The potential consequences of climate change are becoming increasingly apparent. The Sixth Assessment Report (AR6) of the Intergovernmental Panel on Climate Change (IPCC) states that "[i]t is unequivocal that human influence has warmed the atmosphere, ocean and land" and that anthropogenic climate change "has caused widespread adverse impacts and related losses and damages to nature and people, beyond natural climate variability. "1 Given these concerns, policymakers and organizations are increasingly focusing on the necessity and feasibility of a transition to a carbon-neutral economy. Reports from the OECD, the IEA, the White House,

McKinsey & Co., and Princeton University's High Meadows Environmental Institute2 , among many others, emphasize the need to heavily invest in cleaner production methods to reduce current carbon emissions, as well as the need to devote significant resources to R&D for developing new green technology to prevent future carbon emissions. These conclusions are based largely on scenario analysis aimed at achieving climate policy goals such as the 1.5 C ˝GMT temperature anomaly ceiling proposed by the 2015 Paris Climate Agreement. The implementation of a socially efficient carbon-neutral transition, taking into account the various frictions, risks, uncertainties, and endogenous feedbacks and responses related to climate change and climate policy action, is the key economic question that we address in this paper. We develop and solve a dynamic general equilibrium framework with three types of capital: "dirty" capital that generates carbon emissions when used in production, "green" capital that produces no emissions but is initially less productive than dirty capital, and "knowledge" capital that increases the likelihood of a technology shock that augments the productivity of green capital. By focusing on capital stocks, our model incorporates dynamic features that play an important role in the transition to carbon-neutrality. First, the model framework leads to an emissions pathway that is "sticky" or persistent because dirty capital depreciates gradually and is costly to disinvest. Second, convex adjustment costs related to investment capital means that the accumulation of new green capital can take significant time. Finally, the arrival of improved green technology depends on the stock of knowledge capital in the economy, which requires R&D investment across time and substitutes resources away from consumption, as well as from other types of investment. An important consideration in our analysis is the substantial uncertainty about the central mechanisms in our model. Specifically, our framework allows for uncertainty as it pertains to carbon-climate dynamics, or the mapping from carbon emissions into atmospheric carbon into temperature changes; economic damage functions, or the negative impact on output due to changes in atmospheric temperature; and technological innovation, or the likelihood of a technology shock that augments the productivity of green capital via R&D investment. While each of these model components allows for risk in the form of stochastic realizations, we are interested in exploring alternative forms of uncertainty. This includes ambiguity about possible model parameterizations, as well as the possibility that a given model is misspecified in a meaningful and unknown way. We explicitly incorporate these uncertainty considerations into the social planner's decision problem by applying tools from dynamic decision theory. Our analysis highlights the endogenous interdependencies and feedback effects that substantially impact optimal policy choices and social valuations related to green innovation coming from uncertainty aversion. Our general equilibrium framework with multiple endogenous state variables requires solving relatively high-dimensional PDEs with significant non-linearities due to the model uncertainty concerns and stochastic jump processes related to technological change and climate damages. As a result, our analysis requires computational methods that provide global solutions to accurately characterize the endogenous optimal policies and the social valuations of interest. Standard finite difference methods commonly used in economics and finance are not well equipped for such problems. We, therefore, develop and implement an algorithm using deep neural network methods for our multi-dimensional, continuous-time, climate-economic framework. Our algorithm implements an extended deep Galerkin method that is able to handle multiple non-stationary, endogenous state variables with considerable non-linearity in an infinite horizon setting. Our numerical methodology is an important contribution of our work to the fields of macroeconomics, finance, and climate-economics. The algorithm provides a toolbox that considerably expands the ability of researchers to address research questions in these areas that can quickly be overwhelmed by the "curse of dimensionality" when accounting for regional, household, and firm heterogeneities across production technologies, economic frictions, and policy objectives. Our paper builds on and contributes to a number of important areas of research across economics, finance, geoscience, and applied mathematics. The implications of anthropogenic emissions have been a central focus of geoscientists for many decades, beginning with the seminal work of Arrhenius (1896). Recent work has focused on characterizing the dynamic relationship between carbon emissions, atmospheric carbon concentration, and temperature change via pulse experiments (Joos et al. 2013, Geoffroy et al. 2013, Eby et al. 2009), deriving simplified emulators or approximations of these complex relationships for the use of policymakers (Matthews et al. 2009, Pierrehumbert 2014, MacDougall et al. 2017), as well as quantifying the dynamic and stochastic features of carbon-climate dynamics (Ricke andCaldeira 2014, Palmer andStevens 2019). These components are critical inputs into our model framework and uncertainty analysis. The climate economics literature has given substantial focus to estimating the economic consequences of climate change and deriving a Social Cost of Carbon (SCC), or the cost to social welfare of emitting an additional amount of carbon emissions into the atmosphere. Economists have estimated the economic costs generated by observed climate change for numerous economic sectors, regions, and dimensions of the economy (Dell et al. 2012, Hsiang et al. 2017, Colacito et al. 2019, Allen et al. 2019, Carleton and Greenstone 2021). Theoretical modeling and integrated assessment model analysis to derive SCC valuations have considered for numerous economic mechanisms (Golosov et al. 2014, Acemoglu et al. 2016, Nordhaus 2018, Cai et al. 2017, Cai and Lontzek 2019) and proposed climate damage function approximations and frameworks (Lenton et al. 2008, Weitzman 2012, Cai et al. 2015, Drijfhout et al. 2015, Ritchie et al. 2021).
Recent work has begun to examine important dimensions related to interconnected climate change and economic model uncertainty (Olson et al. 2012, Lemoine and Traeger 2014, Hassler et al. 2018, Nordhaus 2018, Dietz and Venmans 2019, Barnett et al. 2020, Rudik 2020, Barnett et al. 2021, Barnett 2023). Importantly, many of these studies of uncertainty have exploited the powerful toolset developed in dynamic decision theory (Anderson et al. 2003, Maccheroni et al. 2006, Hansen and Sargent 2007, Klibanoff et al. 2009, Hansen and Miao 2018, Barnett et al. 2020, Cerreia-Vioglio et al. 2021), allowing researchers to account for model uncertainty explicitly in the decision-maker's problem within the model. This paper also links to an important area of macroeconomic research. This includes the foundational work on endogenous economic growth (Brock and Mirman 1972, Baumol 1986, Lucas Jr 1988, Romer 1990), as well as recent analysis of the social and private benefits of innovation (Bloom et al. 2019, Lucking et al. 2019), and considerations for the transition to a green economy (Acemoglu et al. 2012, 2016, Jaakkola and Van der Ploeg 2019). In addition, the connection between macroeconomics and asset pricing in production-based asset pricing (Brock 1982, Cochrane 1991, Jermann 1998), as well as the importance links to economic growth and innovation (Papanikolaou 2011, Kogan and Papanikolaou 2014, Kung and Schmid 2015) have important implications for the social valuations we derive in our climate-economics-innovation linked framework. In recent years, deep learning algorithms, built on the neural network's remarkable ability to represent and approximate high-dimensional functions and efficient gradient descent optimizers, have been very successful in many areas, ranging from computer vision and speech recognition to scientific computing (see, e.g. Carleo and Troyer (2017), Gao and Duan (2017), Han et al. (2018), Zhang et al. (2018), Han et al. (2019)). The climate change problem considered in this paper aims to study how choices for investment in clean and dirty capital, and R&D for technological innovation, are determined when facing uncertainties, which, mathematically, is a stochastic control problem. Along this direction, deep learning algorithms are roughly categorized into: direct parameterization, partial differential equations (PDEs), and forward-backward stochastic differential equations (FBSDEs) approach. In this first category, the seminal work in high-dimension was proposed by Han and E (2016), which solves a global-in-time minimization problem in accordance with the utility of the control problem. Later, Bachouch et al. (2021) extended to the local-in-time approach combined with dynamic programming techniques. Han and Hu (2021) solve the control problem with aftereffects, modeled by stochastic differential delayed equations, using the same spirit of Han and E (2016) but with advanced network architectures, and Carmona and Laurière (2022) extended to mean-field control problems. In the PDE approach, the stochastic control problem will first be reformulated into a Hamilton-Jacobi-Bellman (HJB) PDE using the dynamic programming principle, and then solved by deep learning algorithms. For generic PDEs, Sirignano and Spiliopoulos (2018)  In the FBSDE approach, the control problem will first be reformulated into a coupled forwardbackward system using the stochastic Pontryagin maximum principle. The system is, in general, hard to solve numerically, let alone in high dimensions, due to its coupled nature and opposing directions: one with an initial condition running forward in time and one with a terminal condition running backward in time. Han and Long (2020) solved the fully-coupled FBSDE by extending the deep BSDE solver with convergence analysis subject to neural networks' universal approximation, and Ji et al. (2020) further extended the results with the Picard iteration method. Recently, attention has also been drawn to stochastic control problems with jumps. The problem is more involved as the PDE becomes partial integro-differential, thus non-local; and the FBSDE system now contains a Lévy process. For such problems, Boussange et al. (2022)  When using neural networks to parameterize the quantity of interest, usually the value function or the control policy, people may want to incorporate domain knowledge, such as guaranteed monotonicity or convexity of the learned function with respect to some of the input variables. There have been fruitful studies on how to build network structures in order to fulfill such requirements, see for instance, Sill (1997), Gupta et al. (2016), Wehenkel and Louppe (2019) and Runje and Shankaranarayana (2022). We now outline the main model for our analysis. The model incorporates components related to climate dynamics, economic elements and dynamics related to damages from climate change, production technologies, preferences, and technological innovation through R&D, as well as model uncertainty aversion. We outline each of these model pieces in what follows, and then derive a number of theoretical results before presenting the numerical solutions to the model. Recent work in geosciences has focused on constructing simplified approximations of climate dynamics generated by large-scale atmosphere-ocean general circulation models (AOGCM) for use in policy analysis. These include a tractable framework demonstrated by Matthews et al. (2009) and Friedlingstein (2015) as a proportional relationship between temperature change and cumulative carbon emissions. This relationship is of the general form: temperature anomaly « climate sensitivity pβ f q ˆcumulative emissions. Not only is this approximate relationship straightforward to incorporate into most continuoustime macroeconomic and asset pricing models, it also directly provides a model comparison set to use in our uncertainty analysis based on the constant of proportionality implied by different large-scale climate models. For our analysis, we use a stochastic variant of this relationship

