dXt=μ(Xt) dt+σ(Xt) dWtdX_t=\mu(X_t)\,dt+\sigma(X_t)\,dW_tdXt​=μ(Xt​)dt+σ(Xt​)dWt​Vt+12σ2S2VSS+rS VS−rV=0V_t+\tfrac12\sigma^2 S^2 V_{SS}+rS\,V_S-rV=0Vt​+21​σ2S2VSS​+rSVS​−rV=0ρV=sup⁡a{f+b Vx+12σ2Vxx}\rho V=\sup_{a}\{f+b\,V_x+\tfrac12\sigma^2 V_{xx}\}ρV=asup​{f+bVx​+21​σ2Vxx​}−ut−ν Δu+H(x,∇u)=f(x,m)-u_t-\nu\,\Delta u+H(x,\nabla u)=f(x,m)−ut​−νΔu+H(x,∇u)=f(x,m)xy=kxy=kxy=kIL(r)=2r1+r−1\mathrm{IL}(r)=\frac{2\sqrt{r}}{1+r}-1IL(r)=1+r2r​​−1CVaRα=11−α∫α1VaRβ dβ\mathrm{CVaR}_\alpha=\frac{1}{1-\alpha}\int_\alpha^1 \mathrm{VaR}_\beta\,d\betaCVaRα​=1−α1​∫α1​VaRβ​dβx˙i=xi[(Ax)i−x⊤Ax]\dot{x}_i=x_i\big[(Ax)_i-x^\top Ax\big]x˙i​=xi​[(Ax)i​−x⊤Ax]C=E[e−rT(ST−K)+]C=\mathbb{E}\big[e^{-rT}(S_T-K)^+\big]C=E[e−rT(ST​−K)+]df=(ft+μfx+12σ2fxx)dt+σfx dWtdf=\big(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\big)dt+\sigma f_x\,dW_tdf=(ft​+μfx​+21​σ2fxx​)dt+σfx​dWt​∂tm−∇ ⁣⋅(m ∇pH)=ν Δm\partial_t m-\nabla\!\cdot(m\,\nabla_p H)=\nu\,\Delta m∂t​m−∇⋅(m∇p​H)=νΔmΔ=∂V∂S,Γ=∂2V∂S2\Delta=\frac{\partial V}{\partial S},\quad \Gamma=\frac{\partial^2 V}{\partial S^2}Δ=∂S∂V​,Γ=∂S2∂2V​λ(δ)=A e−κδ\lambda(\delta)=A\,e^{-\kappa\delta}λ(δ)=Ae−κδd1=ln⁡(S/K)+(r+12σ2)TσTd_{1}=\frac{\ln(S/K)+(r+\tfrac12\sigma^2)T}{\sigma\sqrt{T}}d1​=σT​ln(S/K)+(r+21​σ2)T​u(x)=sup⁡τ E[e−rτg(Xτ)]u(x)=\sup_{\tau}\,\mathbb{E}\big[e^{-r\tau}g(X_\tau)\big]u(x)=τsup​E[e−rτg(Xτ​)]κ=γ σ2/η\kappa=\sqrt{\gamma\,\sigma^2/\eta}κ=γσ2/η​E[dWt2]=dt\mathbb{E}[dW_t^2]=dtE[dWt2​]=dtJ(π)=E[∑tγtrt]J(\pi)=\mathbb{E}\big[\textstyle\sum_t \gamma^t r_t\big]J(π)=E[∑t​γtrt​]dXt=μ(Xt) dt+σ(Xt) dWtdX_t=\mu(X_t)\,dt+\sigma(X_t)\,dW_tdXt​=μ(Xt​)dt+σ(Xt​)dWt​Vt+12σ2S2VSS+rS VS−rV=0V_t+\tfrac12\sigma^2 S^2 V_{SS}+rS\,V_S-rV=0Vt​+21​σ2S2VSS​+rSVS​−rV=0ρV=sup⁡a{f+b Vx+12σ2Vxx}\rho V=\sup_{a}\{f+b\,V_x+\tfrac12\sigma^2 V_{xx}\}ρV=asup​{f+bVx​+21​σ2Vxx​}−ut−ν Δu+H(x,∇u)=f(x,m)-u_t-\nu\,\Delta u+H(x,\nabla u)=f(x,m)−ut​−νΔu+H(x,∇u)=f(x,m)xy=kxy=kxy=kIL(r)=2r1+r−1\mathrm{IL}(r)=\frac{2\sqrt{r}}{1+r}-1IL(r)=1+r2r​​−1CVaRα=11−α∫α1VaRβ dβ\mathrm{CVaR}_\alpha=\frac{1}{1-\alpha}\int_\alpha^1 \mathrm{VaR}_\beta\,d\betaCVaRα​=1−α1​∫α1​VaRβ​dβx˙i=xi[(Ax)i−x⊤Ax]\dot{x}_i=x_i\big[(Ax)_i-x^\top Ax\big]x˙i​=xi​[(Ax)i​−x⊤Ax]C=E[e−rT(ST−K)+]C=\mathbb{E}\big[e^{-rT}(S_T-K)^+\big]C=E[e−rT(ST​−K)+]df=(ft+μfx+12σ2fxx)dt+σfx dWtdf=\big(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\big)dt+\sigma f_x\,dW_tdf=(ft​+μfx​+21​σ2fxx​)dt+σfx​dWt​∂tm−∇ ⁣⋅(m ∇pH)=ν Δm\partial_t m-\nabla\!\cdot(m\,\nabla_p H)=\nu\,\Delta m∂t​m−∇⋅(m∇p​H)=νΔmΔ=∂V∂S,Γ=∂2V∂S2\Delta=\frac{\partial V}{\partial S},\quad \Gamma=\frac{\partial^2 V}{\partial S^2}Δ=∂S∂V​,Γ=∂S2∂2V​λ(δ)=A e−κδ\lambda(\delta)=A\,e^{-\kappa\delta}λ(δ)=Ae−κδd1=ln⁡(S/K)+(r+12σ2)TσTd_{1}=\frac{\ln(S/K)+(r+\tfrac12\sigma^2)T}{\sigma\sqrt{T}}d1​=σT​ln(S/K)+(r+21​σ2)T​u(x)=sup⁡τ E[e−rτg(Xτ)]u(x)=\sup_{\tau}\,\mathbb{E}\big[e^{-r\tau}g(X_\tau)\big]u(x)=τsup​E[e−rτg(Xτ​)]κ=γ σ2/η\kappa=\sqrt{\gamma\,\sigma^2/\eta}κ=γσ2/η​E[dWt2]=dt\mathbb{E}[dW_t^2]=dtE[dWt2​]=dtJ(π)=E[∑tγtrt]J(\pi)=\mathbb{E}\big[\textstyle\sum_t \gamma^t r_t\big]J(π)=E[∑t​γtrt​]dXt=μ(Xt) dt+σ(Xt) dWtdX_t=\mu(X_t)\,dt+\sigma(X_t)\,dW_tdXt​=μ(Xt​)dt+σ(Xt​)dWt​Vt+12σ2S2VSS+rS VS−rV=0V_t+\tfrac12\sigma^2 S^2 V_{SS}+rS\,V_S-rV=0Vt​+21​σ2S2VSS​+rSVS​−rV=0ρV=sup⁡a{f+b Vx+12σ2Vxx}\rho V=\sup_{a}\{f+b\,V_x+\tfrac12\sigma^2 V_{xx}\}ρV=asup​{f+bVx​+21​σ2Vxx​}−ut−ν Δu+H(x,∇u)=f(x,m)-u_t-\nu\,\Delta u+H(x,\nabla u)=f(x,m)−ut​−νΔu+H(x,∇u)=f(x,m)xy=kxy=kxy=kIL(r)=2r1+r−1\mathrm{IL}(r)=\frac{2\sqrt{r}}{1+r}-1IL(r)=1+r2r​​−1CVaRα=11−α∫α1VaRβ dβ\mathrm{CVaR}_\alpha=\frac{1}{1-\alpha}\int_\alpha^1 \mathrm{VaR}_\beta\,d\betaCVaRα​=1−α1​∫α1​VaRβ​dβx˙i=xi[(Ax)i−x⊤Ax]\dot{x}_i=x_i\big[(Ax)_i-x^\top Ax\big]x˙i​=xi​[(Ax)i​−x⊤Ax]C=E[e−rT(ST−K)+]C=\mathbb{E}\big[e^{-rT}(S_T-K)^+\big]C=E[e−rT(ST​−K)+]df=(ft+μfx+12σ2fxx)dt+σfx dWtdf=\big(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\big)dt+\sigma f_x\,dW_tdf=(ft​+μfx​+21​σ2fxx​)dt+σfx​dWt​∂tm−∇ ⁣⋅(m ∇pH)=ν Δm\partial_t m-\nabla\!\cdot(m\,\nabla_p H)=\nu\,\Delta m∂t​m−∇⋅(m∇p​H)=νΔmΔ=∂V∂S,Γ=∂2V∂S2\Delta=\frac{\partial V}{\partial S},\quad \Gamma=\frac{\partial^2 V}{\partial S^2}Δ=∂S∂V​,Γ=∂S2∂2V​λ(δ)=A e−κδ\lambda(\delta)=A\,e^{-\kappa\delta}λ(δ)=Ae−κδd1=ln⁡(S/K)+(r+12σ2)TσTd_{1}=\frac{\ln(S/K)+(r+\tfrac12\sigma^2)T}{\sigma\sqrt{T}}d1​=σT​ln(S/K)+(r+21​σ2)T​u(x)=sup⁡τ E[e−rτg(Xτ)]u(x)=\sup_{\tau}\,\mathbb{E}\big[e^{-r\tau}g(X_\tau)\big]u(x)=τsup​E[e−rτg(Xτ​)]κ=γ σ2/η\kappa=\sqrt{\gamma\,\sigma^2/\eta}κ=γσ2/η​E[dWt2]=dt\mathbb{E}[dW_t^2]=dtE[dWt2​]=dtJ(π)=E[∑tγtrt]J(\pi)=\mathbb{E}\big[\textstyle\sum_t \gamma^t r_t\big]J(π)=E[∑t​γtrt​]
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#mean-field-games

3 posts.

Aug 18, 2025Research

When the Pool Fights Back (Part II): Approximating Equilibria in Complex MFGs

by Agustin Munoz Gonzalez · 6 min

Introduction: From Theory Breakdown to Smart Workarounds In \[Part I], When the Pool Fights Back (Part I): Modeling Liquidity with Transaction Costs we introduced a more realistic mean field game (MFG) model for decentralized trading, where agents interact with a liquidity pool…

Aug 18, 2025Research

When the Pool Fights Back (Part I): Modeling Liquidity with Transaction Costs

by Agustin Munoz Gonzalez · 5 min

Exploring how a simple tweak such as adding transaction costs turns a tractable mean field game into a complex dance of strategic interaction.

Aug 18, 2025Research

What if Uniswap Was a Game?

by Agustin Munoz Gonzalez · 7 min

When I first encountered mean field games , I was struck by their elegance: a tool to study strategic behavior among many agents without modeling each one individually. At the time, I was also working on quantitative strategies inside liquidity pools . It didn’t take long to…