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Concentrated Liquidity Provision: a Reinforcement Learning Perspective
Georgios Chionas, Charalampos Kleitsikas, Stefanos Leonardos, Leandro Sánchez-Betancourt, Carmine Ventre
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
Last extracted: 8/21/2026, 3:08:08 AM
Summary
This paper addresses the dynamic liquidity provision problem in Automated Market Makers (AMMs) with concentrated liquidity, such as UniswapV3. The authors formulate the problem as a stochastic impulse control problem, accounting for gas costs, impermanent loss, and inventory exposure. They employ reinforcement learning (RL) to solve this high-dimensional problem, demonstrating that learned policies adapt to market conditions, risk preferences, and transaction costs. The study highlights a trade-off between narrow positions (higher fees but higher risk/costs) and wide positions (lower fees but more stability), showing that RL agents can compress the left tail of the Profit and Loss distribution and avoid catastrophic outcomes.
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Georgios Chionas → affiliatedwith → University of Liverpool
confidence 99% · Georgios Chionas Affiliation: University of Liverpool
Carmine Ventre → affiliatedwith → King’s College London
confidence 99% · Carmine Ventre Affiliation: King’s College London
UniswapV3 → implements → Concentrated Liquidity
confidence 98% · the introduction of concentrated liquidity (CL) in UniswapV3 significantly expanded their action space
Reinforcement Learning → solves → Stochastic Impulse Control
confidence 95% · We formulate dynamic liquidity provision as a stochastic impulse control problem and use reinforcement learning (RL) to solve it
Narrow Positions → causes → Impermanent Loss
confidence 92% · Narrow-position agents achieve higher fee income in risk-neutral settings but incur higher Impermanent Loss (IL)
Gas Costs → influences → Rebalancing Frequency
confidence 90% · agents adapt their rebalancing frequency to blockchain frictions such as gas costs
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Abstract
Abstract:Automated market makers (AMMs) are a cornerstone of decentralised finance (DeFi). Constant product markets with concentrated liquidity, such as UniswapV3, are now a well-established design. In these markets, liquidity providers (LPs) face a sequential decision problem: they must decide when to rebalance their positions and which price ranges to allocate capital to as market conditions evolve. We formulate dynamic liquidity provision as a stochastic impulse control problem and use reinforcement learning (RL) to solve it, focusing on providing interpretable solutions. We show that learned policies exhibit rich state-dependent behaviour, allocating liquidity according to mispricing, rebalancing costs, uncertainty, inventory exposure, and heterogeneous risk preferences. These behaviours help compress the left tail of the Profit and Loss (PnL) distribution and avoid catastrophic outcomes under high uncertainty. Finally, we benchmark the RL agents against baseline and sophisticated agents from the AMM microstructure literature and analyse their performance.
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- Source: https://arxiv.org/abs/2608.19389v1
- Canonical: https://arxiv.org/abs/2608.19389v1
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Concentrated Liquidity Provision: a Reinforcement Learning Perspective Georgios Chionas Affiliation: University of Liverpool , Liverpool , UK Note: Authors contributed equally to the paper. email: g.chionas@liverpool.ac.uk , Charalampos Kleitsikas Affiliation: King’s College London , London , UK email: charalampos.kleitsikas@kcl.ac.uk , Stefanos Leonardos Affiliation: King’s College London , London , UK email: stefanos.leonardos@kcl.ac.uk , Leandro Sánchez-Betancourt Affiliation: University of Oxford , Oxford , UK email: leandro.sanchezbetancourt@maths.ox.ac.uk and Carmine Ventre Affiliation: King’s College London , London , UK email: carmine.ventre@kcl.ac.uk Abstract. Automated market makers (AMMs) are a cornerstone of decentralised finance (DeFi). Constant product markets with concentrated liquidity, such as UniswapV3, are now a well-established design. In these markets, liquidity providers (LPs) face a sequential decision problem: they must decide when to rebalance their positions and which price ranges to allocate capital to as market conditions evolve. We formulate dynamic liquidity provision as a stochastic impulse control problem and use reinforcement learning (RL) to solve it, focusing on providing interpretable solutions. We show that learned policies exhibit rich state-dependent behaviour, allocating liquidity according to mispricing, rebalancing costs, uncertainty, inventory exposure, and heterogeneous risk preferences. These behaviours help compress the left tail of the Profit and Loss (PnL) distribution and avoid catastrophic outcomes under high uncertainty. Finally, we benchmark the RL agents against baseline and sophisticated agents from the AMM microstructure literature and analyse their performance. 