider φ P K and assume that Assumptions 1, 2 and 4(i) hold. Then,the corresponding function v , defined in Proposition 3.2, satisfies, for all p t, p, p , e q Pr , τ q ˆ R d ˆ R d ˆ R , | v p t, p, e q ´ v p t, p , e q| ď C β ˆ } φ } ? τ ´ t ` l ˙ | p ´ p | . (3.24) Proof.
We proceed as in the proof of Lemma 3.7: We consider first the noisy version ofthe system (for a small parameter ǫ ą ) with smooth terminal condition and coefficientfunctions, namely P t ,pt “ p ` ż tt b p P t ,ps q d s ` d ÿ ℓ “ ż tt σ ¨ ℓ p P t ,ps q d W ℓs (3.25) E t ,p,e,ǫt “ e ` ż tt µ p P t ,ps , E t ,p,e,ǫs q d s ` ǫB s (3.26) Y t ,p,e,ǫt “ φ p P t ,pτ , E t ,p,e,ǫτ q ´ ż τt Z t ,p,e,ǫs d W s (3.27)where B is a Brownian Motion independent from W and, in this smooth setting, we havethat for t ď t ď τ , Y ǫt “ v ǫ p t, P t ,pt , E t ,p,e,ǫt q , Z ǫt “ B p v ǫ p t, P t ,pt , E t ,p,e,ǫt q σ p P t ,pt q , where v ǫ is a classical solution to B t v ǫ ` B p v ǫ b p p q ` T r r a p p qB pp v ǫ s ` ǫ B ee v ǫ ` µ p p, v ǫ qB e v ǫ “ rv ǫ and v ǫ p τ , ¨q “ φ p¨q . (3.28)Note that we do not need to add noise on the P -component as σ is assumed to beuniformly elliptic here, see Assumption 4(i). In particular, we deduce from (3.28), thatfor ď i ď d , B p i v ǫ satisfies the following equation: ru “ B t u ` B p ub p p q ` T r r a p p qB pp u s ` ǫ B ee u ` µ p p, v ǫ qB e u (3.29) ` B p v ǫ B p i b p p q ` T r rB p i a p p qB pp v ǫ s ` B p i µ p p, v ǫ qB e u ` B y µ p p, v ǫ qB e u . (3.30)We also introduce the tangent process associated to P t ,p valued in the set of d ˆ d matrices: B p P t ,pt “ I d ` ż t B p b p P t ,ps qB p P t ,ps d s ` d ÿ ℓ “ ż t B p σ ¨ ℓ p P t ,ps qB p P t ,ps d W ℓs , (3.31) INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 21 where for ď k ď d , we have pB p P t ,pt q .k “ B p k P t ,p and I d is the identity matrix. Thefollowing estimate is well known, see e.g. [32]: for κ ě , E « sup t Pr t ,τ s |B p P t ,pt | κ ff ď C κ . (3.32) C κ depends only on T and L the Lipschitz constant of b , σ and the extra parameter κ .Below, we drop the dependence in p t , p, e q for the reader’s convenience. The remainderof the proof is split into two steps: we first obtain a Bismut-Elworthy-Li type formula forthe gradient of v ǫ with respect to p and then use it to obtain the desired upper bound.1.a Applying Ito’s formula to U ǫ “ B p v ǫ p t, P t , E ǫt q , one gets d U ǫt “ ´ U ǫt ´ B p b p P t q ` p r ` B y µ p P t , Y ǫt qB e v ǫ p t, P t , E ǫt qq I d ¯ d t ´ A ǫt d t ` B p µ p P t , Y ǫt qB e v ǫ p t, P t , E ǫt q d t ` ǫ B pe v ǫ p t, P t , E ǫt q d B t ` d M ǫt where A ǫt “ p T r rB p ℓ a p P t q V ǫt sq ď ℓ ď d , d M ǫt “ pp V ǫt q ℓ ¨ σ p P t q d W t q ď ℓ ď d , and V ǫt “ B pp v ǫ p t, P t , E ǫt q . Introducing, E t : “ exp ´ ż tt B y µ ` P t , v ǫ p t, P t , E ǫt q ˘ B e v ǫ p s, P t , E ǫt q d s ¯ , (3.33)we then apply Ito’s formula to p ¯ U ǫt : “ e ´ r p t ´ t q E t U ǫt B p P t q t ď t ď τ , to get d ¯ U ǫt “ e ´ r p t ´ t q B p µ p P t , Y ǫt qB p P t E t B e v ǫ p t, P t , E ǫt q d t ` d M ǫt , (3.34)where M ǫ is a square integrable martingale. In particular, we observe that, for t ď t ď τ , B p v ǫ p , p, e q “ E ” e ´ r p t ´ t q E t U ǫt B p P t ` Θ t ı (3.35)with Θ t “ ż tt e ´ rs B p µ p P s , Y ǫs qB p P s E s B e v ǫ p s, P s , E ǫs q d s . (3.36)Let us introduce H t τ : “ τ ´ t ˆż τt ´ e ´ r p t ´ t q E t σ p P t q ´ B p P t ¯ J d W t ˙ J , (3.37)which is a random d -dimensional row vector. Integrating (3.35), from t to τ and usingIto Isometry, we get B p v ǫ p t , p, e q “ E „ż τt U ǫt σ p P t q d W t H t τ ` τ ´ t ż τt Θ t d t . Observing that Z ǫ “ U ǫ σ p P q and using (3.27), we deduce from the previous equality B p v ǫ p t , p, e q “ E „ φ p P τ , E ǫτ q H t τ ` τ ´ t ż τt Θ t d t . (3.38) }B p v ǫ } . We first observe that, ď e ´ r p t ´ t q E t ď for all t ď t ď τ , (3.39)as B y µ p¨q ď , recall Assumption 1, and B e v ǫ p¨q ě .Combing (3.32), with Assumption 4(i) and (3.39), we obtain classically E “ | H t τ | κ ‰ κ ď C κ ? τ ´ t , for all κ ě . (3.40)We also compute, using (3.32) and Assumption 1, E r| Θ t |s ď Cl ż τt E s B y µ ` P t , v ǫ p t, P t , E ǫt q ˘ B e v ǫ p s, P s , E ǫs q d s , (3.41)Now, observing that E s B y µ ` P t , v ǫ p t, P t , E ǫt q ˘ B e v ǫ p s, P s , E ǫs q “ E s , we obtain E r| Θ t |s ď Cl . (3.42)Combining (3.42) and (3.40) with (3.38), we conclude for this step }B p v ǫ p t , ¨q} ď C ´ l ` } φ } ? τ ´ t ¯ . (3.43)The proof is concluded by invoking the local uniform convergence of v ǫ to v when remov-ing the noise and smoothness of the coefficient functions. l Remark 3.9.