where Y t is the global mean temperature anomaly, with respect to the preindustrial level, E t is global carbon emissions, W t is a Brownian motion process with filtration F t , ς is the volatility loading for temperature, and β f is the climate sensitivity proportionality constant. This model abstracts from transitory "weather" fluctuations in temperature and assumes that emissions today have a long-lasting, i.e., permanent, impact on future temperature.
The inclusion of stochastic variation, which allows for a meaningful uncertainty analysis by "disguising" the climate model in the statistical sense that a small set of observations do not immediately reveal the true model, is motivated by the argument about climate model predictability put forward in Palmer and Stevens (2019). The parameter β f varies across different carbon-climate dynamics models, and we will denote these different values as β f,ℓ for a given model ℓ P t1, ..., Lu. In the geoscience literature, this parameter is known as the transient climate response to cumulative emissions (TCRE) parameter and serves as a climate sensitivity measure. While the proportionality relationship used here is based on a relationship of carbon emissions to temperature change that is born out at a time horizon of 10 years and beyond, (Dietz and Venmans 2019, Barnett et al. 2021, Barnett 2023) and others have shown that this relationship is well suited for frequencies as short as one year. We follow Barnett et al. (2021) and use pulse experiment results of Joos et al. (2013) and Geoffroy et al. (2013) to build the set of climate sensitivity models for our analysis. Joos et al. (2013) examines carbon dynamics variation and uncertainty based on responses of atmospheric carbon concentration to emission pulses of 100 GtC for several alternative Earth System models. Geoffroy et al. (2013) provides temperature dynamics variation and uncertainty based on approximate dynamics relating the log of atmospheric carbon to future temperature, following Arrhenius equation. We combine these two sets of pulse experiments by taking 9 different atmospheric carbon responses as inputs into 16 temperature dynamics approximations to derive 144 different carbon-climate TCRE models. Figure 1 shows the dynamic pathways and the implied TCRE parameters for these 144 models. We can see from the pathways that the temperature response peaks around 10 years, and flattens out thereafter, and the implied TCRE parameters are quite dispersed, with values ranging from around 1 to just below 3, with the average value being 1.86. Following Barnett et al. (2023), our model of economic damages from climate change, or climate damages, is piece-wise log quadratic. Damages N t are assumed to impact capital, output, investments, and consumption in a proportional manner. This specification allows for a change in the evolution of damages that is triggered by a jump process realization. Uncertainty about climate damages is connected to the values of Ŷ , which is defined by Ŷ " Y `ȳ ´Ỹ for y ă ỹ ď ȳ, and γ m 3 , for which there are alternatives values denoted by m P M. There is a stochastic jump process that determines dynamically which value of γ m 3 will be realized for the damage function and at the value of Ỹ the jump will be realized. Specifically, the jump process has m absorbing states, where each state m corresponds to a value of γ m 3 . Before the realization of the jump, each γ m 3 value has a prior probability π p m . When the jump is realized, the value of γ m 3 is revealed and the true damage function curvature becomes known to the social planner in the model. The likelihood of a damage function jump is determined by a jump intensity I d pyq that is increasing in the endogenous temperature anomaly state variable y and is localized to the range of temperature values ry, ys. At the time of the jump τ , the state variable Ŷ shifts from the temperature anomaly Y " Ỹτ to the damage jump threshold ȳ. Thus Ŷ is a state variable that shifts from capturing just the temperature anomaly to a jump-adjusted measure capturing the derivative shift of the damage function as well. The uncertainty is associated both with the probability distribution on possible γ m 3 model realizations, as well as the probability of when a damage jump will occur in the ry, ys. The full functional form is left for the Appendix, but the evolution of log damages is given by

