# Kolmogorov Backward Equations

##### Ashutosh Bharambe
using Flux, StochasticDiffEq
using NeuralPDE
using Plots
using CUDA

ERROR: LoadError: LoadError: LoadError: LoadError: UndefVarError: CUDA not defined
in expression starting at /root/.cache/julia-buildkite-plugin/depots/a6029d3a-f78b-41ea-bc97-28aa57c6c6ea/packages/Zygote/i1R8y/src/forward/number.jl:6
in expression starting at /root/.cache/julia-buildkite-plugin/depots/a6029d3a-f78b-41ea-bc97-28aa57c6c6ea/packages/Zygote/i1R8y/src/forward/Forward.jl:1
in expression starting at /root/.cache/julia-buildkite-plugin/depots/a6029d3a-f78b-41ea-bc97-28aa57c6c6ea/packages/Zygote/i1R8y/src/Zygote.jl:1
in expression starting at /root/.cache/julia-buildkite-plugin/depots/a6029d3a-f78b-41ea-bc97-28aa57c6c6ea/packages/Flux/qp1gc/src/Flux.jl:1


## Introduction on Backward Kolmogorov Equations

The backward Kolmogorov Equation deals with a terminal condtion. The one dimensional backward kolmogorov equation that we are going to deal with is of the form :

$\frac{\partial p}{\partial t} = -\mu(x)\frac{\partial p}{\partial x} - \frac{1}{2}{\sigma^2}(x)\frac{\partial^2 p}{\partial x^2} ,\hspace{0.5cm} p(T , x) = \varphi(x)$

for all $t \in{ [0 , T] }$ and for all $x \in R^d$

#### The Black Scholes Model

The Black-Scholes Model governs the price evolution of the European put or call option. In the below equation V is the price of some derivative , S is the Stock Price , r is the risk free interest rate and σ the volatility of the stock returns. The payoff at a time T is known to us. And this makes it a terminal PDE. In case of an European put option the PDE is:

$\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}{\sigma^2}{S^2}\frac{\partial^2 V}{\partial S^2} -rV = 0 ,\hspace{0.5cm} V(T , S) = max\{\mathcal{K} - S , 0 \}$

for all $t \in{ [0 , T] }$ and for all $S \in R^d$

In order to make the above equation in the form of the Backward - Kolmogorov PDE we should substitute

$V(S , t) = e^{r(t-T)}p(S , t)$

and thus we get

$e^{r(t-T)}\frac{\partial p}{\partial t} + re^{r(t-T)}p(S , t) = -\mu(x)\frac{\partial p}{\partial x}e^{r(t-T)} - \frac{1}{2}{\sigma^2}(x)\frac{\partial^2 p}{\partial x^2}e^{r(t-T)} + re^{r(t-T)}p(S , t)$

And the terminal condition

$p(S , T) = max\{ \mathcal{K} - x , 0 \}$

We will train our model and the model itself will be the solution of the equation

## Defining the problem and the solver

We should start defining the terminal condition for our equation:

function phi(xi)
y = Float64[]
K = 100
for x in eachcol(xi)
val = max(K - maximum(x) , 0.00)
y = push!(y , val)
end
y = reshape(y , 1 , size(y)[1] )
return y
end

phi (generic function with 1 method)


Now we shall define the problem : We will define the σ and μ by comparing it to the orignal equation. The xspan is the span of initial stock prices.

d = 1
r = 0.04
sigma = 0.2
xspan = (80.00 , 115.0)
tspan = (0.0 , 1.0)
σ(du , u , p , t) = du .= sigma.*u
μ(du , u , p , t) = du .= r.*u
prob = KolmogorovPDEProblem(μ , σ , phi , xspan , tspan, d)

ERROR: UndefVarError: KolmogorovPDEProblem not defined


Now once we have defined our problem it is necessary to define the parameters for the solver.

sdealg = EM()
dt = 0.01
dx = 0.01
trajectories = 100000

ERROR: UndefVarError: EM not defined


Now lets define our model m and the optimiser

m = Chain(Dense(d, 64, elu),Dense(64, 128, elu),Dense(128 , 16 , elu) , Dense(16 , 1))
use_gpu = false
if CUDA.functional() == true
m = fmap(CUDA.cu , m)
use_gpu = true
end