1. Introduction Decentralised finance (DeFi) has introduced financial exchanges in which trading is executed by smart contracts. The most prominent examples are automated market makers (AMMs), popularised by Uniswap. Most AMMs are implemented as constant product markets (CPMs), a special class of constant function market makers (3), where prices are determined algorithmically by an invariant function and the reserves locked in the liquidity pool. In these markets, liquidity providers (LPs) deposit capital into a pool and earn fees from users trading against it. While LPs were largely passive in designs such as UniswapV2, the introduction of concentrated liquidity (CL) in UniswapV3 significantly expanded their action space, allowing them to allocate capital to specific price ranges (2). CL changed the economics of liquidity provision. By concentrating capital around the current price, LPs can earn more trading fees, but narrower positions stop earning fees once the price exits the chosen interval. Hence, LPs face a trade-off between wide positions, which cover more possible prices but earn lower fees, and narrow positions, which are more profitable while active but expose the LP to greater concentration risk. As prices evolve, LPs can dynamically move their capital and create new positions, but doing so incurs blockchain frictions, such as gas fees, and may also require additional trades due to changes in the token composition of the new position. These costs must therefore be weighed against the expected benefits of rebalancing. Thus, we formulate optimal liquidity provision as a stochastic impulse control problem, which includes stochastic order arrivals, price dynamics, and gas costs. Since Hamilton-Jacobi-Bellman-Quasi-Variational Inequality (HJBQVI)-style methods are intractable in such realistic high-dimensional settings, we employ reinforcement learning (RL) to study dynamic liquidity provision strategies. Our objective is to provide a tractable computational representation of state-dependent policies in settings where closed-form impulse-control solutions are unavailable. Our contribution. We make the following contributions. • We show that learned policies generate rich and interpretable liquidity provision strategies: agents adapt their rebalancing frequency to blockchain frictions such as gas costs, while choosing position width and asymmetry in response to their risk profile, inventory holdings, existing deployed position, and the degree and direction of arbitrage pressure in the market. • We demonstrate a trade-off between narrow and flexible learned strategies. Narrow-position agents achieve higher fee income in risk-neutral settings but incur higher Impermanent Loss (IL) and gas costs due to more frequent rebalancing, whereas wider-position agents require less active risk management when inventory risk is considered. Related Work: Academic work on AMMs dates back to (22), but AMMs saw widespread adoption more recently as a computationally cheaper alternative to limit order books in blockchain-based exchanges. Their use in decentralised exchanges was first proposed in (9; 20), the latter introducing a CPM market maker. This mechanism was first analysed by (4), who identified conditions under which such markets can closely track the price of an external reference market. Subsequently, (3) studied the broader class of constant function market makers (CFMMs) and showed that they can incentivise participants to reveal external market prices. Simple AMMs such as UniswapV2 suffer from capital inefficiency. Newer protocols such as UniswapV3 (2) address this issue through CL, which allows LPs to specify the price ranges over which their liquidity is active and gives them more granular control over fee generation. Our work contributes to the work on profitable strategies for LPs in CPMs with CL. In fact, empirical studies such as (19; 14; 21) show that, currently, LPs are on average incurring a loss. (18) study the risk-return trade-off faced by LPs in UniswapV3, while (16) analyse optimal static liquidity provision and show how contract design choices, affect LP profits and trader gas costs. More recently, (10) decompose LP losses in concentrated-liquidity CPMs into convexity costs, arising from the convexity of the pool invariant, and opportunity costs, arising from locking assets in the pool instead of investing them in a risk-free account. For dynamic liquidity provision, (11) derive a closed-form solution for frictionless LP allocations, abstracting gas costs and latency, (17) use approximation techniques to obtain profitable strategies, while (13) introduce a broader family of AMMs, in which LPs participate in price discovery. RL works for CL such as (23; 24) employ training in historical data, while (6) uses a hybrid setup. Contributing to these efforts, our work focuses primarily on a model-based approach that replicates the full CL mechanics instead, allowing for state-dependent and interpretable analyses in multiple economic regimes. 2. Preliminaries Constant Product Markets. Let X be a numéraire asset (e.g. USDC) and Y be a risky asset (e.g. ETH). A liquidity pool for the pair of assets X,YX,Y defines the marginal exchange rate of asset Y in units of asset X, denoted by Z. The state of a CPM is described by the invariant function f(qX,qY)=qXqY=κ2f(q^X,q^Y)=q^Xq^Y=κ^2, where qXq^X and qYq^Y are the reserves of assets X and Y and κ is the liquidity parameter (depth) of the pool, a measure of its available liquidity. The invariant function f connects the state of the CPM before and after a trade is executed. The marginal exchange rate, Z, in a CPM is defined as Z:=qX/qYZ:=q^X/q^Y. In other words, the marginal exchange rate Z is the price quoted by the CPM at its current state for an infinitesimal trade. Moreover, the CPM charges a fee, τ, proportional to the size of the trade and thus, 1−τ1-τ is the net percentage of the trade after the trading fee. Consider now a liquidity taker’s (LT) trade of size Δx x (i.e., the LT provides Δx x units of asset X in exchange for asset Y). The LT receives Δy′ y units of asset Y such that f(qX+(1−τ)Δx,qY−Δy′)=f(qX,qY)f(q^X+(1-τ) x,q^Y- y )=f(q^X,q^Y), which, for a CPM, implies that Δy′=qY−qXqYqX+(1−τ)Δx y =q^Y- q^Xq^Yq^X+(1-τ) x. Similarly, when the LT sells Δy y units of asset Y, she receives Δx′ x units of asset X such that f(qX−Δx′,qY+(1−τ)Δy)=f(qX,qY)f(q^X- x ,q^Y+(1-τ) y)=f(q^X,q^Y), which, for a CPM implies that Δx′=qX−qXqYqY+(1−τ)Δy x =q^X- q^Xq^Yq^Y+(1-τ) y. Concentrated Liquidity. In CPMs with CL, LPs can concentrate their liquidity over a specific range of rates (Zℓ,Zu](Z ,Z^u]. The rates ZℓZ and ZuZ^u take values from a discrete grid (1) Z−N=Z¯,Z−N+1,…,Z0,…,ZN−1,ZN=Z¯\Z^-N= Z,Z^-N+1,…,Z^0,…,Z^N-1,Z^N= Z\ where N∈ℕ0N _0. The smallest possible liquidity range is defined by two consecutive ticks (Zi,Zi+1](Z^i,Z^i+1] and it is called a tick range. Within each tick range, the CPM with CL behaves locally as a CPM. When an LP opens a new position, she specifies a range of rates (Zℓ,Zu](Z ,Z^u] and a liquidity depth κ~ κ (or liquidity units), which determines the quantities x and y in assets X and Y respectively that the LP should provide, namely (2) (x,y)=(0,κ~((Zℓ)−1/2−(Zu)−1/2)),Z≤Zℓ,(κ~(Z1/2−(Zℓ)1/2),κ~(Z−1/2−(Zu)−1/2)),Zℓ<Z≤Zu,(κ~((Zu)1/2−(Zℓ)1/2), 0),Z>Zu. (x,y)= cases (0,\, κ ((Z )^-1/2-(Z^u)^-1/2 ) ),&Z≤ Z ,\\[4.62497pt] ( κ (Z^1/2-(Z )^1/2 ),\, κ (Z^-1/2-(Z^u)^-1/2 ) ),&Z <Z≤ Z^u,\\[4.62497pt] ( κ ((Z^u)^1/2-(Z )^1/2 ),\,0 ),&Z>Z^u. cases To obtain the total liquidity depth in a tick range, one needs to sum the depths of the individual liquidity positions in the same tick range. The formulas in (2) define a reparametrization whereby the state of the CPM with CL can be fully described by the total liquidity depth in each tick range and the marginal exchange rate. This representation is more convenient since an LP action changes the liquidity depths of the respective ticks, while an LT action changes the marginal exchange rate Z.11 1 For a more thorough treatment on the mechanics of CPMs with CL, see (2; 11) Lastly, the trading fees are stored in each tick range, and are distributed pro-rata to each LP. For example, if the LP’s position with depth κ~ κ is in a tick range with total depth κ, then for every liquidity taking trade that pays in fees an amount of p, the LP earns the amount of p~=κ~κp 1Zℓ<Z≤Zu p= κp\, 1_Z <Z≤ Z^u. 3. An impulse control formulation Having introduced the mechanics of CPMs with CL, we now turn to the dynamic liquidity provision problem faced by an individual LP. In this problem, the LP observes the market conditions, which are defined in the next paragraph, and decides when to close her position and open a new one, i.e. when to rebalance, and what the new position should be. When the LP rebalances, she collects the accumulated fees from her last position and then reinvests her initial wealth to the new position. Moreover, each rebalancing incurs a fixed transaction cost, which captures the gas cost paid for interacting with the blockchain, and for that reason, we formulate the problem as an impulse control problem.22 2 The difference between an impulse control problem and a continuous stochastic control problem is that, in the former case, the controller takes actions occasionally (interventions) because of the transaction costs, while in the latter case, the controller takes actions continuously; cf. (15). More formally, the controller is an LP who, at a sequence of intervention times, chooses a range of ticks from (1) to post her liquidity, with Lt=(Lti)i=−N−1L_t=(L^i_t)_i=-N^N-1 denoting the existing liquidity as a vector in the pool, excluding the agent.33 3 For simplicity we take this liquidity vector to be deterministic; however, our RL implementation supports Markovian stochastic counterparts. We consider that the price formation of the risky asset Y is exogenous to the CPM with CL and we denote its external midprice at time t as StS_t. The external midprice StS_t follows the dynamics (3) dSt=μStdt+σStdWt,S0>0,dS_t=μ S_tdt+σ S_tdW_t, S_0>0, with drift μ, volatility σ>0σ>0, and where WtW_t is a Brownian motion. Concurrently, at each time t, the CPM with CL quotes its own price for the risky asset Y, the marginal exchange rate ZtZ_t. Thus, at each time t, there are two prices quoted for the risky asset Y; the marginal exchange rate (the price quoted by the AMM) denoted by ZtZ_t, and the external price, denoted by StS_t. In terms of the dynamics of the marginal exchange rate Z, we assume it lives within the grid (1). In particular, when it is at tick i, an LT trade can only move it from ZiZ^i to Zi+1Z^i+1 or Zi−1Z^i-1 provided it stays in the grid. Hence, the marginal exchange rate Z evolves solely through LT trades, which constitute the order flow. More precisely, let +,iN^+,i and −,iN^-,i denote the counting processes of trades moving the marginal exchange rate from ZiZ^i to Zi+1Z^i+1 and from Zi+1Z^i+1 to ZiZ^i, respectively. The total number of up/down movements ±N^± is given by t±=∑it±,iN^±_t= _iN^±,i_t. We work under a probability measure ℙP under which the counting processes t±N^±_t (counting the number of LT trades that move the marginal exchange rate up/down in the grid (1)) have intensities (4) λt±=maxλ0,λ1±λ2(St−Zt−),λ^±_t= \ _0,\, _1± _2(S_t-Z_t-) \, where λ0>0 _0>0 and λ1,λ2≥0 _1, _2≥ 0. The term λ1 _1 captures the baseline order flow (i.e. the noise traders), while the term ±λ2(St−Zt−)± _2(S_t-Z_t-) captures arbitrage pressure. The term λ0>0 _0>0 is a safeguard parameter to prevent intensities from becoming negative. This model for intensities has been studied previously in (5). The LP’s filtration (the LP’s observable information when taking new actions) will be generated by StS_t and t±N_t^±, since t±,iN_t^±,i is determined by t±N_t^± and the marginal exchange rate ZtZ_t. Hence, the LP’s control is an impulse strategy (5) α:=(rn,Znℓ,Znu)n≥1, α:= (r_n,Z _n,Z^u_n )_n≥ 1, where (rn)n≥1(r_n)_n≥ 1 is an increasing sequence of stopping times and, at each intervention time rnr_n, the LP chooses a lower and an upper tick satisfying Z¯≤Znℓ<Znu≤Z¯ Z≤ Z _n<Z^u_n≤ Z. The pair (Znℓ,Znu](Z _n,Z^u_n] specifies the liquidity range until the next intervention. Equivalently, if iti_t denotes the active tick at time t, the LP’s liquidity vector is (6) L~α,it=κ~αt 1Zℓt<Zit≤Zit+1≤Zut,i=−N,…,N−1, L^α,i_t= κ^α_t\,1_\Z _t<Z^i_t≤ Z^i_t+1≤ Z^u_t\, i=-N,…,N-1, where κ~tα κ^α_t is the liquidity depth deployed in the selected range, determined by the LP’s available capital at rebalancing and the value per liquidity unit in that range. We then define the total liquidity depth in tick range (Zi,Zi+1](Z^i,Z^i+1] by κtα,i:=Lti+L~tα,iκ^α,i_t:=L^i_t+ L^α,i_t. The fees collected by the LP are proportional to her share of liquidity in the current tick. Thus, for a fee rate τ∈(0,1)τ∈(0,1), the cumulative fee process is Ptα P^α_t =∑i=−N−1[∫0tL~s−α,iκs−α,i(τ1−τ)Δxsids+,i+∫0tL~s−α,iκs−α,i(τ1−τ)SsΔysids−,i] = _i=-N^N-1 [ _0^t L^α,i_s-κ^α,i_s- ( τ1-τ ) x^i_s\,dN^+,i_s+ _0^t L^α,i_s-κ^α,i_s- ( τ1-τ )S_s\, y^i_s\,dN^-,i_s ] where Δxsi x^i_s and Δysi y^i_s are the trade sizes required to move the marginal exchange rate by one tick in the corresponding direction. The dependence on the chosen width enters through L~α,i L^α,i: for fixed amounts of capital, a narrower range creates a larger liquidity share in the active tick, while a wider range lowers the share but remains active in more price states. In our setup, each intervention incurs a fixed transaction cost; this is a key distinction from the frictionless case in (11). When the LP changes her position at time rnr_n, she pays a fixed gas cost. We write the cumulative cost up to time t as Ctα=∑rn≤tgC^α_t= _r_n≤ tg, where g>0g>0 is the gas cost for each new LP action. Let VtαV^α_t be the value of the LP’s current position in units of asset X at the current external price, i.e. Vtα=xtα+Stytα.V^α_t=x^α_t+S_ty^α_t\,. Relative to holding the initial token amounts outside the pool, the realised IL term is ILtα=−[Vtα−(x0α+Sty0α)]IL^α_t=- [V^α_t- (x^α_0+S_ty^α_0 ) ]. The risk-neutral LP solves (7) supα∈ℙα[PTα−ILTα−CTα]. _α E^P^α [P^α_T-IL^α_T-C^α_T ]. where PTαP^α_T denotes collected fees, ILTαIL^α_T denotes impermanent loss, and CTαC^α_T denotes rebalancing costs. The problem therefore captures the main trade-off faced by a LP in a CPM with CL; narrow ranges increase the fee share conditional on being active, but expose the position to concentration risk and more frequent rebalancing, while wider ranges lower the fee share but reduce the probability of falling out of range. The impulse structure makes the rebalancing friction explicit. Risk Aversion through Running Inventory Penalty. Besides the risk neutral case defined in (7), we will modulate the performance of the LP under different levels of risk aversion. The simplest way to introduce risk aversion is through an inventory penalty. We therefore extend the objective function in (7) as follows (8) supα∈ℙα[PTα−ILTα−CTα−ϕ∫0T(ytα−y^)2t], _α E^P^α [P^α_T-IL^α_T-C^α_T-φ _0^T(y_t^α- y)^2dt ], where the constant ϕ∈ℝ+φ ^+ determine the running penalty deviating from some level y y (see (13)). In particular, the LP’s performance criterion (8) aims to maintain a target reserve level y y. Note that without a target reserve y y, the performance criterion would essentially force the LP to liquidate the risky asset that she holds. It is well-known in the algorithmic trading literature (as shown in (12)) that the running penalty term in (8) can be interpreted as arising from the agent’s ambiguity aversion with respect to the external price StS_t. As we show in Section 4, the per-step running inventory penalty is the main component pushing the LP towards wider positions. Theoretical Results. Two structural properties of the LP’s PnL follow directly from the CL mechanics of Section 2, and both are used to interpret the learned policies in Section 4. Lemma 3.1 (Fee revenue is linear in own liquidity). Under one-tick jumps, the fees collected by an individual LP are linear in κ~ κ and independent of the other LPs liquidity LiL^i in the pool. Proof Sketch. Moving from ZiZ^i to Zi+1Z^i+1 requires an after-fee trade of size Δxi=κi((Zi+1)1/2−(Zi)1/2) x^i=κ^i ((Z^i+1)^1/2-(Z^i)^1/2 ) in X, and symmetrically Δyi=κi((Zi)−1/2−(Zi+1)−1/2) y^i=κ^i ((Z^i)^-1/2-(Z^i+1)^-1/2 ) in Y in the opposite direction: the volume needed to cross a tick is proportional to the total depth κiκ^i of that tick. Since the LP earns the pro-rata share κ~i/κi κ^i/κ^i of the resulting fee, κiκ^i cancels in the LP’s fee revenue process PtαP^α_t, leaving an integrand proportional to κ~s−i κ^i_s- and free of LiL^i. Deeper liquidity in a range absorbs more volume per tick crossed, and the pro-rata rule returns to the LP exactly her share of that extra volume. ∎ Lemma 3.2 (IL is linear in own liquidity). For a position of depth κ~ κ in (Zℓ,Zu](Z ,Z^u], the IL between posting and withdrawal is linear in κ~ κ. Proof Sketch. Every branch of (2) is positively homogeneous of degree one in κ~ κ, so xtx_t, yty_t, and hence ILt=−[xt+ytSt−(x0+y0St)]IL_t=- [x_t+y_tS_t-(x_0+y_0S_t) ], are linear in κ~ κ, whether or not the position is in range. While Zt∈(Zℓ,Zu]Z_t∈(Z ,Z^u] the boundary terms cancel and ILt=κ~[Z01/2−Zt1/2+St(Z0−1/2−Zt−1/2)]IL_t= κ [Z_0^1/2-Z_t^1/2+S_t (Z_0^-1/2-Z_t^-1/2 ) ], so the range enters only through κ~ κ. For fixed capital, a narrower range implies a larger κ~ κ, which makes the IL curve locally steeper, showing the cost side of concentration. ∎ From Lemma 3.1 and Lemma 3.2, we thus have: Theorem 3.3 (Affinity of the LP’s PnL in liquidity). The LP’s PnL PTα−ILTα−CTαP^α_T-IL^α_T-C^α_T is affine in κ~ κ: it is linear through the fee and IL terms, while the gas cost CTαC^α_T enters as a κ~ κ-independent constant. These results assume an LP whose liquidity is small relative to the pool, so that her actions do not affect the liquidity