For our main setting, we assume ry, ys " r1.5, 2.0s, consistent with the critical temperature thresholds used by the IPCC and the 2015 Paris Climate Agreement. Figure 2 shows the implied climate damages across all γ m 3 values for the jump realization occurring at the temperature anomaly values of ỹ " y " 1.5 and ỹ " ȳ " 2 which bound our range for the damage jumps occurring. The red shaded region shows the range of possible damage outcomes across the possible values of γ m 3 , and the black line gives the mean value. 3 While there is significant discussion about the likelihood of tipping points or thresholds at the global scale captured by discrete jumps or shifts in climate damages 4 , our setting is instead a smooth shift where the planner realizes whether the damage function is more concave than previously known. In this sense, we view our framework as a valuable tool for characterizing the possibility and uncertainty of potentially severe, nonlinear climate damage outcomes that are highly relevant for the optimal policy decisions of a social planner confronting climate change. Moreover, the structure of our uncertainty is novel 3 While these plots are nearly identical to those found in Barnett et al. (2021), there are key differences. Specifically, in Barnett et al. (2021), there is a possibility for discontinuities in the damage function and jumps in the climate damages, which are ignored in their analysis. Our specification removes such discontinuities, and allows only for smooth shifts in the derivative of the damage function. 4 See Brook et al. (2013) and Levitan (2013) for examples on this discussion. with regards to much of the existing climate-economics literature, as the information dynamics in most settings is static, or uncertainty about the damage function is never resolved.
For the economic component of the model, we assume there are two sectors producing perfectly substitutable consumption goods using their own AK technologies:

Each sector has its own capital stock that evolves with logarithmic adjustment costs and Brownian shocks as follows:

For computational tractability, we redefine the state variables characterizing the capital stocks in the model and use log K " logpK d `Kg q and R "

. By Ito's lemma, these two state variables have the following evolution processes:

There are two key differences across these different consumption good production sectors in the economy. First, Sector d is the only sector that generates emissions, so that E t in our temperature evolution equation is given by

where η is the emissions intensity parameter of Sector d production. Second, Sector d is initially more productive than Sector g in that A d ą A g . While Sector d is initially more productive than

Sector g, we also assume that there is the potential for a "green" technology shock that would augment Sector g productivity. The arrival rate of this one-time jump in Sector g productivity is an increasing function of the aggregate knowledge capital stock in the economy κ. The evolution of the aggregate knowledge stock is given by

where I κ is the level of R&D investment made to increase the knowledge capital stock, ζ is the decay rate of the knowledge stock, ψ 0 and ψ 1 capture the effectiveness of R&D investment, and σ κ is the volatility associated with the knowledge capital stock. We include uncertainty about the realization of the technology shock in the form of a discrete set of possible realizations for the post-technology jump Sector g productivity A j g . The dynamic realization of the technological change shock that determines the value of A j g is through a stochastic jump process. There are j absorbing states, where alternative values are denoted by j P J , with each state j corresponding to a particular value of A j g and each value of A j g having a prior probability π j g . For each potential realization of the technology shock, the value of A j g is such that A j g ě A g . The likelihood of a technological change jump is dependent upon a jump intensity I g pκq. We choose the simple functional form I g pκq " κϱ, where κ ą 0 is a constant, so that the arrival rate is increasing in the endogenous knowledge stock state variable κ. The information dynamics and framework and structure of the technological change jump are similar to the damage function jump. Once the jump is realized, the true technological change outcome for the value of A j g is known to the social planner. In addition, the uncertainty pertains to the probability distribution of potential A j g realizations and the arrival rate of the technological change shock. This structure focuses on the economic implications related to the uncertainty of breakthrough green technologies, again a relatively novel channel of innovation compared to the climate-economics literature. We find this specification appealing as it allows for a broader interpretation of the uncertainty of "green" technology change that further enriches our analysis and uncertainty quantification as it relates to optimal policy considerations related to technological innovation and climate change. Finally, we assume that flow utility in the model is a log function over damaged aggregate consumption, where exponential-quadratic damages multiplicatively scale consumption. The utility function is therefore given by

where δ is the subjective discount rate, the two types of consumption goods are perfectly substitutable. Investment decisions are made by optimally dividing the output net of consumption across investment into the three types of capital in the economy (dirty production capital, green production capital, and knowledge capital), so that the market clearing final output constraint is given by the relationship

where i d " I d {K d is the dirty investment-to-dirty capital ratio, i g " I g {K g is the green investmentto-green capital ratio, and i κ " I κ {pK d `Kg q is the R&D investment-to-total capital ratio. With the model framework laid out, we can not turn to solving the Hamilton-Jacobi-Bellman equations that characterize the solution to the Social Planner's problem in our model. The solution to our model is a recursive Markovian equilibrium that solves the Hamilton-Jacobi-Bellman equation characterizing the planner's social welfare function, or value function. Therefore, the equilibrium solution is determined by optimal investment choices: ti g , i d , i κu that maximize the planner's discounted, lifetime expected utility as functions of the state variables