ERROR: UndefVarError: Dense not defined


And then finally call the solver

@time sol = solve(prob, NeuralPDE.NNKolmogorov(m, opt, sdealg, ensemblealg), verbose = true, dt = dt,
dx = dx , trajectories = trajectories , abstol=1e-6, maxiters = 1000 , use_gpu = use_gpu)

ERROR: UndefVarError: NeuralPDE not defined


## Analyzing the solution

Now let us find a Monte-Carlo Solution and plot the both:

monte_carlo_sol = []
x_out = collect(85:2.00:110.00)
for x in x_out
u₀= [x]
g_val(du , u , p , t) = du .= 0.2.*u
f_val(du , u , p , t) = du .= 0.04.*u
dt = 0.01
tspan = (0.0,1.0)
prob = SDEProblem(f_val,g_val,u₀,tspan)
output_func(sol,i) = (sol[end],false)
ensembleprob_val = EnsembleProblem(prob , output_func = output_func )
s = reduce(hcat , sim_val.u)
mean_phi = sum(phi(s))/length(phi(s))
global monte_carlo_sol = push!(monte_carlo_sol , mean_phi)
end

ERROR: UndefVarError: SDEProblem not defined


##Plotting the Solutions We should reshape the inputs and outputs to make it compatible with our model. This is the most important part. The algorithm gives a distributed function over all initial prices in the xspan.

x_model = reshape(x_out, 1 , size(x_out)[1])
if use_gpu == true
m = fmap(cpu , m)
end
y_out = m(x_model)
y_out = reshape(y_out , 13 , 1)

ERROR: UndefVarError: use_gpu not defined


And now finally we can plot the solutions

plot(x_out , y_out , lw = 3 ,  xaxis="Initial Stock Price", yaxis="Payoff" , label = "NNKolmogorov")
plot!(x_out , monte_carlo_sol , lw = 3 ,  xaxis="Initial Stock Price", yaxis="Payoff" ,label = "Monte Carlo Solutions")

ERROR: UndefVarError: plot not defined


## Appendix

These tutorials are a part of the SciMLTutorials.jl repository, found at: https://github.com/SciML/SciMLTutorials.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this tutorial, do the following commands:

using SciMLTutorials
SciMLTutorials.weave_file("tutorials/advanced","03-kolmogorov_equations.jmd")

Computer Information:

Julia Version 1.6.2
Commit 1b93d53fc4 (2021-07-14 15:36 UTC)
Platform Info:
OS: Linux (x86_64-pc-linux-gnu)
CPU: AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-11.0.1 (ORCJIT, znver2)
Environment:
JULIA_DEPOT_PATH = /root/.cache/julia-buildkite-plugin/depots/a6029d3a-f78b-41ea-bc97-28aa57c6c6ea


Package Information:

      Status /var/lib/buildkite-agent/builds/8-amdci4-julia-csail-mit-edu/julialang/scimltutorials-dot-jl/tutorials/advanced/Project.toml
[2169fc97] AlgebraicMultigrid v0.4.0
[6e4b80f9] BenchmarkTools v1.0.0
[052768ef] CUDA v2.6.3
[2b5f629d] DiffEqBase v6.62.2
[9fdde737] DiffEqOperators v4.26.0
[0c46a032] DifferentialEquations v6.17.1
[587475ba] Flux v0.12.1
[961ee093] ModelingToolkit v5.17.3
[2774e3e8] NLsolve v4.5.1
[315f7962] NeuralPDE v3.10.1
[1dea7af3] OrdinaryDiffEq v5.56.0
[91a5bcdd] Plots v1.15.2
[0bca4576] SciMLBase v1.13.4
[30cb0354] SciMLTutorials v0.9.0
[47a9eef4] SparseDiffTools v1.13.2
[684fba80] SparsityDetection v0.3.4
[789caeaf] StochasticDiffEq v6.34.1
[37e2e46d] LinearAlgebra
[2f01184e] SparseArrays