Let us mention that (3.38) is a Bismut-Elworthy-Li formula for this par-tially degenerate setting and it differs from the representations obtained in Proposition5.1 in [6] . It is well expected since we impose non-degeneracy of the P -component. More-over, the control obtained in Lemma 3.8 indicates that existence could be obtained withless regularity than the Lipchitz assumption on φ in the p -variable imposed in Proposition3.2. Parametrised one-period FBSDE.
In the sequel, we need to consider termi-nal conditions parametrised by an extra variable, representing the initial value of theemissions process.
Definition 3.10.
Let p K be the class of bounded measurable functions R d ˆ R ˆ R Q p p, e, e q Ñ Φ p p, e, e q P R such that, ‚ for each e P R , Φ p¨ , ¨ , e q P K with a Lipschitz constant L Φ that is independent of e i.e. there is L Φ ą such that | Φ p p , e, e q ´ Φ p p , e, e q| ď L Φ | p ´ p | , (3.44) for every p , p P R d and every e, e P R . ‚ for each e P R the function R d ˆ R Q p p, e q ÞÑ Φ p p, e ` e , e q P R is in K . Note that there is a natural injection from K into p K : Indeed, it is clear that for everyfunction φ P K , the function p φ P p K given by R d ˆ R ˆ R Q p p, e, e q ÞÑ p φ p p, e, e q “ φ p p, e q P R , (3.45)belongs to p K . INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 23
Remark 3.11.
In the multi period model, the cap on emissions in a given period maydepend on the level of emissions made up until the start of that period. In a typical setup,we may have a regulator releasing a pre-determined quantity of allowances at the start ofevery period. The cap on emissions in a period is then determined by the total numberof allowances available in the market at the time of compliance, taking into account allemissions made and allowances already surrendered in previous periods. The typical formof a terminal condition Φ in p K is Φ p p, e, e q “ " h p p, e q , if e ă Λ p e q , , otherwise. (3.46) Here, e can be considered to be a parameter representing the total recorded emissions atthe start of the compliance period. The function Λ maps this quantity to the total numberof allowances available in the market at the start of the period. This function accounts forthe regulator’s releases of allowances, see e.g. Example 1. So, Λ p e q represents the numberof allowances available to use for compliance and is therefore the cap on emissions. In (3.46) , p and e will represent, respectively, the value of the noise process and of thecumulative emissions process, both at the end of the compliance period. The penalty hereis, as stated before, set equal to , and it is incurred when total emissions exceed the cap.The function h represents the value that the allowance price will take at the start of thefollowing compliance period if over-emission has not occurred. Typically, h is determinedby the dynamics of the following period. In this paper, we want to consider terminalconditions of the form (3.46) with Λ P Θ and h satisfying appropriate conditions inheritedfrom the properties of the value function for the FBSDE for a single period model andallow us to prove, in a recursive manner, that the multi period pricing problem is wellposed. This is what motivates the definition of p K . The following result is the version of Proposition 3.2 for terminal conditions dependingon a parameter, e . It follows directly from the proof of Proposition 3.2. Corollary 3.12.
Let Assumption 1 hold. For Φ P ˆ K , there exists a unique function v Φ defined by r , τ q ˆ R d ˆ R ˆ R Q p t , p, e, e q Ñ v Φ p t , p, e, e q “ Y t ,p,e, e t P R , where p P t ,p , E t ,p,e, e , Y t ,p,e, e q is a solution to (3.4) - (3.5) with terminal condition Φ p¨ , ¨ e q .It is continuous and satisfies(1) For any t P r , τ q , the function v Φ p t, ¨ , ¨ , ¨q is {p l p τ ´ t qq -Lipschitz continuouswith respect to e .(2) For any t P r , τ q , the function v Φ p t, ¨ , ¨ , ¨q is C -Lipschitz continuous with respectto p , where C ą depends on L , L Φ and τ only.(3) Given p p, e q P R d ˆ R and e P R , for any family p p t , e t q ď t ă τ converging to p p, e q as t Ò τ , we have Φ ´ p p, e, e q ď lim inf t Ñ τ v Φ p t, p t , e t , e q ď lim sup t Ñ τ v Φ p t, p t , e t , e q ď Φ ` p p, e, e q (3.47) uniformly for p in compact subset of R d .(4) For any parameters p t , p, e, e q P r , τ q ˆ R d ˆ R ˆ R , v Φ p t, P t ,pt , E t ,p,e, e t , e q “ Y t ,p,e, e t for t ď t ă τ where p P t ,p , E t ,p,e, e , Y t ,p,e, e q . The following lemma is key to show the existence of a solution to the multi-period model.
Lemma 3.13.
Let Φ P ˆ K and v Φ be given by Corollary 3.12. Then, for every t Pr , τ q , p P R d , the function e ÞÑ v Φ p t, p, e, e q , is monotone increasing and satisfies lim e Ñ´8 v Φ p t, p, e, e q “ . (3.48) Proof.
