Add analytical solutions, fast mean and variance calculations

[ChrisRackauckas-Claude]

Summary

This PR addresses issue #2 by implementing analytical solutions, fast mean calculations, and fast variance calculations for the financial models in DiffEqFinancial.jl.

New Functions Added

Geometric Brownian Motion (GBM):

• gbm_mean(μ, u0, t) - E[X(t)] = x₀ exp(μt)
• gbm_variance(μ, σ, u0, t) - Var[X(t)] = x₀² exp(2μt)(exp(σ²t) - 1)
• gbm_std(μ, σ, u0, t) - Standard deviation

Ornstein-Uhlenbeck (OU):

• ou_mean(a, r, u0, t) - E[X(t)] = r + (x₀ - r)exp(-at)
• ou_variance(a, σ, t) - Var[X(t)] = (σ²/2a)(1 - exp(-2at))
• ou_std(a, σ, t) - Standard deviation
• ou_stationary_mean(r) - Long-run mean = r
• ou_stationary_variance(a, σ) - Long-run variance = σ²/2a

Cox-Ingersoll-Ross (CIR):

• cir_mean(κ, θ, u0, t) - E[r(t)] = θ + (r₀ - θ)exp(-κt)
• cir_variance(κ, θ, σ, u0, t) - Full CIR variance formula
• cir_std(κ, θ, σ, u0, t) - Standard deviation
• cir_stationary_mean(θ) - Long-run mean = θ
• cir_stationary_variance(κ, θ, σ) - Long-run variance = θσ²/2κ

Black-Scholes Log-Price:

• bs_log_mean(r, σ, u0, t) - E[ln S(t)] = ln S₀ + (r - σ²/2)t
• bs_log_variance(σ, t) - Var[ln S(t)] = σ²t
• bs_log_std(σ, t) - Standard deviation

Heston Model:

• heston_mean(μ, u0, t) - E[S(t)] = S₀ exp(μt) (independent of volatility path)
• heston_variance_mean(κ, Θ, v0, t) - Mean of variance process (follows CIR)
• heston_variance_variance(κ, Θ, σ, v0, t) - Variance of variance process

Benefits

• Performance: These analytical functions provide exact results without Monte Carlo simulation
• Accuracy: No discretization or sampling error
• Convenience: Easy-to-use API for common moment calculations
• Type Stability: All functions are type-stable (verified with JET.jl)

Tests Added

• Unit tests verifying analytical formulas
• Monte Carlo validation tests confirming convergence to analytical values
• JET.jl static analysis tests for type stability

Test plan

• All existing tests pass
• New analytical solution tests pass
• Monte Carlo validation tests verify formulas match simulation
• JET.jl type stability tests pass

Closes #2

cc @ChrisRackauckas

🤖 Generated with Claude Code

[ChrisRackauckas-Claude]

Rebase and Documentation Update

Rebased onto current master (b4eed69) and made the following fixes:

Conflict Resolution

• Resolved merge conflict in test/runtests.jl — kept upstream's GROUP-based test structure and integrated the analytical solution tests into the core test group
• Moved JET type stability tests for analytical functions into test/nopre/jet_tests.jl (following upstream's nopre test separation pattern)
• Kept upstream's explicit import style in src/DiffEqFinancial.jl (using specific imports from packages)

Formatting

• Applied Runic.jl formatting to all changed files (fixes the Format Check CI failure)

Documentation

• Added docs/src/analytics.md with dedicated documentation page for all analytical functions (GBM, OU, CIR, Black-Scholes, Heston)
• Updated docs/pages.jl to include the new "Analytical Solutions" page in the docs navigation
• The @autodocs in diffeqfinancial.md also picks up all exported functions automatically

Tests

• All 106 tests pass locally (Interface Compatibility + Core Tests + Analytical Solutions + Explicit Imports + JET Tests + AllocCheck)
• Docs build succeeds locally

Co-Authored-By: Chris Rackauckas accounts@chrisrackauckas.com