And the full manifest:

      Status /var/lib/buildkite-agent/builds/8-amdci4-julia-csail-mit-edu/julialang/scimltutorials-dot-jl/tutorials/advanced/Manifest.toml
[c3fe647b] AbstractAlgebra v0.16.0
[621f4979] AbstractFFTs v1.0.1
[1520ce14] AbstractTrees v0.3.4
[2169fc97] AlgebraicMultigrid v0.4.0
[ec485272] ArnoldiMethod v0.1.0
[4fba245c] ArrayInterface v3.1.15
[4c555306] ArrayLayouts v0.7.0
[13072b0f] AxisAlgorithms v1.0.0
[ab4f0b2a] BFloat16s v0.1.0
[aae01518] BandedMatrices v0.16.9
[6e4b80f9] BenchmarkTools v1.0.0
[8e7c35d0] BlockArrays v0.15.3
[ffab5731] BlockBandedMatrices v0.10.6
[764a87c0] BoundaryValueDiffEq v2.7.1
[fa961155] CEnum v0.4.1
[00ebfdb7] CSTParser v2.5.0
[052768ef] CUDA v2.6.3
[7057c7e9] Cassette v0.3.6
[082447d4] ChainRules v0.7.65
[d360d2e6] ChainRulesCore v0.9.44
[944b1d66] CodecZlib v0.7.0
[35d6a980] ColorSchemes v3.12.1
[3da002f7] ColorTypes v0.11.0
[5ae59095] Colors v0.12.8
[861a8166] Combinatorics v1.0.2
[a80b9123] CommonMark v0.8.1
[38540f10] CommonSolve v0.2.0
[bbf7d656] CommonSubexpressions v0.3.0
[34da2185] Compat v3.30.0
[aa819f21] CompatHelper v1.18.6
[8f4d0f93] Conda v1.5.2
[88cd18e8] ConsoleProgressMonitor v0.1.2
[187b0558] ConstructionBase v1.2.1
[d38c429a] Contour v0.5.7
[a8cc5b0e] Crayons v4.0.4
[8a292aeb] Cuba v2.2.0
[667455a9] Cubature v1.5.1
[9a962f9c] DataAPI v1.6.0
[82cc6244] DataInterpolations v3.3.1
[864edb3b] DataStructures v0.18.9
[e2d170a0] DataValueInterfaces v1.0.0
[bcd4f6db] DelayDiffEq v5.31.0
[2b5f629d] DiffEqBase v6.62.2
[459566f4] DiffEqCallbacks v2.16.1
[5a0ffddc] DiffEqFinancial v2.4.0
[aae7a2af] DiffEqFlux v1.37.0
[c894b116] DiffEqJump v6.14.2
[77a26b50] DiffEqNoiseProcess v5.7.3
[9fdde737] DiffEqOperators v4.26.0
[055956cb] DiffEqPhysics v3.9.0
[41bf760c] DiffEqSensitivity v6.45.0
[163ba53b] DiffResults v1.0.3
[b552c78f] DiffRules v1.0.2
[0c46a032] DifferentialEquations v6.17.1
[c619ae07] DimensionalPlotRecipes v1.2.0
[b4f34e82] Distances v0.10.3
[31c24e10] Distributions v0.24.18
[ffbed154] DocStringExtensions v0.8.4
[e30172f5] Documenter v0.26.3
[d4d017d3] ExponentialUtilities v1.8.4
[e2ba6199] ExprTools v0.1.3
[8f5d6c58] EzXML v1.1.0
[c87230d0] FFMPEG v0.4.0
[7a1cc6ca] FFTW v1.4.1
[9aa1b823] FastClosures v0.3.2
[1a297f60] FillArrays v0.11.7
[6a86dc24] FiniteDiff v2.8.0
[53c48c17] FixedPointNumbers v0.8.4
[587475ba] Flux v0.12.1
[59287772] Formatting v0.4.2
[f6369f11] ForwardDiff v0.10.18
[069b7b12] FunctionWrappers v1.1.2