We split the proof into two steps. We first use a change of variables to prove themonotonicity property and then use bounds for value functions for singular FBSDEs toprove the limit property (3.48).1. For a given t P r , τ q , p P R d and e P R , we have v Φ p t , p, e , e q “ Y t , where Y ispart of the solution p P t , E t , Y t , Z t q t Pr ,τ s of (3.4)-(3.5) started at time t and with terminalcondition Φ p¨ , ¨ , e q . Explicitly, the relaxed terminal condition (3.5) in this setting is P „ Φ ´ p P τ , E τ , e q ď lim t Ò τ Y t ď Φ ` p P τ , E τ , e q “ . (3.49)Now, set ¯ E t “ E t ´ e for every t P r , τ q . Then, p P t , ¯ E t , Y t , Z t q t Pr t ,τ s satisfies the samedynamics (3.4) with p P t , ¯ E t q “ p p, q . Moreover, (3.49) leads to P „ ¯Φ ´ p P τ , ¯ E τ , e q ď lim t Ò τ Y t ď ¯Φ ` p P τ , ¯ E τ , e q “ , (3.50)where ¯Φ p p, ¯ e, e q : “ Φ p p, ¯ e ` e , e q , (3.51)for every p P R d and ¯ e, e P R . Notice that ¯Φ belongs to ˆ K . Consequently, for the valuefunction, v ¯Φ associated to the terminal condition ¯Φ , we must have, by uniqueness, that Y “ v Φ p t , p, e , e q“ v ¯Φ p t , p, , e q . (3.52)For each fixed p P R d and ¯ e P R , the real function e ÞÑ ¯Φ p p, ¯ e, e q is monotone increasingbecause Φ P ˆ K . Therefore, viewing e as a parameter, we can apply the comparisonproperty, Proposition 3.5 (iii) along with (3.52) to show that the first part of the lemmaholds; v Φ has the stated monotonicity property.2. For the second part, we use the inequality (A.2) which is stated in the appendixas part of Proposition A.1. For any e P R , the function p p, e q ÞÑ ¯Φ p p, e, e q belongs to K and so, by (A.2) we have, for any Λ ą , v ¯Φ p t , p, e q ď e ´ r p τ ´ t q „ E “ ¯Φ p P τ , Λ , e q ‰ ` C ˆ L p τ ´ t q Λ ˙ , (3.53)where C depends on L , L Φ , τ and p only. Now, noting that ¯Φ p P τ , Λ , e q converges to as e tends to ´8 , we take limits in (3.53), and use the bounded convergence theorem togive lim e Ñ´8 v ¯Φ p t , p, , e q ď Ce ´ r p τ ´ t q ˆ L p τ ´ t q Λ ˙ . (3.54)Since this holds for any Λ ą , we conclude that lim e Ó´8 v ¯Φ p t , p, e q “ . Combiningthis observation with (3.52) completes the proof. l INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 25
The following corollary is useful in the next section as it allows us to link twotrading periods.
Corollary 3.14.
Using the notation of Lemma 3.13, for Λ P Θ and a given t P r , τ q ,consider the function ψ defined by ψ p p, e, e q “ v Φ p t , p, e, e q if e ă Λ p e q , otherwise. (3.55) Then ψ P ˆ K . Proof.
For the first part of the definition of ˆ K , fix a value e P R . By Corollary 3.12, foreach fixed e P R , the function p ÞÑ v Φ p t , p, e, e q is C -Lipschitz continuous for a constant C depending only on L , τ and L Φ , the Lipschitz constant of Φ . Note that this C isindependent of e and the parameter, e . Using Lemma 3.13 and the definition of ψ , wealso see that p p, e q ÞÑ ψ p p, e, e q is monotone increasing in e , has limit as e tends to ´8 and limit as e tends to `8 .Next, for the second part of the definition of ˆ K , fix a value of e P R . For any p P R d and e P R , we will have ψ p p, e ` e , e q “ v Φ p t , p, e ` e , e ` e q if e ă Γ p e q , otherwise, (3.56)where, as in Definition 2.1, Γ p e q “ Λ p e q ´ e and Γ is a monotone decreasing function suchthat lim e Ñ`8 Γ p e q “ ´8 . Now, using the same arguments as above, we see that, foreach e P R , the function p p, e q ÞÑ ψ p p, e ` e , e q is in K , recall Definition 3.1. l The multi period model.
The aim of this section is to prove Theorem 2.3 andsome key properties of the multi-period model that will be used for the infinite periodmodel.The proof of Theorem 2.3 is based on the existence of a pricing function, which isgiven essentially by the following proposition.
Proposition 3.15.
Let θ P p K with Lipschitz constant L θ , and q be a positive integerdenoting the number of periods. Let Λ , Λ ,..., Λ q ´ be cap functions in Θ and ă T ă T ă ... ă T q “ T be a sequence of times.There exists a bounded, measurable function v q : r , T q ˆ R d ˆ R ˆ R Ñ R such that(1) For each e P R , v q p¨ , ¨ , ¨ , e q is continuous on r T k ´ , T k q ˆ R d ˆ R , ď k ď q .(2) For any k such that ď k ď q , for any t P r T k ´ , T k q , the function v q p t, ¨ , ¨ , ¨q is {p ℓ p T k ´ t qq -Lipschitz continuous with respect to e .(3) For any t P r T k ´ , T k q , the function v q p t, ¨ , ¨ , ¨q is C qk -Lipschitz continuous withrespect to p , where, for each k , C qk is a constant depending on L , L θ , T and q only.(4) for ď k ă q , define, Φ q,k p p, e, e q “ " v q p T k , p, e, e q , if e ă Λ k p e q , otherwise, (3.57) and set Φ q,q p p, e, e q “ θ p p, e, e q , (3.58) for every p p, e q P R d ˆ R and e P R . Then, for any integer k with ď k ď q andany p p, e q P R d ˆ R , we will have Φ q,k ´ p p, e, e q ď lim inf t Ò T k v q p t, p t , e t , e q ď lim sup t Ò T k v q p t, p t , e t , e q ď Φ q,k ` p p, e, e q , (3.59) for any family p p t , e t q ď t ă T k converging to p p, e q as t Ò T k .(5) For ď k ď q , and any parameters p t , p, e, e q P r T k ´ , T k q ˆ R d ˆ R ˆ R , we have v q p t, P t ,pt , E t ,p,e, e t , e q “ Y t ,p,e, e t for t ď t ă T k where p P t ,p , E t ,p,e, e , Y t ,p,e, e q issolution to (3.4) - (3.5) over the time interval r t , T k s with p P t ,pt , E t ,p,e, e t q “ p p, e q and terminal condition Φ q,k p¨ , ¨ , e q . Remark 3.16. (1) In general, each of the constants C qk in part (3) of Proposition3.15 depends on the time period k and the number of periods q . For each q , wewill generally have C q ě C q ě .... ě C qq “ C , where C is the Lipschitz constantfor a one period model on r T q ´ , T s with terminal condition θ , as described inpart (2) of Corollary 3.12. Controlling the constants C qk is key to studying theinfinite period model, as we shall see in the next section.(2) Using standard arguments for FBSDEs, one can show that item (5) of Proposition3.15 holds when p and e are replaced by a pair of square integrable F t randomvariables of the appropriate dimension. The same holds true for part (5) inCorollary 3.17 below. Proof.