taking activity in the future. Learning Agents as Dynamic LPs. The impulse control problems in (7) and (8) are analytically intractable in realistic settings, since the LP must jointly decide when to intervene and where to deploy liquidity under stochastic prices, order arrivals, gas costs, inventory exposure, and a discrete tick grid. RL provides a natural numerical approach for learning admissible strategies of the form (5), with rewards defined as sample-path counterparts of the objectives inside the supremum. For a risk-neutral LP, the reward is realised PnL from fees, IL, and rebalancing costs, corresponding to (7); for a risk-averse LP, we use the inventory-penalised criterion in (8), where ϕφ controls the degree of risk aversion. The mechanics and frictions of CL, together with Theorem 3.3, imply the key economic trade-off studied in our experiments. Narrower ranges increase fee income conditional on being active but expose the LP to concentration risk and costly rebalancing; wider ranges remain active over more price states and reduce out-of-range and inventory risk, but dilute liquidity across ticks and lower the LP’s fee share per tick. In the risk-neutral case, the linearity of fee revenue and IL in the LP’s own liquidity implies a strong incentive to concentrate liquidity in the expected most profitable tick range, providing a theoretical benchmark for studying risk aversion, inventory penalties, and rebalancing frictions. We therefore consider two action parametrisations. A flexible agent that freely chooses the lower and upper bounds (Znℓ,Znu](Z _n,Z^u_n] and a narrow agent with the minimum fixed position width of two ticks (Zni,Zni+2](Z^i_n,Z^i+2_n] that learns only where to place her liquidity. The narrow specification isolates the value of learning where to post liquidity from the value of learning how wide the position should be. Decisions are made only at intervention times, consistent with the impulse-control formulation and with the fact that continuous rebalancing is unrealistic under blockchain frictions. 4. Experimental Results Based on the above, in the following experiments, we implement two RL agents with the Proximal Policy Optimisation (PPO) algorithm, namely the PPO and PPO_narrow. For the action space, when (Zn+1ℓ,Zn+1u]=(Znℓ,Znu](Z _n+1,Z^u_n+1]=(Z _n,Z^u_n], the agent chooses to hold her previous position, and thus no extra gas costs are charged. For the first action, if (Z0ℓ,Z0u]=(0,0](Z_0 ,Z_0^u]=(0,0] the agent chooses not to provide liquidity. The observation space contains the main variables needed for liquidity provision: mispricing St−ZtS_t-Z_t, distances from the current pool price to the position boundaries, time, gas cost, cumulative fee income, and the LP’s current holdings in the risky and numéraire assets. In our experiments, each trained policy and benchmark strategy is evaluated on 1000 independent simulations using random seeds distinct from those used during training. We compare strategies across scenarios and profile the learned policies as state-dependent functions, relating their behaviour to empirical liquidity-provision patterns. The aim is to understand how learned LP policies adapt their position width, asymmetry, and rebalancing frequency to the LP’s risk appetite and to the stochastic and blockchain-specific properties of the environment. Experimental Setup: We train and evaluate PPO and PPO_narrow, in the stochastic environment of Section 3. The external price of the risky asset Y (e.g. ETH) follows the geometric Brownian motion in (3). The liquidity-taking arrivals follow the linear intensity model in (4), for which we set λ1=15 _1=15 and λ2=4000 _2=4000. These values represent an order-flow intensity that is mainly driven by arbitrage pressure from deviations between the pool price and the external price. Equations (3) and (4) are the two sources of stochasticity in the experiments. For the external price, we set S0=1000S_0=1000, dt=0.001dt=0.001 (assuming T=1T=1), μ=0μ=0, and vary volatility over σ∈0.005,0.01,0.015,0.02,0.025,0.03σ∈\0.005,0.01,0.015,0.02,0.025,0.03\. Higher σ increases the likelihood of sharp price movements, IL, and positions moving out of range. The main blockchain friction is the gas cost paid at each rebalance, for which we consider g∈2,4,6g∈\2,4,6\, corresponding to low, medium, and high operational costs. We train both risk-neutral and risk-averse LPs: risk-neutral agents use the cumulative PnL reward in (7), while risk-averse agents use the inventory-penalised reward in (8), with ϕ∈20,50,80φ∈\20,50,80\ controlling the strength of inventory risk aversion. As observed in practice, the pool price ZtZ_t follows closely StS_t due to arbitrage. Therefore, we set the parameters of StS_t within the range of empirical estimations calculated in (11). The rest of the parameters were calibrated to be economically sensible based on the above within our model-based setup. The action range is restricted to [−15,15][-15,15] ticks around the current tick. Each LP starts with 1000 numéraire units, and the proportional fee tier is set to τ=0.3%τ=0.3\%, a standard UniswapV3 fee tier for token pairs. We benchmark PPO and PPO_narrow against rule-based and model-based strategies operating within the same [−15,15][-15,15] tick window. The DeployOnceAgent deploys capital at the start and never rebalances. We use two variants: DeployNarrow, which deploys over [−1,1][-1,1] around the initial pool price, and DeployWide, which deploys over [−15,15][-15,15]. The former represents the narrow single-shot allocation motivated by the concentration incentives in Section 3, while the latter maximises the probability of remaining active throughout the simulation. The ArrivalRebalanceAgent also posts liquidity over [−1,1][-1,1], but