[d9f16b24] Functors v0.2.1
[0c68f7d7] GPUArrays v6.4.1
[61eb1bfa] GPUCompiler v0.10.0
[28b8d3ca] GR v0.57.4
[a75be94c] GalacticOptim v1.2.0
[5c1252a2] GeometryBasics v0.3.12
[bc5e4493] GitHub v5.4.0
[af5da776] GlobalSensitivity v1.0.0
[42e2da0e] Grisu v1.0.2
[19dc6840] HCubature v1.5.0
[cd3eb016] HTTP v0.9.9
[eafb193a] Highlights v0.4.5
[0e44f5e4] Hwloc v2.0.0
[7073ff75] IJulia v1.23.2
[b5f81e59] IOCapture v0.1.1
[7869d1d1] IRTools v0.4.2
[615f187c] IfElse v0.1.0
[d25df0c9] Inflate v0.1.2
[83e8ac13] IniFile v0.5.0
[a98d9a8b] Interpolations v0.13.2
[c8e1da08] IterTools v1.3.0
[42fd0dbc] IterativeSolvers v0.9.1
[82899510] IteratorInterfaceExtensions v1.0.0
[692b3bcd] JLLWrappers v1.3.0
[682c06a0] JSON v0.21.1
[98e50ef6] JuliaFormatter v0.13.7
[e5e0dc1b] Juno v0.8.4
[5ab0869b] KernelDensity v0.6.3
[929cbde3] LLVM v3.7.1
[b964fa9f] LaTeXStrings v1.2.1
[2ee39098] LabelledArrays v1.6.1
[23fbe1c1] Latexify v0.15.5
[a5e1c1ea] LatinHypercubeSampling v1.8.0
[73f95e8e] LatticeRules v0.0.1
[5078a376] LazyArrays v0.21.4
[d7e5e226] LazyBandedMatrices v0.5.7
[093fc24a] LightGraphs v1.3.5
[d3d80556] LineSearches v7.1.1
[2ab3a3ac] LogExpFunctions v0.2.4
[e6f89c97] LoggingExtras v0.4.6
[bdcacae8] LoopVectorization v0.12.23
[1914dd2f] MacroTools v0.5.6
[a3b82374] MatrixFactorizations v0.8.3
[739be429] MbedTLS v1.0.3
[442fdcdd] Measures v0.3.1
[e89f7d12] Media v0.5.0
[c03570c3] Memoize v0.4.4
[e1d29d7a] Missings v1.0.0
[78c3b35d] Mocking v0.7.1
[961ee093] ModelingToolkit v5.17.3
[4886b29c] MonteCarloIntegration v0.0.2
[f9640e96] MultiScaleArrays v1.8.1
[ffc61752] Mustache v1.0.10
[d41bc354] NLSolversBase v7.8.0
[2774e3e8] NLsolve v4.5.1
[872c559c] NNlib v0.7.19
[77ba4419] NaNMath v0.3.5
[315f7962] NeuralPDE v3.10.1
[8913a72c] NonlinearSolve v0.3.8
[6fe1bfb0] OffsetArrays v1.9.0
[429524aa] Optim v1.3.0
[bac558e1] OrderedCollections v1.4.1
[1dea7af3] OrdinaryDiffEq v5.56.0
[90014a1f] PDMats v0.11.0
[65888b18] ParameterizedFunctions v5.10.0
[d96e819e] Parameters v0.12.2
[69de0a69] Parsers v1.1.0
[995b91a9] PlotUtils v1.0.10
[91a5bcdd] Plots v1.15.2
[e409e4f3] PoissonRandom v0.4.0
[f517fe37] Polyester v0.3.1
[85a6dd25] PositiveFactorizations v0.2.4
[21216c6a] Preferences v1.2.2
[33c8b6b6] ProgressLogging v0.1.4
[92933f4c] ProgressMeter v1.6.2
[8a4e6c94] QuasiMonteCarlo v0.2.2
[74087812] Random123 v1.3.1
[fb686558] RandomExtensions v0.4.3
[e6cf234a] RandomNumbers v1.4.0
[c84ed2f1] Ratios v0.4.0