Throughout this proof, we fix the number of periods, q and denote the resultingfunction by v instead of v q . We use induction to show that v has the stated propertieson each period r T k ´ , T k s , for ď k ď q .1. On the last interval, the terminal condition is simply given by θ , which is in p K .Using Corollary 3.12, one obtains a function u defined on the time interval r , T q forthe terminal condition θ . We simply set v p t, p, e, e q “ u p t, p, e, e q for every t P r T q ´ , T q , p P R d and e, e P R , to define v on the last period. All of the properties of v p t, ¨ , ¨ , ¨q for t P r T q ´ , T q follow from Corollary 3.12. Finally, we apply Corollary 3.14 to see that Φ q,q ´ P p K .2. Induction step. Assume that, for some k with ă k ă q , we have that Φ q,k P ˆ K andthat v p t, ¨ , ¨ , ¨q has been defined for t P r T k , T q and satisfies the properties in the statementof the proposition. Using Corollary 3.12 with terminal condition Φ q,k and terminal time T k , one obtains a function u q,k : r , T k q ˆ R d ˆ R ˆ R satisfying the properties statedthere. We set v p t, p, e, e q “ u q,k p t, p, e, e q for every t P r T k ´ , T k q , p P R d and e, e P R ,to define v on the time period r T k ´ , T k q . The properties of v p t, ¨ , ¨ , ¨q for t P r T k ´ , T k q follow directly from the corresponding properties of u q,k in Corollary 3.12. Lastly, weapply Corollary 3.14 to see that Φ q,k ´ P p K . l Proof of Theorem 2.3.
We now turn to the proof of the main result for the multi-period setting and which has been announced in Section 2.Fix a value of q , a starting point p p , e q and set v “ v q , the function from Proposition3.15.1. Existence part: Firstly, by classical SDE results [30] we have existence of a process, P , on r , T s as defined in (2.4). To build E and Y , we use then a forward induction.On r T , T s , the first interval, consider Y ,p ,e ,e given by Proposition 3.15 above. Inparticular E T is well defined. INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 27 On r T , T s , we can define, for t ă T , E ξt “ E T ` ż tT µ p P s , v p s, P s , E ξs , ξ qq d s for any (bounded) ξ that is F T measurable. Indeed, the random drift coefficient e ÞÑ µ p P s , v p s, P s , e, ξ qq is Lipschitz continuous on r T , t s , t ă T and adapted. E isthus well defined on r T , T q and has a limit at T by the Cauchy criterion. Let now, Y ξt “ v p t, P t , E ξt , ξ q and observe that e ´ rt Y ξt is a martingale. Indeed, consider ξ n a simplerandom variable with ξ n Ñ ξ . Then e ´ rt Y ξ n t is a martingale from the properties of v q inProposition 3.15. To obtain the result, one can then pass to the limit and apply the dom-inated convergence theorem (the processes Y ξ n are uniformly bounded as v is bounded).Setting, in particular, ξ “ E T here, we obtain a process E and the corresponding process Y both on r T , T q . One can also verify that the condition (2.12) which links the twoperiods is satisfied, thanks to the properties of the function v q in Proposition 3.15.2. Uniqueness. First, notice that uniqueness of the P process follows by classical re-sults for SDEs driven by Lipschitz continuous coefficients. Now, for t P r , T q , thetuple of processes p P, E, Y, Z q has been constructed so that it is the solution of FB-SDE (3.4)-(3.5) with starting point p p , e q and terminal condition Φ q, p¨ , ¨ , e q . Usingthe same computation as that used to prove uniqueness of solutions in Theorem 2.2in [6] (see also Theorem A.2 in the appendix), we will find that p E t , Y t , Z t q t Pr ,T s isunique in S , pr , T sq ˆ S , pr , T qq ˆ H ,d pr , T sq . Indeed, this shows that existenceand uniqueness in Theorem 2.3 holds when q “ . Similarly, for any t , T P r , T s with t ă T , using Corollary 3.12, we obtain existence and uniqueness for processes p P t , E t , Y t , Z t q t Pr t ,T s satisfying (3.4)-(3.5) with a given starting point p P t , E t q and aterminal condition Φ P ˆ K and the relevant relaxed terminal condition for Y T . Usingstandard FBSDE arguments and the value function v Φ in Corollary 3.12, we see that thesame result holds when p , e and e are replaced by F t measurable and square integrablerandom variables; in that case we will still have Y t “ v Φ p t, P t , E t q for t P r t , T q .Now, proceeding to the time period r T , T s , we see that for t P r T , T s , the tuple p P t , E t , Y t , Z t q has been constructed to be the solution to (3.4)-(3.5) with the F T mea-surable starting point p P T , E T q and terminal condition Φ q, p¨ , ¨ , E T q . By the aboveargument, existence and uniqueness is seen to hold on the time period r T , T s . Theproof is completed by an easy induction.3.4. Properties of the multi-period model with constant cap functions.
Wecollect here some key properties of the multi-period model that will be used in the nextsection. We restrict to a non-parametrised version of the multi-period model, assumingthat the cap function are constant. More precisely, we will work throughout this sectionwith Assumption 2.The following results are deduced directly from Theorem 2.3 and Proposition 3.15 inthis restricted setting. They are stated here for the reader’s convenience.
Corollary 3.17.
Let Assumptions 1 and 2 hold. Let θ P K with Lipschitz constant L θ ,and q be a positive integer denoting the number of periods. We consider a q period multiperiod model with caps Λ , Λ ,..., Λ q ´ and terminal condition θ in the final period, over r , T q s and time intervals r T , T q , r T , T q ,..., r T q ´ , T q q .There exists a bounded measurable function v q : r , T q ˆ R d ˆ R Ñ R such that(1) For each ď k ď q , the function p t, p, e q ÞÑ v q p t, p, e q is continuous on r T k ´ , T k q ˆ R d ˆ R , ď k ď q ; (2) For any t P r T k ´ , T k q , the function v q p t, ¨ , ¨q is {p l p T k ´ t qq -Lipschitz continuouswith respect to e .(3) For any t P r T k ´ , T k q , the function v q p t, ¨ , ¨q is C qk -Lipschitz continuous withrespect to p , where, for each k , C qk is a constant depending on L , L θ , T and q only.(4) for ď k ă q , p e, e q P R ˆ R , denoting Φ q,k p p, e q “ " v q p T k , p, e q , if e ă Λ k , otherwise. (3.60) and setting Φ q,q p p, e q “ θ p p, e q , (3.61) for every p p, e q P R d ˆ R , we have, given p p, e q P R d ˆ R and k such that ď k ď q ,for any family p p t , e t q ď t ă T k converging to p p, e q as t Ò T k , we have, Φ q,k ´ p p, e q ď lim inf t Ò T k v q p t, p t , e t q ď lim sup t Ò T k v q p t, p t , e t q ď Φ q,k ` p p, e q . (3.62) (5) For ď k ď q , any parameters p t , p, e q P r T k ´ , T k q ˆ R d ˆ R ˆ R , v q p t, P t ,pt , E t ,p,et q “ Y t ,p,et for t ď t ă T k where p P t ,p , E t ,p,e , Y t ,p,e q is so-lution to (3.4) - (3.5) over the time interval r t , T k s with p P t ,pt , E t ,p,et q “ p p, e q and terminal condition Φ q,k .Finally, assume that we are given two deterministic starting points P P R d , E P R .Then there exists a unique càdlàg process p Y t q ď t ď T P S , pr , T sq , a continuous process p E t q ď t ď T P S , c pr , T sq and a continuous process p P t q ď t ď T P S , c pr , T sq satisfyingthe dynamics (2.4) on each period r T k ´ , T k q , ď k ď q . For each k , the process Y iscontinuous on r T k ´ , T k q ; it can have a jump at T k . There it satisfies, for every ď k ď q ,almost surely, lim t Ò T k Y t “ Y T k , if E T k ă Λ k , lim t Ò T k Y t “ , if E T k ą Λ k ,Y T k ď lim t Ò T k Y t ď , if E T k “ Λ k , (3.63) with, as above, Y T q “ . From now on, unless stated otherwise, for θ in Proposition 3.15, we always set θ p p, e, e q “ e ě Λ q p e q , p p, e q P R d ˆ R , (3.64)and similarly for constant caps and the setting of Corollary 3.17: θ p p, e q “ e ě Λ q . (3.65)This is the standard setting of a multi period model as in Theorem 2.3. Indeed, for asingle period model for a carbon market, the standard terminal condition is, when thecap on emissions is Λ and penalty for over emission is π > 0, given by φ p e q “ π e ě Λ .Terminal conditions of this form are standard for singular FBSDEs modelling carbonmarkets in the literature; see [5, 7, 9, 10, 11, 14, 29] among others.The final period of a multi period model is no different to a single period model andso, taking into account the variable caps and the fact that we set the penalty equal to throughout this paper, (3.64) is the corresponding terminal condition to be used for thefinal period. INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 29
Lemma 3.18.
Let Assumptions 1 and 2 hold. For any positive integer q , denote by v q the function defined in Corollary 3.17 for θ given by (3.65) . Then, we have, for q ě v q p T k , p, e q “ v q ´ p T k ´ , p, e ´ λ q , (3.66) for every integer ď k ď q ´ and p p, e q P R d ˆ R . Proof.
1. Let x P R be fixed. For φ P K , we define ˜ φ p p, e q : “ φ p p, e ´ x q , for p p, e q P R d ˆ R . Note that ˜ φ P K . Let v (respectively ˜ v ) be the function given inProposition 3.2 and associated to φ (respectively ˜ φ ). The goal of this step is to showthat v p t, p, e ´ x q “ ˜ v p t, p, e q for p t, p, e q P r , τ q ˆ R d ˆ R . (3.67)For a t ă τ , let p P t ,pt q t ď t ď τ be the solution of (2.2) satisfying P t ,pt “ p . Denote by p P t ,pt , ˜ E t ,p,et , ˜ Y t ,p,et q the p P, E, Y q part of the unique solution of (3.4) given in Proposi-tion 3.2 over time interval r t , τ s with p P t ,pt , ˜ E t ,p,et q “ p p, e q and with terminal condition ˜ φ . Similarly, let p P t ,pt , E t ,p,et , Y t ,p,et q be the p P, E, Y q part of the solution of (3.4) withthe same initial condition, over the same time interval, and with terminal condition φ .For the first solution tuple, the following relaxed terminal condition is satisfied. P ” ˜ φ ´ p P t ,pτ , ˜ E t ,p,eτ q ď ˜ Y t ,p,eτ ´ ď ˜ φ ` p P t ,pτ , ˜ E t ,p,eτ q ı “ . (3.68)We observe that the previous condition rewrites P ” φ ´ p P t ,pτ , ˜ E t ,p,eτ ´ x q ď ˜ Y t ,p,eτ ď φ ` p P t ,pτ , ˜ E t ,p,eτ ´ x q ı “ . (3.69)by definition of ˜ φ . By the uniqueness result of Proposition 3.2, we observe then that p P t ,pt , ˜ E t ,p,et ´ x, ˜ Y t ,p,et q “ p P t ,pt , E t ,p,e ´ xt , Y t ,p,e ´ xt q t ď t ă τ , where p E t ,p,e ´ x , Y t ,p,e ´ x q are parts of the solution p P t ,p , E t ,p,e ´ x , Y t ,p,e ´ x , Z t ,p,e ´ x q ,of (3.4), over r t , τ s with initial condition p P t ,pt , E t ,p,e ´ xt q “ p p, e ´ x q and terminal con-dition φ , as they have the same dynamics and relaxed terminal condition. In particular,we obtain (3.67) for t “ t .2. Now, for any φ P K and T ą , we will denote here by u φ,T the value function v fromProposition 3.2 for the system (3.4) over r , T s and with terminal condition φ . For thisstep, we note that for any φ P K , T ą and ∆ T ą , we have u φ,T p t, p, e q “ u φ,T ` ∆ T p t ` ∆ T, p, e q , (3.70)for every t P r , T q and every p p, e q P R d ˆ R . Indeed, this property is easily seen to holdfor functions v ǫ,n satisfying the PDE system (3.7). Approximating φ by a sequence ofsmooth functions and using Corollary 3.3, we conclude that (3.70) holds.3. In this step, we prove (3.66) by induction on the period number, k .3a. Induction step. For a k such that ď k ď q ´ , assume that (3.66) holds. In thissetting, this leads to Φ q ´ ,k ´ p p, e ´ λ q “ Φ q,k p p, e q , (3.71)where the terminal condition functions Φ ¨ , ¨ are as in Corollary (3.17).By construction, recall the proof of Proposition 3.15, v q is given on r T k ´ , T k q by u Φ q,k ,T k over the same time interval. That is, v q p t, p, e q “ u Φ q,k ,T k p t, p, e q , (3.72) for every t P r T k ´ , T k q and p p, e q P R d ˆ R . Similarly, v q ´ on the time interval r T k ´ , T k ´ q matches u Φ q ´ ,k ´ ,T k ´ on the same time interval. Using step 2 followedby step 1, we compute v q p T k ´ , p, e q “ u Φ q,k ,T k p T k ´ , p, e q“ u Φ q,k ,T k ´ p T k ´ , p, e q“ u Φ q ´ ,k ´ ,T k ´ p T k ´ , p, e ´ λ q“ v q ´ p T k ´ , p, e ´ λ q , for every p p, e q P R d ˆ R , completing the induction step.3b. For the initialization step, we note that Φ q,q p p, e q : “ e ě Λ q “ e ´ λ ě Λ q ´ “ Φ q ´ ,q ´ p p, e ´ λ q . Using the same computation as in part 3a, this leads to (3.66) for k “ q ´ . l Remark 3.19.