These optimal controls must satisfy the evolution equations for the state variables, as well as the market clearing conditions given in the model set-up above and the first order conditions from the HJB equations we outline below. To arrive at the full model solution, we must derive solutions sequentially working from the post-technology jump, post-damage-function-jump state back to the pre-technology jump, pre-damage-function-jump state, accounting for the different possible combinations of intermediate states, which are the pre-technology jump, post-damage-functionjump state and the post-technology jump, pre-damage-function-jump state. The first jump-state we must solve is after the technology jump and damage jump have already occurred, or the post-technology jump, post-damage-function-jump state. We denote this value function by V pm,jq pK d , K g , Ŷ , log N q, indicating that we are at a given realization of γ m 3 and in the second technology state with green capital productivity A j g .
We can guess-and-verify that the value function solution in this stage can be analytically simplified as follows

The remaining HJB equation characterizing the simplified value function is given by:

Taking first order conditions with respect to i d and i g , we end up with the following two expressions for optimal investment choices

δprA d ´id sp1 ´Rq `rA 1 g ´ig sRq ´1 " Γ g θ g p1 `θg i g q ´1rv pm,jq log K `p1 ´Rqv pm,jq R s.

Each expression equates the marginal benefit of an additional unit of a given type of capital to the marginal utility of consumption, which is the utility loss of forgoing consumption for an additional unit of investment. Our next two HJB equations come from the intermediate jump states. The first case is when the technology jump has already occurred, but the damage function jump has not. In this case, we can apply a similar analytical simplification to the value function for the social planner. We denote this value function by V pjq pK d , K g, Y, log N q, indicating that we have not had a realization of γ m 3 , but we are in the second technology state with green capital productivity A j g . The value function in this stage is given by

There are two adjustments made to the simplified HJB equation in this case relative to the post-technology and -damage jump case. First, because the damage jump has not yet occurred, the equation capturing the evolution of damages is altered to be

Second, the expectation of damage function jump introduces the additional term

π m d pv pm,jq ´vpjq q. We again take first order conditions with respect to i d and i g . These expressions are unchanged from the post-damage and technology jump case, equating the marginal benefit of an additional unit of a given type of capital to the utility loss of forgoing consumption for an additional unit of investment, which is the marginal utility of consumption. The second intermediate case is when the technology jump has not yet occurred, but the damage function jump has. We can again apply the same analytical simplification to the value function for the social planner. We denote this value function by

indicating that we have had a realization of γ m 3 , but we are in the technology state with green capital productivity A g . The value function in this stage is given by

Relative to the post-technology and -damage jump case, we introduce the following new terms to the simplified HJB equation

π j g rv pm,jq ´vpmq s.

The first two terms account for the evolution of the knowledge capital and the possibility for R&D investment to increase the knowledge capital stock. The last term accounts for the possibility of the green technology jump that alters green capital productivity to A j g and is increasingly more likely as the knowledge capital stock increases. In addition, allowing for R&D investment alters the output constraint in this setting, so that our flow utility term net of climate damages is now given by δ logprA d ´id sp1 ´Rq `rA g ´ig sR ´iκ q. We now take first order conditions with respect to i d , i g , and i κ , which give us expressions for optimal investment choices δprA d ´id sp1 ´Rq `rA g ´ig sR ´iκ q ´1 " Γ d θ d p1 `θd i d q ´1rv pmq log K ´Rv pmq R s, δprA d ´id sp1 ´Rq `rA g ´ig sR ´iκ q ´1 " Γ g θ g p1 `θg i g q ´1rv pmq log K `p1 ´Rqv pmq R s, δprA d ´id sp1 ´Rq `rA g ´ig sR ´iκ q ´1 " ψ 0

As before, these expressions equate the marginal benefit of an additional unit of a given type of capital to the marginal utility of consumption, which is the utility loss of forgoing consumption for an additional unit of investment. However, given the altered output constraint expression, the marginal utility of consumption on the left-hand side of these equations now also incorporates the fact that some output is dedicated to R&D investment instead of consumption. Finally, we have the components needed for the value function associated with pre-technology and -damage jumps. We denote this value function by V pK d , K g, Y, log κ, log N q, indicating that neither a realization of γ m 3 nor the jump to the green capital productivity has occurred. The same analytical simplification can be applied here, and so the value function in this stage is given by

The simplified HJB equation incorporates each of the changes applied to the two intermediate jump state cases. In particular, relative to the post-damage and technology jump state HJB equation, we introduce the following additional terms

The first line accounts for the evolution of the knowledge capital and the possibility for R&D investment to increase the knowledge capital stock. The second line captures the altered evolution of damages because the damage jump has not yet occurred. The last line accounts for the possibility of the green technology jump that alters green capital productivity to A j g and is increasingly more likely as the knowledge capital stock increases, as well as the expectation of damage function jump that becomes more likely as temperature increases. We again alter the output constraint in this setting relative to the post-damage and technology jump case due to the possibility of R&D investment.
As a result, the first order conditions with respect to i d , i g , and i κ match those of the intermediate jump state before the technology jump and after the damage jump, except that the relevant value function derivatives do not pertain to v. As in each case, the FOCs equate the marginal benefit of an additional unit of a given type of capital to the marginal utility of consumption, which is the utility loss of forgoing consumption for an additional unit of investment. With the baseline model results in place, we now introduce the various channels of model uncertainty of focus in our analysis. We explore uncertainty aversion in the form of model misspecification and/or smooth ambiguity across the following channels:

• Climate Sensitivity to Carbon Emissions

• "Green" Technological Change Aversion to these different channels of uncertainty are introduced into our baseline model using the tools of dynamic decision theory. Importantly, the planner in our model is wary of model uncertainty of these different forms and channels, and thus adopts a preference structure to identify potential worst-case models, constrained by a penalization function, and make optimal policy decisions that are robust to these possible outcomes. We outline the different forms of model uncertainty for our framework in what follows, focusing specifically on misspecification concerns about jump and diffusion processes. Further elaboration on alternative settings, including smooth ambiguity aversion, and complete details on the HJB and FOC equations characterizing the model solutions, is provided in the appendix. We first consider the setting with misspecification concerns about the damage function and technological change jump channels. Before the jumps occur, the relevant contributions to the HJB equation related to the technological change and damage function jumps are