[3cdcf5f2] RecipesBase v1.1.1
[01d81517] RecipesPipeline v0.3.2
[731186ca] RecursiveArrayTools v2.11.4
[f2c3362d] RecursiveFactorization v0.1.12
[189a3867] Reexport v1.0.0
[ae029012] Requires v1.1.3
[ae5879a3] ResettableStacks v1.1.0
[37e2e3b7] ReverseDiff v1.9.0
[79098fc4] Rmath v0.7.0
[7e49a35a] RuntimeGeneratedFunctions v0.5.2
[476501e8] SLEEFPirates v0.6.20
[1bc83da4] SafeTestsets v0.0.1
[0bca4576] SciMLBase v1.13.4
[30cb0354] SciMLTutorials v0.9.0
[6c6a2e73] Scratch v1.0.3
[efcf1570] Setfield v0.7.0
[992d4aef] Showoff v1.0.3
[699a6c99] SimpleTraits v0.9.3
[ed01d8cd] Sobol v1.5.0
[2133526b] SodiumSeal v0.1.1
[b85f4697] SoftGlobalScope v1.1.0
[a2af1166] SortingAlgorithms v1.0.0
[47a9eef4] SparseDiffTools v1.13.2
[684fba80] SparsityDetection v0.3.4
[276daf66] SpecialFunctions v1.4.1
[860ef19b] StableRNGs v1.0.0
[aedffcd0] Static v0.2.4
[90137ffa] StaticArrays v1.2.0
[82ae8749] StatsAPI v1.0.0
[2913bbd2] StatsBase v0.33.8
[4c63d2b9] StatsFuns v0.9.8
[789caeaf] StochasticDiffEq v6.34.1
[7792a7ef] StrideArraysCore v0.1.11
[09ab397b] StructArrays v0.5.1
[d1185830] SymbolicUtils v0.11.2
[0c5d862f] Symbolics v0.1.25
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.4.2
[5d786b92] TerminalLoggers v0.1.3
[f269a46b] TimeZones v1.5.5
[a759f4b9] TimerOutputs v0.5.9
[0796e94c] Tokenize v0.5.16
[3bb67fe8] TranscodingStreams v0.9.5
[592b5752] Trapz v2.0.2
[a2a6695c] TreeViews v0.3.0
[5c2747f8] URIs v1.3.0
[3a884ed6] UnPack v1.0.2
[1986cc42] Unitful v1.7.0
[3d5dd08c] VectorizationBase v0.20.11
[81def892] VersionParsing v1.2.0
[19fa3120] VertexSafeGraphs v0.1.2
[44d3d7a6] Weave v0.10.8
[efce3f68] WoodburyMatrices v0.5.3
[ddb6d928] YAML v0.4.6
[c2297ded] ZMQ v1.2.1
[a5390f91] ZipFile v0.9.3
[e88e6eb3] Zygote v0.6.11
[700de1a5] ZygoteRules v0.2.1
[6e34b625] Bzip2_jll v1.0.6+5
[83423d85] Cairo_jll v1.16.0+6
[3bed1096] Cuba_jll v4.2.1+0
[7bc98958] Cubature_jll v1.0.4+0
[5ae413db] EarCut_jll v2.1.5+1
[2e619515] Expat_jll v2.2.10+0
[b22a6f82] FFMPEG_jll v4.3.1+4
[f5851436] FFTW_jll v3.3.9+7
[a3f928ae] Fontconfig_jll v2.13.1+14
[d7e528f0] FreeType2_jll v2.10.1+5
[559328eb] FriBidi_jll v1.0.5+6
[0656b61e] GLFW_jll v3.3.4+0
[d2c73de3] GR_jll v0.57.2+0
[78b55507] Gettext_jll v0.21.0+0
[7746bdde] Glib_jll v2.68.1+0
[e33a78d0] Hwloc_jll v2.4.1+0
[1d5cc7b8] IntelOpenMP_jll v2018.0.3+2
[aacddb02] JpegTurbo_jll v2.0.1+3
[c1c5ebd0] LAME_jll v3.100.0+3
[dd4b983a] LZO_jll v2.10.1+0
[dd192d2f] LibVPX_jll v1.9.0+1