Focusing on v q in the first period r , τ q , we observe that it can be computedby the following iteration:i) for n “ , set ω “ on r , τ q .ii) for n ě , ω n is obtained by solving the one-period FBSDE with terminal conditiongiven by Φ n : p p, e q “ ω n ´ p , p, e ´ λ q , if e ă λ, , otherwise, (3.73) using Proposition 3.2.Then, it follows from Lemma 3.18 above, that, for q ě , t P r , τ q and p p, e q P R d ˆ R ,we have v q p t, p, e q “ ω q p t, p, e q . It is then completely natural to ask if the functions ω q converges when q Ñ 8 . Afirst result in this direction is the following.
Proposition 3.20.
The sequence p ω q q q ě is bounded and increasing, in the sense that,for q ě , ω q p t, p, e q ě ω q ´ p t, p, e q , @p t, p, e q P r , τ q ˆ R d ˆ R . Their limit w is well defined. Moreover, if r ą , it satisfies, for all t ă τ , sup p P R d ż ´8 w p t, p, e q d e ă `8 . (3.74) Proof.
1. The monotonicity of the sequence is proved by induction. Assume that, forsome n ě , that w n ´ p t, p, e q ď w n p t, p, e q for every p t, p, e q P r , τ q ˆ R d ˆ R . Then Φ n ´ p p, e q ď Φ n p p, e q for every p p, e q P R d ˆ R by construction, recall (3.73). Now, usingthe comparison principle, Proposition 3.5(iii), we obtain that w n p t, p, e q ď w n ` p t, p, e q for every p t, p, e q P r , τ q ˆ R d ˆ R . Observe that a direct application of the comparisonprinciple leads also to w p t, p, e q ď w p t, p, e q for every p t, p, e q P r , τ q ˆ R d ˆ R .2. The pointwise convergence comes from the fact that the p ω q q q ě are trivially boundedby one. By monotone convergence, lim n Ñ8 ż ´8 ω n p , p, e q d e “ ż ´8 w p , p, e q d e . INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 31
Now, using (3.11), we obtain sup p P R d ż ´8 ω n p t, p, e q d e ď e ´ r p τ ´ t q ˜ sup p P R d ż ´8 Φ n p p, e q d e ` C ¸ . (3.75)This leads, in particular at t “ , sup p P R d ż ´8 ω n p , p, e q d e ď e ´ rτ ˜ sup p P R d ż ´8 ω n ´ p , p, e q d e ` C ¸ , recalling the definition of Φ n and the non-negativity of ω n ´ . At the limit, we thus obtain(3.74) at t “ . Combining this with (3.75) and a monotone convergence argumentconcludes the proof. l The goal of the next section is to identify precisely the limit of the sequence of functions p ω q q q ě with the definition of the infinite period model.4. Study of the Infinite Period Model
This section is dedicated to the proof Theorem 2.6. In a first part, we will buildthe function w and the associated FBSDE. We show that it is the limit of a q periodpricing function when q goes to infinity. In a second part, we will prove uniqueness ofthis function.As usual, in order to build the solution to the FBSDE, we first exhibit a pricingfunction (decoupling field). Proposition 4.1.
Let Assumptions 1, 2 hold. Assume also that Assumption 3 or As-sumption 4 holds.There exists a continuous function w : r , τ q ˆ R d ˆ R Ñ R satisfying(1) For any t P r , τ q , the function w p t, ¨ , ¨q is {p ℓ p τ ´ t qq -Lipschitz continuous withrespect to e ,(2) For any t P r , τ q , the function w p t, ¨ , ¨q is C -Lipschitz continuous with respect to p , where C ą is a constant depending on L , τ and β only.(3) Given p p, e q P R d ˆ R , for any family p p t , e t q ď t ă τ converging to p p, e q as t Ò τ ,we have Φ w ´ p p, e q ď lim inf t Ò τ w p t, p t , e t q ď lim sup t Ò τ w p t, p t , e t q ď Φ w ` p p, e q (4.1) where Φ w p p, e q “ " w p , p, e ´ λ q , if e ă λ, , otherwise. (4.2) (4) The following holds sup p ż ´8 w p , p, e q d e ă 8 (5) For any p t , p, e q P r , τ q ˆ R d ˆ R , denoting P t ,p the unique strong solution to (2.2) , the unique strong solution p E t ,p,et q t ď t ă τ to E t ,p,et “ e ` ż tt µ p P t ,ps , w p s, P t ,ps , E t ,p,es qq d s, t ď t ă τ (4.3) is such that p e ´ rt Y t q t ď t ă τ , with Y t “ w p t, P t ,pt , E t ,p,et q , is a r , s -valued F -martingale. Then, the limit lim t Ò τ Y t exists and it satisfies P ˆ Φ w ´ p P t ,pτ , E t ,p,eτ q ď lim t Ò τ Y t ď Φ w ` p P t ,pτ , E t ,p,eτ q ˙ “ . (4.4) Proof.
Recall that p ω n q n ě denotes the sequence introduced in Remark 3.19 and that w “ lim n ω n is well defined by Proposition 3.20.Part (1): By Proposition 3.2 each function w n satisfies item 1 in the statement of theproposition and, therefore, the same holds for w .Part (2): Let L n be the Lipschitz constant in p of ω n p , ¨ , ¨q .i) Under Assumption 3: By Proposition 3.2, it depends only on L , L Φ n and τ . Precisely,from Lemma 3.7, we have that L n ď Ll ` e ´ βτ L n ´ . This leads to lim n Ñ8 L n ă 8 .Moreover, using Lemma 3.7 again, we also have that | ω n p t, p, e q ´ ω n p t, p , e q| ď ˆ Ll ` L n ´ ˙ | p ´ p | . By taking limits we deduce that w is C -Lipschitz in p with C “ L { l ` lim n Ñ8 L n .ii) Under Assumption 4: From Lemma 3.8, we have that L n ď C β ˆ } Φ n } ? τ ` l ˙ ď ˜ C , recall (3.73) and ď ω n ´ ď . The constant ˜ C depends on β , L and τ but,importantly, it is uniform in n . Now, by Proposition 3.2, each ω n is Lipschitzcontinuous with a Lipschitz constant depending on ˜ C and so is their limit ω .Part (4) follows directly from Proposition 3.20.Finally, for the functions Φ n given by (3.73) we see that lim n Ñ8 Φ n “ Φ w . UsingProposition A.3 from the appendix, we conclude that w is a value function for the singularFBSDE (3.4) with terminal condition Φ w . Then, parts (3) and (5) follow from Proposition3.2 and the original result, Proposition A.1, in the appendix. l Proposition 4.2.