Assuming a common robustness parameter ξ across each channel, these terms are replaced by the following uncertainty-adjusted contributions min g j ,fm

π j g p1 ´gj `gj log g j q `Id pyq

The adjustment introduces probability distortions g j and f m to the jump processes for technological change and climate damages. These distortions impact expectations about the arrival rate and distribution of post-jump outcomes related to these jumps. To constrain these distortions, the relative entropy terms ξ " I g pκq ř J j"1 π j g p1 ´gj `gj log g j q `Id pyq

ı are introduced into preferences, penalizing distortions that are "too large" based on the uncertainty aversion parameter ξ. Note that g j , and f m are endogenous objects solved for by the decision maker by minimizing discounted lifetime expected utility, subject to the relative entropy penalties. The FOCs resulting from this minimization objective are given by

Each of these represents probability distortions that adjust the distributions associated with the jump processes related to the economic and climate channels for which there are concerns about model uncertainty. The inclusion of these probability distortions into the planner's problem leads to robustly-altered optimal policy choices that account for the distorted likelihood of outcomes based on potential worst-case outcomes. We note that in each of the various jump realization states, there are modifications that need to be made to these components. After the technology and damage jumps have both taken place, jump uncertainty is no longer relevant for the social planner. After the technology jump has occurred, but not the damage jump, we replace A g with A j g and the relevant uncertainty adjustments to the HJB equation and minimization FOC pertaining to the climate model and damage function jump are given by min fm

Before the technology jump has occurred, but after the damage jump, the relevant uncertainty adjustments to the HJB equation and minimization FOC pertaining to the climate model and technology jump are as follows min

π j g g j pv pm,jq ´vpmq q `ξI g pκq J ÿ j"1 π j g p1 ´gj `gj log g j q , g j " exp ˆ´1 ξ pv pm,jq ´vpmq q ˙. We now consider the setting with misspecification concerns about the climate and capital stocks diffusion channels. Before the technology and damage jumps occur, the relevant contributions to the HJB equation related to the diffusion components are of the form

Assuming the same common robustness parameter ξ for each channel as before, these terms are replaced by the following uncertainty-adjusted contributions

This adjustment introduces a drift distortion h to the diffusion processes for the different types of capitals and for temperature, disguised by the Brownian shocks. This type of drift distortion is a known result from the Girsanov theorem, and captures a distributional change represented by a likelihood ratio. To constrain this distortion, again, a relative entropy term is introduced into preferences, in this setting, the quadratic expression ξ|h| 2 {2, penalizing distortions that are "too large" based on the uncertainty aversion parameter ξ. As with g j and f m , h is an endogenous object solved by the decision maker by minimizing discounted lifetime expected utility, subject to the relative entropy penalties. The FOC resulting from this minimization objective is given by

The inclusion into the planner's problem of this probability distortion in the form of a drift distortion leads to further robustly-altered optimal policy choices that account for the distorted likelihood of outcomes based on potential worst-case outcomes. As with the jump misspecification concerns, in each of the various jump realization states there are modifications that need to be made to these components. After the technology and damage jumps have both taken place, the remaining uncertainty implications come from the climate sensitivity channel and the green and dirty capital stocks.
Therefore, in addition to replacing A g with A j g , and denoting the realized γ m 3 value, the uncertainty-adjusted contribution to the HJB and the relevant minimization FOC are given by min

Before the technology jump has occurred, but after the damage jump, the relevant uncertainty adjustments to the HJB equation and minimization FOC pertaining to the diffusion terms are

We have so far shown the introduction of the jump and diffusion uncertainty separately in our analysis. However, in our analysis, we incorporate each of the uncertainty channels simultaneously, allowing for broadly conceived uncertainty considerations in our results.
There are explicit contributions related to diffusion uncertainty and jump uncertainty in the model. However, allowing for all of the uncertainty channels simultaneously allows for important interaction effects as the jump misspecification distortions f m and g j , and the diffusion misspecification distortion h, will alter the value functions and their derivatives across jump states and states of nature.
These more implicit effects will influence the optimized distortions arising from uncertainty aversion, and as a result, impact the social valuations and optimal policy choices of the social planner. Before delving into numerical results, we outline the numerical method used to derive those results.
The computational algorithm developed here is a critical contribution of our paper to the literature. Much of the theoretical work in climate economics and climate finance requires, by necessity, the use of computational methods to derive solutions.
The integrated structure of these models requires a multi-dimensional state space, often times with non-linear dynamic relationships and functional forms, in order to rigorously account for relevant model features. The result of specifying such models is the need to confront the "curse of dimensionality", or the fact that the complexity of deriving numerical solutions exponentially increases in the number of dimensions.
As noted by (Han et al. 2018) We address these issues in our framework by implementing an extended deep Galerkin method algorithm that considers critical parameters as additional (pseudo) variables in the network input, an approach not previously explored in the literature.
Because of this, we are able to derive global numerical solutions for a continuous-time, infinite horizon setting with significant nonlinearities and multiple endogenous state variables. This is critical to our analysis, as exploring transition dynamics for endogenous optimal policy responses is at the heart of understanding and analyzing outcomes related to climate change, model uncertainty, and the transition to carbonneutrality.
Furthermore, our algorithm can still handle high-dimensional PDEs, avoiding the "curse of dimensionality" in such cases. As such, the resiliency of our algorithm to various modeling complexities, while still being able to scale to higher-dimensional settings, opens the door to exploring models with regional, household, and firm heterogeneities across technologies, economic frictions, policy objectives, and other sources.
Thus, our numerical algorithm not only enriches our ability to explore a more extensive set of questions and models in climate-economics and climate finance than before, but also more broadly in economics, finance, and other areas of research that require solving dynamic stochastic optimal control problems. We now outline our algorithm, and provide pseudo-code and further details about implementation in the appendix.
We use the deep Galerkin method-policy improvement algorithms (DGM-PIA)

proposed in Al-Aradi et al. (2022) to solve the aforementioned HJB equations.
We first illustrate the algorithm on a generic HJB equation for V pxq:

´δV pxq `sup αPA tL α V pxq `f px, αqu " 0, where x and α denote the state and control variables, A is the control space, the differential operator L α is the infinitesimal generator of the controlled state process X α , f is the utility function and δ is the discount factor. DGM-PIA solves for the value function V and the optimal control α simultaneously by parameterizing both as deep neural networks V θ and α φ . Then, the networks are trained by taking alternating stochastic gradient descent steps for the two functions. Let α φ 0 (as a function of x) be the initial control parameterized by the neural net, at stage n, the algorithm contains two steps:

Step 1. Find a solution to the linear PDE ´δV θn pxq `Lα φn V θn pxq `f px, α φn pxqq " 0, for the fixed control α φn , by updating θ n via minimizing

Step 2. Update the policy corresponding to

for the fixed value function V θn , by update φ n`1 via minimizing

where νpxq is a probability measure on the domain Ω of x characterizing the different regions' relative importance. In our problem, depending on the setting we are solving, the state processes could contain plog K, R, Y t , log κ, log N t q, the control variables could contain pi d , i g , i κ , g j , f m , hq. In addition, in order to keep the value function's monotonicity with respect to γ m 3 , we take γ m 3 as an input or "pseudo-state" of the parameterized neural network. We next present and discuss the numerical model solution results, which are derived using the numerical algorithm outlined above. Before getting into the results, we briefly outline a few details for completeness regarding model assumptions, functional forms, and parameter values. After presenting and discussing the numerical results, we discuss details about how we validate our neural-network-based solutions. First, for tractability we consider the case of independent Brownian shocks, i.e.,

Second, we use the following functional forms for our jump arrival rates. For the technology change jump, we assume an arrival rate that is proportional to the knowledge stock: I g pκq " κ{ϱ. The parameter ϱ scales the knowledge capital stock variable to change the units into arrival rate units, and is chosen based on expected green policy implementation timelines proposed by various countries. Section 6.2 provides further details about the choice of this parameter value. The damage jump intensity I d pyq follows the arrival rate proposed in Barnett et al. (2021):

Figure 3: Intensity function, r 1 " 1.5 and r 2 " 2.5. With this intensity function, the probability of a jump at an anomaly of 1.6 is approximately .02 per annum, increasing to about .08 per annum at an anomaly of 1.7, increasing further to approximately .18 per annum at an anomaly of 1.8 and then to about one third per annum when the anomaly is 1.9. Figure 3 shows the increasing nature of the arrival rate as temperature anomaly y increases. The calibration is such that the probability of a jump taking place by y " 2 is essentially 1. We now outline how we chose the parameter values used in our numerical analysis. The economic parameter values are given in Table 1. We note that the special case of our model without climate change and technological innovation is the model given in Eberly and Wang (2009). We therefore  pα g , Γ g , θ g , σ g q ( -0.035, 0.025, 100, 0.15) pA d , A g q (0.12, 0.10) tA j g u A g ˆt1 `pj´1q J´1 u j"1,...,3 pζ, ψ 0 , ψ 1 , σ κ q (0, 0.10583, 0.5, 0.016) ϱ 448

We assume ζ " 0. Lucking et al. (2019) and Bloom et al. (2019) have provided estimates for the returns to R&D investment which guide our choice of pψ 0 , ψ 1 q. The choice of ϱ translates our initial value of knowledge stock, which is based on estimates from the BLS of the total US R&D stock values (scaled up to World values), to an expected arrival time of a green technological innovation occurring between 30 and 80 years. The value of σ κ is chosen to match the other capital volatilities, which are based on estimates from the World Bank database. The climate dynamics and climate damage parameter values are given in Table 2. The values of β f,ℓ come from pulse experiments estimates produced by Barnett et al. (2021)  For our computations, we must also specify ranges and initial values for our state variables. These values are given in Table 3. The initial value of total capital K 0 matches estimates from the World Bank of World GDP. The initial value of global mean temperature anomaly Y 0 matches the estimated current value from Masson-Delmotte et al. (2021b). The initial value of the green capital-to-total capital ratio R 0 is based on estimates of clean and dirty capital splits from the EIA and IEA. The initial value of knowledge stock κ 0 matches estimates from the BLS of Total US R&D stock values (scaled up to World values). We now discuss the computational results of our model. The results shown are simulation pathway outcomes based on the solutions to the HJB equations and are initialized at today's values of the state variables and shown out to 30 years, which is near the time when the temperature anomaly hits 1.5 ˝C and the probability of a damage jump occurring becomes non-zero. For the probability of the technology and damage jumps occurring, we examine the outcomes out to 40 years.
The results provided are for the following specifications related to model uncertainty:

• β f as the average of the 144 climate models

• 20 damage models γ 3 P t0, ..., 1{3u

• Misspecification over technology and damage jumps and the climate model dynamics