[e9f186c6] Libffi_jll v3.2.2+0
[d4300ac3] Libgcrypt_jll v1.8.7+0
[7e76a0d4] Libglvnd_jll v1.3.0+3
[94ce4f54] Libiconv_jll v1.16.1+0
[4b2f31a3] Libmount_jll v2.35.0+0
[89763e89] Libtiff_jll v4.1.0+2
[38a345b3] Libuuid_jll v2.36.0+0
[856f044c] MKL_jll v2021.1.1+1
[e7412a2a] Ogg_jll v1.3.4+2
[458c3c95] OpenSSL_jll v1.1.1+6
[efe28fd5] OpenSpecFun_jll v0.5.4+0
[91d4177d] Opus_jll v1.3.1+3
[2f80f16e] PCRE_jll v8.44.0+0
[30392449] Pixman_jll v0.40.1+0
[ea2cea3b] Qt5Base_jll v5.15.2+0
[f50d1b31] Rmath_jll v0.3.0+0
[fb77eaff] Sundials_jll v5.2.0+1
[a2964d1f] Wayland_jll v1.17.0+4
[2381bf8a] Wayland_protocols_jll v1.18.0+4
[02c8fc9c] XML2_jll v2.9.12+0
[aed1982a] XSLT_jll v1.1.34+0
[4f6342f7] Xorg_libX11_jll v1.6.9+4
[0c0b7dd1] Xorg_libXau_jll v1.0.9+4
[935fb764] Xorg_libXcursor_jll v1.2.0+4
[a3789734] Xorg_libXdmcp_jll v1.1.3+4
[1082639a] Xorg_libXext_jll v1.3.4+4
[d091e8ba] Xorg_libXfixes_jll v5.0.3+4
[a51aa0fd] Xorg_libXi_jll v1.7.10+4
[d1454406] Xorg_libXinerama_jll v1.1.4+4
[ec84b674] Xorg_libXrandr_jll v1.5.2+4
[ea2f1a96] Xorg_libXrender_jll v0.9.10+4
[c7cfdc94] Xorg_libxcb_jll v1.13.0+3
[cc61e674] Xorg_libxkbfile_jll v1.1.0+4
[12413925] Xorg_xcb_util_image_jll v0.4.0+1
[2def613f] Xorg_xcb_util_jll v0.4.0+1
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
[35661453] Xorg_xkbcomp_jll v1.4.2+4
[33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
[c5fb5394] Xorg_xtrans_jll v1.4.0+3
[8f1865be] ZeroMQ_jll v4.3.2+6
[3161d3a3] Zstd_jll v1.5.0+0
[0ac62f75] libass_jll v0.14.0+4
[f638f0a6] libfdk_aac_jll v0.1.6+4
[b53b4c65] libpng_jll v1.6.38+0
[a9144af2] libsodium_jll v1.0.20+0
[f27f6e37] libvorbis_jll v1.3.6+6
[1270edf5] x264_jll v2020.7.14+2
[dfaa095f] x265_jll v3.0.0+3
[d8fb68d0] xkbcommon_jll v0.9.1+5
[56f22d72] Artifacts
[2a0f44e3] Base64
[8bb1440f] DelimitedFiles
[8ba89e20] Distributed
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
[b27032c2] LibCURL
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[ca575930] NetworkOptions
[44cfe95a] Pkg
[de0858da] Printf
[9abbd945] Profile
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays
[10745b16] Statistics
[4607b0f0] SuiteSparse
[fa267f1f] TOML
[a4e569a6] Tar
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll
[deac9b47] LibCURL_jll
[29816b5a] LibSSH2_jll
[c8ffd9c3] MbedTLS_jll
[14a3606d] MozillaCACerts_jll
[4536629a] OpenBLAS_jll
[bea87d4a] SuiteSparse_jll
[83775a58] Zlib_jll
[8e850ede] nghttp2_jll
[3f19e933] p7zip_jll