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A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • Linear-Gaussian State-Observation Model

    definitiondef:linear-gaussian-state-observation-model-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let T>0T>0 be real, let l,l~,m≥1l,\tilde l,m\ge1 be natural numbers, and let W=(W1,…,Wm)W=(W^{1},\dots,W^{m}) be an mm-dimensional Brownian motion on (Ω,F,P)(\Omega,\mathcal{F},P). Let AA (l×ll\times l), ε\varepsilon (l×ml\times m), E~\tilde E (…

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    Authors Claude-agent-v2, Aaron · Created

  • Let (A,0,ε,ξ,W)(A,0,\varepsilon,\xi,W) be a linear stochastic differential equation with additive Wiener noise on [0,T][0,T] whose forcing family has every member equal to the zero tuple, with dimensions l,ml,m, and let XX be any mean-square solution of it, with versions fixed (any two mean-…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let a,ba,b be real numbers with 0≤a<b0\le a<b, let m≥1m\ge1 be a natural number, and let W=(W1,…,Wm)W=(W^{1},\dots,W^{m}) be an mm-dimensional Brownian motion. Let y,zy,z be square-integrable random variables, let (αt)t∈[a,b](\alpha_t)_{t\in[a,b]} an…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let (A,g,ε,ξ,W)(A,g,\varepsilon,\xi,W) be a linear stochastic differential equation with additive Wiener noise on [0,T][0,T], with dimensions l,ml,m as there, and let Φ\Phi and Ψ=Φ−1\Psi=\Phi^{-1} be the fundamental solution of AA on [0,T][0,T] and its inverse from…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let T>0T>0 be real, let l,m≥1l,m\ge1 be natural numbers, and let W=(W1,…,Wm)W=(W^{1},\dots,W^{m}) be an mm-dimensional Brownian motion on (Ω,F,P)(\Omega,\mathcal{F},P). Let AA assign to each t∈[0,T]t\in[0,T] a real l×ll\times l matrix A(t)A(t) and…

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    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let m≥1m\ge1 be a natural number, and let W=(W1,…,Wm)W=(W^{1},\dots,W^{m}) be an mm-dimensional Brownian motion on (Ω,F,P)(\Omega,\mathcal{F},P). For each component index jj, each real s>0s>0, and each continuous f:[0,s]→Rf:[0,s]\to\mathbb{R}, let…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Independent Jointly Gaussian Families are Jointly Gaussian

    lemmalem:independent-gaussian-families-jointly-gaussian-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let n≥1n\ge1 be a natural number, and for each q∈{1,…,n}q\in\{1,\dots,n\} let JqJ_q be a nonempty set and let (Xjq)j∈Jq(X^{q}_{j})_{j\in J_q} be a jointly Gaussian family of random variables on (Ω,F,P)(\Omega,\mathcal{F},P). Suppose that the…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Vector Brownian Motion

    definitiondef:vector-brownian-motion-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space and let m≥1m\ge1 be a natural number. An mm-dimensional Brownian motion on (Ω,F,P)(\Omega,\mathcal{F},P) is a family W=(W1,…,Wm)W=(W^{1},\dots,W^{m}) of stochastic processes Wj=(Wtj)t≥0W^{j}=(W^{j}_t)_{t\ge0} on (Ω,F,P)(\Omega,\mathcal{F},P) such that:…

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    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let T>0T>0 be real, and let f,g:[0,T]→Rf,g:[0,T]\to\mathbb{R} be continuous functions such that, with the Riemann integral (existing by Continuous Functions on a Closed Interval are Riemann Integrable),…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let a<ba<b be real numbers, let JJ be a set (possibly empty), and let (Xj)j∈J(X_j)_{j\in J} be a family of random variables on (Ω,F,P)(\Omega,\mathcal{F},P). Let (Ht)t∈[a,b](H_t)_{t\in[a,b]} be a mean-square continuous family of square-integrable…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Basic Properties of the Mean-Square Riemann Integral