• Comparison across uncertainty parameters ξ P t0.1, 8u

We focus on the simulated pathways from the pre-damage jump and pre-technology jump state for emissions, R&D investment, green and dirty capital investment, as well the distribution of climate and damage models, and the probabilities of a damage or technology jump occurring. We examine each of these results across different uncertainty aversion parameters to highlight the implications of increased model uncertainty on the optimal policy choices by our planner. The red lines, or red histogram bars, represent the uncertainty neutral case when ξ " 8, and the blue lines, or blue histogram bars, represent the uncertainty averse case when ξ " 0.1. Figure 4 provides the main economic policy outcomes of interest. Figures 4a and4b show the pathways for emissions and R&D investment across the uncertainty averse and uncertainty neutral cases. First, in the top left panel we see that emissions in each case start out similarly, and though they are increasing over time for both cases, we can see that the emissions diverge and are lower for the uncertainty averse case of ξ " 0.1. In the top right panel, we see R&D investment as a percent of total output. Note first that the magnitude of R&D is fairly substantial in each case, ranging between 2.5% and 3.5% of total output initially. For comparison's sake, Stine (2008) notes that expenditure on R&D directed towards the Manhattan, Apollo, and the Federal Energy Technology programs reach magnitudes near 0.5%, with total R&D levels reaching over 2%. We can also see that when the planner is concerned about model uncertainty, i.e., for ξ " 0.1, the fraction of output committed to R&D investment starts substantially higher, and though the R&D investment beings to decrease before the uncertaintry neutral case, it always remains higher when there are concerns about model uncertainty. Figures 4c and4d show the levels of dirty and clean investment across the uncertainty aversion cases. Comparing the dirty investment in the bottom left panel and the clean investment in the bottom right panel, we see three key points. First, the level of investment in green capital is substantially higher than in dirty capital. At the beginning of the simulation pathway, the green investment level is about 50% higher, but by the end of the pathway, it is nearly four to five times higher. This highlights the second key observation related to investment, that green investment persistently increases during the simulation pathway, whereas dirty investment begins to taper off and diminish over time. Finally, we see that uncertainty concerns lead to initially higher dirty investment, that decreases more rapidly than in the uncertainty neutral case, whereas there is only a slight amplification of green investment due to difference in uncertainty aversion. From Figure 5 we can see why the planner responds the way they do. The histograms in In summary, it is clear that climate concerns lead to substantial policy action, where the social planner allocates significant resources to R&D investment and green capital investment, while diverting some of those resources away from dirty investment. However, concerns about uncertainty aversion have relatively modest impacts on investment choices in clean production capital, instead focusing somewhat more on reducing dirty production capital and leaning even more heavily into the policy response of amplifying R&D investment. The social optimality of this response is driven by the potentially significant payoff from technological innovation in this setting, the fact that emissions are sticky in the sense that they are proportional to output from production using dirty capital and there is no direct mechanism for removing or transforming capital, and the costly nature of accumulating new green capital that is less productive than dirty capital until a technology jump occurs. As a result, the planner places a substantial social valuation on R&D investment as a policy response tool in order to potentially initiate the technology jump as relying mainly on emissions reductions through additional dirty capital investment reduction or further investment in green capital. An important issue to address for our numerical solutions is the validation of the accuracy of our results. We address this in three ways, with the results shown in Figure 6. First, we examine the training loss for our neural net solutions to determine the magnitude of the HJB equation error for our solution from the DGM-PIA method. Figure 6a shows the training error for the post-technology, post-damage jump state solution across epochs. We are able to reach a training error of approximately 10 ´4, suggesting our solution is reasonably accurate. In addition, because the post-jump model solutions require only 3 state variables, we can compare and validate these results with solutions derived using the solution derived from more standard finite difference methods. In particular, we use the false-transient, conjugate gradientbased numerical algorithm implemented in Barnett et al. (2020) and Barnett et al. (2021) for our finite difference solutions. Figure 6b shows the neural net and finite difference solutions of the value function and optimal clean and dirty investment for different values of R, and values of log K and Y fixed at the midpoints of our state space range (log K " 5.5, Y " 1.5). We see that the two solutions methods provide consistent outcomes. Finally, for the pre-technology, pre-damage jump state, we note that the training errors are similar to the post-damage, post-technology jump state, but we cannot derive finite difference method solutions for this setting. Therefore, we propose the DGM-PIA solution at a given point in the state space to the solution derived from another deep learning method proposed in Han and E (2016), in which the authors directly parameterize the control by neural nets and obtain the optimal parameters by maximizing the approximated utility.
This method is known to be very stable, though significantly slower, because the solution is derived point-by-point via training neural nets across repeated simulations of the state variables. We provide these results in future work, though preliminary results show the solutions for the two methods are roughly consistent. Model uncertainty along multiple dimensions is a central issue when studying optimal climate policy for a carbon-neutral transition. We integrate dynamic decision theory under uncertainty into a climate-economics-innovation framework with multiple capital stocks to guide uncertainty quantification of optimal R&D investment, and investment in dirty and clean production capital. We examine different forms of uncertainty, including diffusion and jump process model misspecification, as well as the implications of uncertainty from climate sensitivity, climate damages, and technological innovation. Concerns about model uncertainty feed into the endogenous equilibrium policy responses in our model by "tilting" the stochastic discounting implicitly used in constructing the marginal valuations related to the externalities in our framework. This results in first-order implications for the socially optimal outcomes in our model as a result of incorporating aversion to model uncertainty in the planner's decision problem. In particular, uncertainty aversion amplifies the incentive to invest in technological innovation as the major policy mechanism. Given the richness of our economic and geoscientific model components, we develop and implement a deep learning-based algorithm to solve the Hamilton-Jacobi-Bellman equations that characterize the planner's value function in our setting. The algorithm allows us to derive global solutions for models that have many endogenous and non-stationary state variables, maximize lifetime expected utility over an infinite horizon, are in continuous-time, and have potentially significant non-linearities. Because the computational method can handle such computationally difficult problems without being bogged down by "the curse of dimensionality" or other numerical issues, we see significant potential for our algorithm in addressing other economically and geoscientifically rich climate-economic problems. For the same reasons, our computational method also has significant promise for recursive, dynamic general equilibrium models used across economics and finances. We leave exploration of such alternative settings for future work. The HJB equation for the post-damage and pre-technology jumps state is given by

`Ig pκqpv pm,jq ´vpmq q.

A.1.3 Pre Damage and Technology Jumps HJB Equation

The HJB equation for the pre damage and technology jumps setting is given by δv " max ig,i d δ logprA d ´id sp1 ´Rq `rA g ´ig sR ´iκ q `δ log K

The HJB equation for the post damage and pre technology jumps setting is given by

δ logprA d ´id sp1 ´Rq `rA g ´ig sR ´iκ q `δ log K

π j g g j ´vpm,jq ´vpmq ¯`ξI g pκq J ÿ j"1 π j g p1 ´gj `gj log g j q . FOC

The HJB equation for the pre damage and technology jumps setting is given by δv " max ig,i d ,iκ min g j ,fm,h δ logprA d ´id sp1 ´Rq `rA g ´ig sR ´iκ q `δ log K

`ξI g pκq J ÿ j"1 π j g p1 ´gj q `ξI d pyq ÿ π m d p1 ´fm q . Consider allowing for carbon abatement in the following way. First, emissions are given by

where β t is the emissions intensity of dirty output. Note that ι t is the choice of abatement and can be solved for as a function of emissions:

The cost of abatement is given as a fraction of the dirty output, given by J " rA d K d `Ag K g sϕ 0 pιq ϕ 1 . Finally, market clearing gives us the output constraint C " A d K d `Ag K g ´Id ´Ig `Iκ ´J " rA d p1 ´ϕ0 p1 ´E λ t A d K d q ϕ 1 q ´id sK d `rA g p1 ´ϕ0 p1 ´E λ t A d K d q ϕ 1 q ´ig sK g ´iκ K.