rebalances after every NarrN_arr liquidity-taking arrivals, adding a simple timing rule relative to DeployNarrow. Finally, we compare against the CarteaDrissiMongaAgent (CDM), a model-based closed-form strategy for concentrated AMMs based on (11). Each trained RL agent corresponds to an independent environment instance, giving 72 scenarios in total (6⋅3⋅46· 3· 4) across σ, g, and risk-aversion settings. To ensure statistical robustness across the training variability of the RL algorithms, each scenario is assessed across 10 different training seeds. To reduce policy-gradient variance and expose policies to a broad state distribution, we collect rollouts using Ntraj,train=200N_traj,train=200 parallel trajectories advanced synchronously. Each trajectory has Nsteps=1000N_steps=1000 environment steps, with intervention times every Kdec=100K_dec=100 steps. Thus, each rollout contains Ntraj,trainNsteps/Kdec=2000N_traj,trainN_steps/K_dec=2000 agent-decision transitions. We set Etrain=300E_train=300, yielding a total training budget of 6×1056× 10^5 Table 1. Mean PnL and 5%5\% CVaR in USDC over 1000 common evaluation trajectories. Values following ± denote the standard deviation across 10 independently trained policies. Bold marks the best RL and benchmark method. p tests their difference, treating training seeds as replication units and additionally propagating trajectory sampling error. σ=0.01,g=2σ=0.01,\ g=2 σ=0.02,g=4σ=0.02,\ g=4 Agent PnL CVaR PnL CVaR DeployNarrow 15.85 -14.52 6.48 -35.37 DeployWide 12.78 -15.11 4.09 -36.48 ArrivalRebalance 49.72 6.86 5.69 -36.23 CDM 35.18 11.55 15.01 -26.23 Risk neutral PPO 42.31± 0.97 6.08± 0.97 5.33± 0.65 -36.51± 0.77 PPO_narrow 49.91± 0.38 9.15± 0.65 6.51± 0.20 -35.69± 0.42 p 0.81 0.0014 <10−10<10^-10 <10−4<10^-4 Risk averse (ϕ=50φ=50) PPO 30.36± 1.60 3.07± 1.56 1.49± 0.16 -9.14± 0.46 PPO_narrow 46.11± 0.67 9.32± 0.74 2.27± 0.40 -8.86± 0.87 p <10−10<10^-10 0.04 <10−10<10^-10 <10−4<10^-4 Table 2. Under adversarial conditions (σ=0.03σ=0.03, g=6g=6), the risk-averse RL agents learn mainly not to deploy. σ=0.03,g=6σ=0.03,\ g=6 Agent PnL CVaR DeployNarrow 0.10 -56.17 DeployWide -2.12 -57.33 ArrivalRebalance -5.38 -62.64 CDM -3.10 -56.93 Risk neutral Risk averse (ϕ=50φ=50) PPO -1.42± 0.80 -54.78± 6.61 0.00± 0.04 -0.68± 1.47 PPO_narrow -0.06± 0.56 -52.68± 7.73 -0.01± 0.03 -0.16± 0.51 p 0.64 0.19 0.88 <10−4<10^-4 agent-decision transitions for the PPO optimisation objective, after which average returns stabilise without material performance gains from further training. For the RL architecture we set: learning rate =3×10−4=3× 10^-4, batch size =200=200, γ=0.99γ=0.99, GAE λ=0.95λ=0.95, clip range =0.15=0.15, entropy =0.03=0.03 and two hidden MLP layers of 256 units. Performance of Learned Policies: Tables 1 and 2 compare the performance of the RL agents and the benchmark strategies across Figure 1. Differences in the mean decomposed PnL components between the RL agents across volatility, by risk profile (rows) and gas cost (columns). ΔNetPnL=ΔFees+(−ΔIL)+(−ΔGas)=PPO_narrow(PnL)−PPO(PnL) \ PnL= +(- )+(- )=PPO\_narrow(PnL)-PPO(PnL) denotes the difference in mean PnL and is shown by the diamond marker. A positive ΔFees (green), negative −ΔIL- (purple), and negative −ΔGas- (orange) indicate that PPO_narrow earns more fees, incurs greater IL, and pays more gas than PPO, respectively. Each quantity is averaged over 1,000 trajectories. Missing bars indicate that one or both agents did not deploy. representative market settings chosen as a coherent regime ladder, corresponding to low, medium, and high combinations of σ and g, under both risk neutrality and an intermediate level of risk aversion (ϕ=50φ=50). The significance tests for PnL use a hierarchical paired t-test and for CVaR, a paired bootstrap over both seeds and trajectories (10410^4 replicates). CDM does not directly account gas costs. Thus, for an optimistic frictionless analytical benchmark version against the RL agents, we consider that the strategy rebalances every KdecK_dec free of costs. Its risk aversion lies in its formulation that depends on σ. For the ArrivalRebalance agent, we finetuned NarrN_arr for each scenario from a gridsearch ranging from Narr=10N_arr=10 to 800800, optimising both PnL and CVaR (Conditional Value at Risk). Under the low-volatility, low-gas setting, for the risk neutral case, PPO_narrow achieves similar PnL with ArrivalRebalance, however with higher CVaR than it, resulting in a better overall strategy. CDM achieves slightly improved CVaR than the risk averse agent, however the PnL of PPO_narrow is better even though gas costs were paid for its positions. As volatility and gas costs increase, profitability declines across all strategies, and the mean-PnL advantage of the RL agents becomes less pronounced. CDM outperforms (with significant p values) in PnL both risk-neutral and risk averse agents under the medium volatility condition. However, the risk aversion case clearly demonstrates the benefits of the RL agents under these market conditions. At σ=0.02σ=0.02 and g=4g=4, the risk-averse RL agents achieve substantially improved CVaR than CDM. Relative to their risk-neutral counterparts under the same market conditions, they sacrifice approximately 4 USDC (a) Mean Spreads in ticks of the two RL agents. The shaded area corresponds to the ±1± 1 standard deviation around the mean. (b) Rebalancing Rate (%) of the RL agents. Figure 2. Position width and rebalancing behaviour of PPO and PPO_narrow. Panel (2(a)) shows mean quoted width, while Panel (2(b)) shows per-decision rebalance rates. Labels and Legends as in Figure 1. in mean PnL while improving CVaR by approximately 27 USDC, illustrating their ability to trade modest expected profitability for substantially stronger