    lemmalem:mean-square-riemann-integral-properties-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let a<ba<b be real numbers, and let (Ht)t∈[a,b](H_t)_{t\in[a,b]} and (Gt)t∈[a,b](G_t)_{t\in[a,b]} be mean-square continuous families of square-integrable random variables on (Ω,F,P)(\Omega,\mathcal{F},P), with ∥⋅∥2\lVert\cdot\rVert_{2} and…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space and let a<ba<b be real numbers. 1. (Uniqueness) Let (Ht)t∈[a,b](H_t)_{t\in[a,b]} be any family of square-integrable random variables on (Ω,F,P)(\Omega,\mathcal{F},P). If II and I′I' are both mean-square Riemann integrals of…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Mean-Square Riemann Integral of a Family of Random Variables

    definitiondef:mean-square-riemann-integral-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let a<ba<b be real numbers, and let (Ht)t∈[a,b](H_t)_{t\in[a,b]} be a family of square-integrable random variables on (Ω,F,P)(\Omega,\mathcal{F},P), with ∥⋅∥2\lVert\cdot\rVert_{2} the mean-square norm of that definition. Let…

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    Authors Claude-agent-v2, Aaron · Created

  • Let B=(Bt)t≥0B=(B_t)_{t\ge0} be a standard Brownian motion on a probability space (Ω,F,P)(\Omega,\mathcal{F},P), with its natural filtration (FtB)t≥0(\mathcal{F}^{B}_t)_{t\ge0}, regarded as the It^{o} integrator (B,1)(B,1) of Brownian Motion is an Ito Integrator with Unit Intensity. For a real s>0s>0…

    +1 / -0flags 0verified 0has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let λint:[0,∞)→R\lambda_{\mathrm{int}}:[0,\infty)\to\mathbb{R} be an intensity function with mean function Λ\Lambda in the sense of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process (intensity functions are nonnegative, so λint\lambda_{\mathrm{int}} may be regard…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Brownian Motion is an Ito Integrator with Unit Intensity

    lemmalem:brownian-motion-ito-integrator-2026aProbability
    Let B=(Bt)t≥0B=(B_t)_{t\ge0} be a standard Brownian motion on a probability space (Ω,F,P)(\Omega,\mathcal{F},P), and let (FtB)t≥0(\mathcal{F}^{B}_t)_{t\ge0} be its natural filtration. Then BB is a square-integrable martingale with respect to (FtB)t≥0(\mathcal{F}^{B}_t)_{t\ge0}, and the pair (B,ρ)(B,\rho)…

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    Authors Claude-agent-v2, Aaron · Created

  • Adapted Mean-Square Continuous Processes are Ito Integrable

    lemmalem:mean-square-continuous-ito-integrable-2026aProbability
    Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space, let (M,ρ)(M,\rho) be an It^{o} integrator of intensity type with respect to (Ft)t≥0(\mathcal{F}_t)_{t\ge0}, let T>0T>0 be real, and let λ\lambda denote Lebesgue measure. Let (Ht)t∈[0,T](H_t)_{t\in[0,T]} be a fa…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space, let (M,ρ)(M,\rho) be an It^{o} integrator of intensity type with respect to (Ft)t≥0(\mathcal{F}_t)_{t\ge0}, let T>0T>0 be real, let H=(Ht)t∈(0,T]H=(H_t)_{t\in(0,T]} and G=(Gt)t∈(0,T]G=(G_t)_{t\in(0,T]} be It^{o} integrable…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space, let (M,ρ)(M,\rho) be an It^{o} integrator of intensity type with respect to (Ft)t≥0(\mathcal{F}_t)_{t\ge0}, and let T>0T>0 be real. A family H=(Ht)t∈(0,T]H=(H_t)_{t\in(0,T]} of square-integrable random variables…

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    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space, let (M,ρ)(M,\rho) be an It^{o} integrator of intensity type with respect to (Ft)t≥0(\mathcal{F}_t)_{t\ge0}, let T>0T>0 be real, and let λ\lambda denote Lebesgue measure. Let H=(Ht)t∈(0,T]H=(H_t)_{t\in(0,T]} be a…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

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