With that, we can construct the HJB equation for this setting, which is given as

The FOC for investment and R&D are given by 0 " ´δprA d p1 ´ϕ0 p1 ´E λ t A d K d q ϕ 1 q ´id sK d `rA g p1 ´ϕ0 p1 ´E λ t A d K d q ϕ 1 q ´ig sK g ´iκ Kq

`Γd θ d p1 `θd i d q ´1V d 0 " ´δprA d p1 ´ϕ0 p1 ´E λ t A d K d q ϕ 1 q ´id sK d `rA g p1 ´ϕ0 p1 ´E λ t A d K d q ϕ 1 q ´ig sK g ´iκ Kq

`Γg θ g p1 `θg i g q ´1V g 0 " ´δprA d p1 ´ϕ0 p1 ´E λ t A d K d q ϕ 1 q ´id sK d `rA g p1 ´ϕ0 p1 ´E λ t A d K d q ϕ 1 q ´ig sK g ´iκ Kq ´1K

`ψ0 ψ 1 i ψ 1 ´1 κ exppψ 1 plog K ´log κqqV log κ 0 " δprA d p1 ´ϕ0 p1 ´E λ t A d K d q ϕ 1 q ´id sK d `rA g p1 ´ϕ0 p1 ´E λ t A d K d q ϕ 1 q ´ig sK g ´iκ Kq

Now we redefine the states to get it all correct. We use log K " logpK d `Kg q and R " Kg K d `Kg . Define dW g as the green capital shock and dW d as the dirty capital shock.
Having assumed the shocks are independent, then our cross-partial terms should drop out and we end up with the following: The FOC for investment and R&D are given by 0 " ´δprA d p1 ´Rq `Ag Rsp1 ´ϕ0 p1 ´E λ t A d Kp1 ´Rq q ϕ 1 q ´id p1 ´Rq ´ig R ´iκ q

`Γd θ d p1 `θd i d q ´1rv k ´p1 ´Rqv R s 0 " ´δprA d p1 ´Rq `Ag Rsp1 ´ϕ0 p1 ´E λ t A d Kp1 ´Rq q ϕ 1 q ´id p1 ´Rq ´ig R ´iκ q

`Γg θ g p1 `θg i g q ´1rv k `p1 ´Rqv R s 0 " ´δprA d p1 ´Rq `Ag Rsp1 ´ϕ0 p1 ´E λ t A d Kp1 ´Rq q ϕ 1 q ´id p1 ´Rq ´ig R ´iκ q ´1 `ψ0 ψ 1 i ψ 1 ´1 I exppψ 1 plog K ´log κqqv log Ig 0 " δprA d p1 ´Rq `Ag Rsp1 ´ϕ0 p1 ´E λ t A d Kp1 ´Rq q ϕ 1 q ´id p1 ´Rq ´ig R ´iκ q So far, the model presented here abstracts from jumps and uncertainty. We could introduce jumps as before with different realizations of γ m 3 and A j g possible, a jump, or jumps, in the value of λ t , and jump and diffusion misspecification concerns across the various different channels as outlined in the main text of the paper. To train the neural nets, we run 250000 epochs with a batch size of 32. The learning rates lr V " 10e ´5 and lr α " 10e ´5 and we use the ADAM optimizer proposed in Kingma and Ba (2014). The training is done on Google Colab with a total runtime of 2.29 hours. For the Han and E (2016) method solution, we use feedforward neural networks with 4 hidden layers of size 32 and tanh activation function to approximate the optimal controls. To train the neural nets, we run N = 100,000 epochs with a batch size of 2 7 . The infinite time horizon is approximated by r0, 1000s, and the time step size is 0.1. Algorithm 1 The DGM-PIA algorithm for solving the generic HJB (5.1) Require: a maximum number of epochs N , initial neural networks for the value function V px; θ 0 q and the control αpx; φ 0 q, learning rates for the value function and the control lr V , lr α 1: for n " 0, 1, 2, . . . , N do

Generate random samples tx m u M m"1 from the domain Ω according to ν 3:

Compute the value function loss in (5.1) using samples tx m u M m"1 :

´δV px m ; θ n q `Lαpxm;φnq V px m ; θ n q `f px m , αpx m ; φ n qq ı 2

Take a gradient descent step to update θ n`1 :

θ n`1 " θ n ´lr V ∇ θ L V pθ n ; tx m u M m"1 q 5:

Compute the control loss in (5.1) using samples tx m u M m"1 :

m"1 " L αpxm;φnq V px m ; θ n q `f px m , αpx m ; φ n qq ı 2

Take a gradient descent step to update φ n`1 :

φ n`1 " φ n ´lr α ∇ φ L α pφ n ; tx m u M m"1 q 7: end for

y ´tγ 1 `γ2 Ŷ `γm 3 p Ŷ ´ỹqu ¯. pjq R ¯p1 ´Rqσ d `´v

, numerical solutions are almost always unavailable for problems where the number of dimensions is greater than or equal to 4 when solving with standard finite difference or finite element methods. Incorporating model uncertainty, which introduces nonlinear endogenous responses to model uncertainty, such as exponential tilting to the distribution of expected future model outcomes, can further exacerbate the computational burden. As we will outline below, we provide a novel implementation of deep neural networks for deriving the numerical solutions to our HJB equations.Only very recently have deep learning and neural network methods been applied to solving dynamic economic models. Because of the ability of these methods to provide global solutions for problems with high dimensions where the "curse of dimensionality" can make computational solutions infeasible for other methods, or where significant non-linearities can cause other computational methodologies to fail, deep learning solution methods have been seen as a potentially "game-changing" toolset. By approximating the representative agent's value function and optimal controls with deep neural networks, the problems of exponentially increasing complexity from high-dimensional state spaces can be ameliorated due to the representation of functions in ology allows us to go beyond these settings, which is essential to analyzing our climate-economics model with model

, numerical solutions are almost always unavailable for problems where the number of dimensions is greater than or equal to 4 when solving with standard finite difference or finite element methods. Incorporating model uncertainty, which introduces nonlinear endogenous responses to model uncertainty, such as exponential tilting to the distribution of expected future model outcomes, can further exacerbate the computational burden. As we will outline below, we provide a novel implementation of deep neural networks for deriving the numerical solutions to our HJB equations. See Masson-Delmotte et al. (2021a) and Pörtner et al. (2022). See OECD (2019), Bouckaert et al. (2021), White House (2021), McKinsey Global Initiative (2022), and Jenkins et al. (2021).