Let Assumptions 1 and 2 hold and assume r ą . The function w defined in Proposition 4.1 is unique. Proof.
Let w and w be two continuous functions satisfying the items in the statementof Proposition 4.1. Set ∆ w : “ w ´ w and ∆Φ : “ Φ w ´ Φ w . Using Lemma 3.6 ˆż | ∆ w |p , p, e q d e ˙ ď e ´ rτ E „ˆż | ∆Φ |p P ,pτ , e q d e ˙ ď e ´ rτ sup p ˆż | ∆Φ |p p, e q d e ˙ ă 8 (4.5)We have used here that ∆Φ p p, e q “ ∆ w p , p, e ´ λ q t e ă λ u and (3.74). This, combinedwith (4.5) leads to p ´ e ´ rτ q sup p ˆż | ∆ w |p , p, e q d e ˙ ď . INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 33
By continuity of ∆ w , we obtain that w p , p, e q “ w p , p, e q for every p p, e q P R d ˆ R .Then, Φ w and Φ w are equal, and, w and w are value functions, in the sense ofProposition 3.2 for the same FBSDE with the same terminal condition. By uniquenessof such value functions, which follows from the uniqueness in Proposition 3.2 or by (3.12),we conclude that w p t, p, e q “ w p t, p, e q for every p p, e q P R d ˆ R and all t P r , τ q . l We conclude this section by the proving the main result for the infinite period model.
Proof of Theorem 2.6 . We build the tuple of processes p P, E, Y, Z q by a forwardinduction, similarly to the proof of Theorem 2.3.1.a For the first period, we set p E t q t Pr ,τ q to be the solution to E t “ e ` ż t µ p P s , w p s, P s , E s qq d s, with P being the strong solution to (2.2) on time interval r , τ s with P “ p . By theCauchy criterion, we can take the limit as t Ò τ to define E t continuously at t “ τ .For t P r , τ q , we set Y t : “ w p t, P t , E t q . Notice that, by construction, w is the uniquevalue function described in Propositions 4.1 and 4.2, and it satisfies all the properties de-scribed therein. In particular, the process p e ´ rt Y t q ď t ă τ as a r , s valued F -martingale.Therefore, for ˆ Y t : “ e ´ rt Y t , we see that lim t Ò τ ˆ Y t exists and is well defined. Then, thesame holds for lim t Ò τ Y t because Y t “ e rt ˆ Y t . For this process, the terminal condition (4.4)will be satisfied. For t P r , τ q we define p Z t q ď t ă τ such that p e ´ rt Z t q ď t ă τ is theintegrand in the martingale representation for p e ´ rt Y t q ď t ă τ as a stochastic integral withrespect to W .1.b For a positive integer k , assume that p P t q ď t ď T k , p E t q ď t ď T k , p Y t q ď t ă T k and p Z t q ď t ă T k are well defined and satisfy the properties described in the statement of The-orem 2.6. On r T k , T k ` q , we define E as the solution to E t “ E T k ` ż tT k µ p P s , w p s ´ T k , P s , E s ´ Λ k qq d s , T k ď t ă T k ` and define E T k ` continuously, using the Cauchy criterion. Above, P is solution to(2.2) on r T k , T k ` s with starting point at T k equal to P T k , coming from the inductionhypothesis. For t P r T k , T k ` q , we simply set Y t : “ w p t ´ T k , P t , E t ´ Λ k q . By Proposition4.1, the process p e ´ rt Y t q T k ď t ă T k ` is a r , s valued F -martingale; we define Z t for t Pr T k , T k ` q such that p e ´ rt Z t q T k ď t ă T k ` is the integrand in the martingale representationfor p e ´ rt Y t q T k ď t ă T k ` as a stochastic integral with respect to W .By induction, this concludes the proof of the existence of a process p P, E, Y, Z q satisfying(2.14) on each period r T k , T k ` q , k ě , in lieu of terminal condition.2. To conclude the proof, we now check (2.15). Set k ě . For t P r T k ´ , T k q , we thushave, by the previous step, that Y t “ w p t ´ T k ´ , P t , E t ´ Λ k ´ q . If E T k ´ Λ k ´ ă λ , thenby continuity of p P, E q and the properties (4.1) and (4.4), we obtain that lim t Ò T k Y t “ w p , P T k , E T k ´ Λ k ´ ´ λ q “ Y T k , by construction. Similarly, by (4.1) and (4.4), we willfind that lim t Ò T k Y t “ when E T k ą Λ k , and that Y T k ď lim t Ò T k Y t ď when E T k “ Λ k .This proves (2.15). l Acknowledgements
The authors would like extend their gratitude to Dr Mirabelle Muûls, whose collabo-ration and input greatly influenced the direction and applicability of this research. Theauthors would also like to thank the Engineering and Physical Sciences Research Council (EPSRC), Climate-KIC and everybody involved with the Centre for Doctoral Trainingin the Mathematics of Planet Earth at Imperial College London and the University ofReading. The EPSRC and Climate-KIC funded all parts of Hinesh Chotai’s PhD pro-gramme in the centre for doctoral training, and provided many opportunities for travelto conferences and events during the programme. Dan Crisan would also like to thankÉcole Polytechnique and Université Paris Diderot for its visitor grants.
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Appendix
A.The proofs of some results for the multi period model in Section 3 rely on somecorresponding results for a single period model. A one period model is, as describedin Proposition 3.2, a FBSDE of the form (3.4) with φ P K and b , σ , µ satisfying theconditions in Assumption 1. This setting is very similar to the setting considered in [6],Section 2. The main difference is the fact that the terminal condition φ , is a functionof two variables, p and e and that φ is L φ Lipschitz continuous in the p variable. Incontrast, in [6], φ was assumed to be a real valued function φ : R Ñ r , s ,e ÞÑ φ p e q which is monotone increasing and satisfies lim e Ó´8 φ p e q “ , lim e Ò`8 φ p e q “ . Consequently, the proof of Proposition 3.2 can be proved using very similar argumentsas those in Section 2 of [6].Proposition 3.2 is a result of the following two results, which are the versions ofTheorem 2.2 and Proposition 2.10 from [6] in this setting. The proofs are very similarto the corresponding proofs there and are omitted [15].
Proposition A.1.