downside protection, reducing extreme losses as market conditions deteriorate. At σ=0.03σ=0.03 and g=6g=6, the risk-averse agents learn mostly not to deploy liquidity, approximating a "DoNothing" benchmark that would report PnL=0=0 and CVaR=0=0. Thus, under sufficiently adverse conditions, the economically meaningful learned behaviour is endogenous market exit rather than superior active liquidity management. Finally, the relatively small standard deviations across independently trained policies indicate stable training outcomes. As discussed, PPO_narrow generally achieves better performance than PPO because its fixed narrow range gives it the theoretically favourable concentration implied by Theorem 3.3. However, by decomposing the PnL of the agents, we notice that this advantage comes with higher IL and gas costs (Figure 1), since narrow positions require more frequent rebalancing. Therefore, when risk aversion is taken into account; by learning wider positions, PPO has lower fee intensity but less active risk management. This mechanism is confirmed in Figure 2. PPO systematically increases its average width as volatility, gas costs, and risk aversion rise, provided that Figure 3. Liquidity provision behaviour of all strategies for one random simulation from the evaluation trajectories (scenario: σ=0.01,g=2,σ=0.01,g=2, risk neutral agents). The red thin vertical lines show when agents rebalance. Figure 4. Illustration of the empirical CL shapes around pool prices (left) in our model-based environment (right). Left figure shows the normalised liquidity distribution around the current tick for Optimism (red), Arbitrum (black) and Ethereum (purple) as observed from real on-chain data calculated in (1). Right figure shows a superposition of all four PPO risk-profile agents, PPO_narrow, and a wide background liquidity. Each agent’s position is her reported average width for σ=0.02σ=0.02 and g=2g=2 with indicative relative wealths. the average rebalancing rate is not near zero. Both agents reduce their rebalancing frequency across these dimensions. PPO_narrow rebalances more often on average, reflecting the greater management required by fixed concentrated positions. Note that PPO_narrow incurs lower IL and gas costs in the few instances where PPO had a higher rebalance rate, as shown in Figure 2(b). Figure 3 provides an example of how agents provide liquidity, for high level intuition. A Qualitative Illustration of Aggregate Liquidity Shapes: The results of Figure 2(a) show that the learned width of the PPO agent varies systematically with the LP’s risk profile and the frictions of the environment. This provides a natural qualitative interpretation of the liquidity concentration observed empirically around the pool prices in real concentrated AMM pools as the aggregate result of heterogeneous LPs with different risk appetites. (a) The agent repositions liquidity closer to the external price due to the expected arbitrage flow. (b) The agent’s size of the chosen width depends on the amount of risky token held, functioning as a risk-management device. (c) For S−Z>0S-Z>0 the agent chooses to hold her position to get rid of the risky asset from the expected incoming trades. Figure 5. The full action response that a risk-averse PPO agent has for fixed wealth, with ϕ=50φ=50 for gas_cost=4gas\_cost=4 and σ=0.01σ=0.01 at t=0.4t=0.4, across two important dimensions: the amount of the risky asset she holds (e.g. ETH), and the adversarial signal (mispricing). It is assumed that the current action of the agent is a symmetrical narrow position (-1,1) around the current tick. The shaded areas in the first two plots are where the agent holds her existing position and the chosen actions can be ignored. Figure 6. Action responses of the risk-neutral PPO_narrow agent for σ=0.01σ=0.01 with gas cost g=2g=2 (left panel) and g=6g=6 (right panel) at time t=0.4t=0.4. The x-axis shows the mispricing (USDC), while the y-axis shows the policy output, i.e. the chosen range centre. The colour bar indicates the current centre (ticks). Both the chosen action and current state are expressed as centre offsets. Dark purple denotes a position well below the current tick, bright yellow well above it, and teal positions partially in range. Solid line segments indicate rebalancing; dashed segments indicate that the agent ignores its centre output and holds its position. Figure 4 illustrates this mechanism by superimposing the average positions of PPO agents trained with different risk profiles and a wide passive existing background liquidity component. The resulting aggregate allocation forms a bell-shaped liquidity profile around the current pool price, providing a model-based illustration of the real liquidity concentration patterns observed in DeFi pools. Flatter empirical distributions, such as the one observed on Ethereum, can therefore be associated with stronger risk aversion, higher gas costs or heterogeneous individual LP objectives. Analogously, the liquidity distribution of Optimism and Arbitrum can be thought qualitatively as the superposition of LPs that are more risk seeking in their approach of capturing the fee share of incoming trades by narrowing their quotes, resembling our theory informed RL agent. In this sense, the shape of liquidity in concentrated AMMs reflects both the local fee incentives of CL and the idiosyncratic risk-management behaviours of the LP population. Profiling Learned Policies as State-Dependent Functions: Figure 5 and 6 examine the policies as state functions to explain how the agents condition their actions on the observed state. Figure 6 shows the response of the optimal risk-neutral PPO_narrow agent. Since this agent has a fixed two-tick width, the relevant decisions are