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Mathematical definition

A diffusion process is a Markov process with continuous sample paths for which the Kolmogorov forward equation is the Fokker–Planck equation.¹

A diffusion process is defined by the following properties. Let a i j ( x , t ) {\displaystyle a^{ij}(x,t)} be uniformly continuous coefficients and b i ( x , t ) {\displaystyle b^{i}(x,t)} be bounded, Borel measurable drift terms. There is a unique family of probability measures P a ; b ξ , τ {\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }} (for τ ≥ 0 {\displaystyle \tau \geq 0} , ξ ∈ R d {\displaystyle \xi \in \mathbb {R} ^{d}} ) on the canonical space Ω = C ( [ 0 , ∞ ) , R d ) {\displaystyle \Omega =C([0,\infty ),\mathbb {R} ^{d})} , with its Borel σ {\displaystyle \sigma } -algebra, such that:

1. (Initial Condition) The process starts at ξ {\displaystyle \xi } at time τ {\displaystyle \tau } : P a ; b ξ , τ [ ψ ∈ Ω : ψ ( t ) = ξ  for  0 ≤ t ≤ τ ] = 1. {\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }[\psi \in \Omega :\psi (t)=\xi {\text{ for }}0\leq t\leq \tau ]=1.}

2. (Local Martingale Property) For every f ∈ C 2 , 1 ( R d × [ τ , ∞ ) ) {\displaystyle f\in C^{2,1}(\mathbb {R} ^{d}\times [\tau ,\infty ))} , the process M t [ f ] = f ( ψ ( t ) , t ) − f ( ψ ( τ ) , τ ) − ∫ τ t ( L a ; b + ∂ ∂ s ) f ( ψ ( s ) , s ) d s {\displaystyle M_{t}^{[f]}=f(\psi (t),t)-f(\psi (\tau ),\tau )-\int _{\tau }^{t}{\bigl (}L_{a;b}+{\tfrac {\partial }{\partial s}}{\bigr )}f(\psi (s),s)\,ds} is a local martingale under P a ; b ξ , τ {\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }} for t ≥ τ {\displaystyle t\geq \tau } , with M t [ f ] = 0 {\displaystyle M_{t}^{[f]}=0} for t ≤ τ {\displaystyle t\leq \tau } .

This family P a ; b ξ , τ {\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }} is called the L a ; b {\displaystyle {\mathcal {L}}_{a;b}} -diffusion.

SDE Construction and Infinitesimal Generator

It is clear that if we have an L a ; b {\displaystyle {\mathcal {L}}_{a;b}} -diffusion, i.e. ( X t ) t ≥ 0 {\displaystyle (X_{t})_{t\geq 0}} on ( Ω , F , F t , P a ; b ξ , τ ) {\displaystyle (\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\mathbb {P} _{a;b}^{\xi ,\tau })} , then X t {\displaystyle X_{t}} satisfies the SDE d X t i = 1 2 ∑ k = 1 d σ k i ( X t ) d B t k + b i ( X t ) d t {\displaystyle dX_{t}^{i}={\frac {1}{2}}\,\sum _{k=1}^{d}\sigma _{k}^{i}(X_{t})\,dB_{t}^{k}+b^{i}(X_{t})\,dt} . In contrast, one can construct this diffusion from that SDE if a i j ( x , t ) = ∑ k σ i k ( x , t ) σ j k ( x , t ) {\displaystyle a^{ij}(x,t)=\sum _{k}\sigma _{i}^{k}(x,t)\,\sigma _{j}^{k}(x,t)} and σ i j ( x , t ) {\displaystyle \sigma ^{ij}(x,t)} , b i ( x , t ) {\displaystyle b^{i}(x,t)} are Lipschitz continuous. To see this, let X t {\displaystyle X_{t}} solve the SDE starting at X τ = ξ {\displaystyle X_{\tau }=\xi } . For f ∈ C 2 , 1 ( R d × [ τ , ∞ ) ) {\displaystyle f\in C^{2,1}(\mathbb {R} ^{d}\times [\tau ,\infty ))} , apply Itô's formula: d f ( X t , t ) = ( ∂ f ∂ t + ∑ i = 1 d b i ∂ f ∂ x i + v ∑ i , j = 1 d a i j ∂ 2 f ∂ x i ∂ x j ) d t + ∑ i , k = 1 d ∂ f ∂ x i σ k i d B t k . {\displaystyle df(X_{t},t)={\bigl (}{\frac {\partial f}{\partial t}}+\sum _{i=1}^{d}b^{i}{\frac {\partial f}{\partial x_{i}}}+v\sum _{i,j=1}^{d}a^{ij}\,{\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}{\bigr )}\,dt+\sum _{i,k=1}^{d}{\frac {\partial f}{\partial x_{i}}}\,\sigma _{k}^{i}\,dB_{t}^{k}.} Rearranging gives f ( X t , t ) − f ( X τ , τ ) − ∫ τ t ( ∂ f ∂ s + L a ; b f ) d s = ∫ τ t ∑ i , k = 1 d ∂ f ∂ x i σ k i d B s k , {\displaystyle f(X_{t},t)-f(X_{\tau },\tau )-\int _{\tau }^{t}{\bigl (}{\frac {\partial f}{\partial s}}+L_{a;b}f{\bigr )}\,ds=\int _{\tau }^{t}\sum _{i,k=1}^{d}{\frac {\partial f}{\partial x_{i}}}\,\sigma _{k}^{i}\,dB_{s}^{k},} whose right‐hand side is a local martingale, matching the local‐martingale property in the diffusion definition. The law of X t {\displaystyle X_{t}} defines P a ; b ξ , τ {\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }} on Ω = C ( [ 0 , ∞ ) , R d ) {\displaystyle \Omega =C([0,\infty ),\mathbb {R} ^{d})} with the correct initial condition and local martingale property. Uniqueness follows from the Lipschitz continuity of σ , b {\displaystyle \sigma \!,\!b} . In fact, L a ; b + ∂ ∂ s {\displaystyle L_{a;b}+{\tfrac {\partial }{\partial s}}} coincides with the infinitesimal generator A {\displaystyle {\mathcal {A}}} of this process. If X t {\displaystyle X_{t}} solves the SDE, then for f ( x , t ) ∈ C 2 ( R d × R + ) {\displaystyle f(\mathbf {x} ,t)\in C^{2}(\mathbb {R} ^{d}\times \mathbb {R} ^{+})} , the generator A {\displaystyle {\mathcal {A}}} is A f ( x , t ) = ∑ i = 1 d b i ( x , t ) ∂ f ∂ x i + v ∑ i , j = 1 d a i j ( x , t ) ∂ 2 f ∂ x i ∂ x j + ∂ f ∂ t . {\displaystyle {\mathcal {A}}f(\mathbf {x} ,t)=\sum _{i=1}^{d}b_{i}(\mathbf {x} ,t)\,{\frac {\partial f}{\partial x_{i}}}+v\sum _{i,j=1}^{d}a_{ij}(\mathbf {x} ,t)\,{\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}+{\frac {\partial f}{\partial t}}.}

See also

• Stochastic differential equation
• Itô calculus
• Fokker–Planck equation
• Markov process
• Diffusion
• Itô diffusion
• Jump diffusion
• Sample-continuous process

References

1. "9. Diffusion processes" (PDF). Retrieved October 10, 2011. http://math.nyu.edu/faculty/varadhan/stochastic.fall08/sec10.pdf ↩