Consider the system (3.4) with a terminal condition φ P K and set t “ to simplify notation (the analogous result holds for any t P r , τ q ). There existsa unique continuous function v : r , τ q ˆ R d ˆ R Ñ r , s , p t, p, e q ÞÑ v p t, p, e q , called avalue function or decoupling field, such that(1) For any t P r , τ q , the function v p t, ¨ , ¨q is {p ℓ p τ ´ t qq -Lipschitz continuous withrespect to e ,(2) For any t P r , τ q , the function v p t, ¨ , ¨q is C -Lipschitz continuous with respect to p , where C is a constant depending on L , L φ and τ only,(3) For any initial parameters p p, e q P R d ˆ R , let p P t q ď t ď τ be the unique strongsolution of the forward equation for P in (3.4) with P “ p . Then, the uniquestrong solution, p E t q ď t ă τ of E t “ e ` ż t µ p P s , v p s, P s , E s qq d s, ď t ă τ , (A.1) is such that the process p e ´ rt v p s, P t , E t qq ď t ă τ is a r , s -valued martingale withrespect to the complete filtration generated by W .The limit lim t Ò τ E t exists almost surely and so does lim t Ò τ p e ´ rt v p s, P t , E t qq ,being the limit of a bounded martingale.(4) Let Z be such that the process p e ´ rt Z t q ď t ď τ is the integrand in the martingalerepresentation for p e ´ rt v p t, P t , E t qq ď t ď τ as a stochastic integral with respect to W . Then Z satisfies the boundedness conditions in Theorem A.2 below: if σ isbounded by L then there exists a constant C depending only on L , L φ and τ ,such that | Z t | ď C for almost every p t, ω q P r , τ s ˆ Ω . On the other hand, forgeneral σ satisfying only the linear growth and Lipschitz continuity conditions inAssumption 1, then, given an initial value p P R d for the forward equation for P in (3.4) , and any a ě , Z satisfies (A.8) in Theorem A.2 below, for almostevery p t, ω q P r , τ s ˆ Ω , where C ą in (A.8) is a constant depending only on L , L φ , τ and a .(5) Given ξ ą and a ě , there exists a constant C p ξ, a q ą depending only on a , ξ , L and τ such that for all t P r , τ q , all e, Λ P R and every p P R d such that | p | ď ξ , we have e ą Λ ùñ v p t, p, e q ě e ´ r p τ ´ t q ˜ E r φ p P t,pT , Λ q|s ´ C p ξ, a q ˆ e ´ Λ L p τ ´ t q ˙ ´ a ¸ ,e ă Λ ùñ v p t, p, e q ď e ´ r p τ ´ t q ˜ E r φ p P t,pT , Λ qs ` C p ξ, a q ˆ Λ ´ eL p τ ´ t q ˙ ´ a ¸ . (A.2) INGULAR FBSDES AND MULTI-PERIOD CARBON MARKETS 37
The upper bound can also be given in the following form. Given a ě , there isa constant C ą depending only on L and τ such that, for every t P r , τ q and p P R d we have v p t, p, e q ď e ´ r p τ ´ t q ˜ E r φ p P t,pT , Λ qs ` min ˜ C p ` | p | a q ˆ Λ ´ eL p τ ´ t q ˙ ´ a , ¸¸ , (A.3) whenever e ă Λ .(6) For any t P r , τ q , p P R d the function e ÞÑ v p t, p, e q is monotone increasing andsatisfies lim e Ñ8 v p t, p, e q “ e ´ r p τ ´ t q , lim e Ñ´8 v p t, p, e q “ . (A.4) (7) Given p p, e q P R d ˆ R , for any family p p t , e t q ď t ă τ converging to p p, e q as t Ò τ ,we have φ ´ p p, e q ď lim inf t Ñ τ v p t, p t , e t q ď lim sup t Ñ τ v p t, p t , e t q ď φ ` p p, e q . (A.5) (8) The limit lim t Ò τ v p t, P t , E t q exists and satisfies P „ φ ´ p P τ , E τ q ď lim t Ò τ v p s, P t , E t q ď φ ` p P τ , E τ q “ , (A.6) where φ ´ and φ ` are, respectively, the left continuous and right continuous ver-sion of φ , as defined by (1.1) .(9) [Comparison principle] For two different terminal conditions φ, φ P K , denote by v φ and v φ , respectively, the corresponding value functions. If φ and φ are suchthat φ p p, e q ě φ p p, e q for every p p, e q P R d ˆ R , then v φ p t, p, e q ě v φ p t, p, e q forevery t P r , τ q ˆ R d ˆ R . Theorem A.2.
Consider (3.4) and set t “ to simplify notation (the analogous resultholds for any t P r , τ q ). Given any initial condition p p, e q P R d ˆ R , there exists aunique progressively measurable 4-tuple of processes p P t , E t , Y t , Z t q ď t ď τ P S ,d c pr , τ sq ˆ S , pr , τ sq ˆ S , pr , τ qq ˆ H ,d pr , τ sq satisfying the dynamics in (3.4) with p P , E q “p p, e q and such that P “ φ ´ p P τ , E τ q ď Y τ ´ ď φ ` p P τ , E τ q ‰ “ , (A.7) where the functions φ ´ and φ ` are the left and right continuous versions, respectively, of φ , as defined in (1.1) .Finally, if σ is bounded by L then there exists a constant C depending only on L , L φ and τ , such that | Z t | ď C for almost every p t, ω q P r , τ s ˆ Ω . In general, when σ onlysatisfies the conditions in Assumption 1, then, given an initial condition p p, e q P R d ˆ R and any a ě , Z satisfies E « sup t Pr ,τ s | Z t | a ff ď C p ` | p | a q , (A.8) for almost every p t, ω q P r , τ s ˆ Ω , where C ą is a constant depending on L , L φ , τ and a . We also have the following result, which was used in the proof of Proposition 4.1, andis an extended version of Corollary 2.11 from [6]. Its proof follows along the same linesas the proofs in Section 2 of that paper. The full proof can be found in [15].
Proposition A.3.
Let p φ n q n ě be a sequence of terminal conditions for (3.4) in the senseof Theorem A.2. That is φ n P K for every n . For each n , denote by L φ n a Lipschitzconstant for φ n . Suppose that the sequence p φ n q n ě converges pointwise to a function φ .Also assume that sup n ě L φ n ă 8 . (A.9) Then, φ P K also. For each n ě , let v φ n and v φ be the value functions, v , fromProposition A.1 for (3.4) with terminal condition φ n and φ , respectively. Then v φ n Ñ v φ as n Ñ 8 . Convergence is uniform on compact subsets of r , τ q ˆ R d ˆ R . r X i v : . [ q -f i n . M F ] A ug ??? ??? ??? e-mail: ???1