where to centre the position and whether or not to rebalance. In the low gas-cost case (left subplot), the agent rebalances positions that are far out of range and places liquidity in the direction of expected arbitrage flow. Specifically, when mispricing is relatively small (-1,1) the positions that are deeply out of range (bold yellow and purple) get rebalanced near the current centre, adjusted to the direction of the arbitrage. However, when the current position is already favourably placed, the agent often chooses to hold, since rebalancing would require paying an additional gas cost. When mispricing is large, the out of range positions get rebalanced and their liquidity is placed again in the direction of the arbitrage however, with heavier asymmetry towards the external price to achieve better rates while collecting the fees. The positions that are already in the direction of the arbitrage (e.g. purple lines with S−Z<0S-Z<0 and, conversely, yellow lines with S−Z>0S-Z>0) are not rebalanced. The existing placement is expected to yield higher fees collected than rebalancing closer to S, due to incurred gas costs. The high gas-cost case (right subplot) demonstrates that this behaviour changes sharply when intervention becomes expensive. All lines are dashed, which means that the agent holds her current position (even with large mispricing signal) and the chosen centre output can be ignored. Thus, high gas costs not only reduce PnL mechanically but also expand the inaction region. The cost of rebalancing to correct her position will be larger than the expected profit, in contrast to the left subplot, where the gas cost was lower. Figure 5 reports the corresponding state-function analysis for a risk-averse PPO agent. PPO controls with her actions Znℓ,ZnuZ _n,Z^u_n essentially both where (i.e. the centre defined as Znc:=Znℓ+Znu2Z^c_n:= Z _n+Z^u_n2) and how wide (i.e. a symmetrical half-width |Znu−Znℓ|2 |Z^u_n-Z _n|2 around ZcZ^c) to place her liquidity. The figure shows how the policy responds jointly to mispricing and inventory holdings. Specifically, Figure 5(a) shows that when S−Z<0S-Z<0, the agent repositions the centre of her liquidity towards the external price, independently of the amount of ETH held. In doing so, she collects fees from the arbitrage flow, while avoiding exchanging her liquidity in the worst possible rate (the current one) due to the expected incoming arbitrage trades. However, as shown in Figure 5(b) with S−Z<0S-Z<0, the chosen width varies with the amount of ETH held. For low ETH (i.e. more USDC) holdings, the agent widens her position reflecting her risk management. She dilutes USDC capital across the arbitrage direction, reducing exposure to unfavourable prices. For higher ETH holdings (less USDC), she identifies the profitable rates, resulting in a more concentrated but still wide position. Lastly, Figure 5(c) shows that for S−Z>0S-Z>0 the agent chooses to hold her asset at (-1,1) around the current tick and wait for trades towards the direction of the arbitrage. As a risk-averse agent, she prefers to take the trade because she will exchange the risky token for the safe token (numéraire). Overall, Figures 5 and 6 show that the learned policies are economically interpretable: they use mispricing to infer order-flow direction, gas costs to decide whether intervention is worthwhile, and inventory holdings to control risk. 5. Limitations Our analysis is deliberately scoped to a controlled, model-based setting. Each RL policy is evaluated in the same stationary economic regime in which it is trained, allowing us to isolate and interpret learned economic mechanisms without confounding them with regime-detection or adaptation errors. The results should therefore be understood as conditional on correctly identified, approximately stationary transition dynamics. The benchmarks are constrained by the concentrated-liquidity literature. We compare the RL agents with tuned static and rebalancing heuristics and with the principal analytical strategy available for CPM liquidity provision. We do not claim universal superiority of RL; rather, we show that it can produce interpretable, state-dependent policies where closed-form solutions are generally unavailable. These policies may serve as reproducible model-based references for future work. The continuous-time impulse-control problem is implemented with discrete decision opportunities, reflecting the discrete execution of on-chain transactions. The reported policies, therefore, solve the discretized problem induced by the chosen decision frequency, not the exact continuous-time formulation. Finally, the superposition experiment shows that heterogeneous risk preferences can resemble liquidity patterns like those in concentrated-liquidity pools, without identifying them as a unique explanation. An empirical approach would require on-chain data analysis, which is beyond the scope of this work. 6. Conclusion We studied dynamic liquidity provision in AMMs with CL through an RL perspective. We formulated the LP’s problem as a stochastic impulse control problem, where the LP jointly decides when to rebalance and where to allocate her capital under price uncertainty, arbitrage trades, and operational frictions while managing inventory risk. Due to the analytical intractability of the problem we employed RL algorithms to obtain interpretable state-dependent policies. The results show that learned policies recover economically meaningful behaviour with complex state-dependent strategies. Higher volatility and gas costs reduce average PnL, while stronger risk aversion compresses the left tail of the PnL distribution by inducing wider positions and lower rebalancing rates. 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