Diffusion process

In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion processes are stochastic in nature and hence are used to model many real-life stochastic systems. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diffusion processes. It is used heavily in statistical physics, statistical analysis, information theory, data science, neural networks, finance and marketing.

A sample path of a diffusion process models the trajectory of a particle embedded in a flowing fluid and subjected to random displacements due to collisions with other particles, which is called Brownian motion. The position of the particle is then random; its probability density function as a function of space and time is governed by a convection–diffusion equation.

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 ${\displaystyle a^{ij}(x,t)}${\displaystyle a^{ij}(x,t)} be uniformly continuous coefficients and ${\displaystyle b^{i}(x,t)}${\displaystyle b^{i}(x,t)} be bounded, Borel measurable drift terms. There is a unique family of probability measures ${\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }}${\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }} (for ${\displaystyle \tau \geq 0}${\displaystyle \tau \geq 0}, ${\displaystyle \xi \in \mathbb {R} ^{d}}${\displaystyle \xi \in \mathbb {R} ^{d}}) on the canonical space ${\displaystyle \Omega =C([0,\infty ),\mathbb {R} ^{d})}${\displaystyle \Omega =C([0,\infty ),\mathbb {R} ^{d})}, with its Borel ${\displaystyle \sigma }${\displaystyle \sigma }-algebra, such that:

1. (Initial Condition) The process starts at ${\displaystyle \xi }${\displaystyle \xi } at time ${\displaystyle \tau }${\displaystyle \tau }: ${\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }[\psi \in \Omega :\psi (t)=\xi {\text{ for }}0\leq t\leq \tau ]=1.}${\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 ${\displaystyle f\in C^{2,1}(\mathbb {R} ^{d}\times [\tau ,\infty ))}${\displaystyle f\in C^{2,1}(\mathbb {R} ^{d}\times [\tau ,\infty ))}, the process

${\displaystyle M_{t}^{[f]}=f(\psi (t),t)-f(\psi (\tau ),\tau )-\int _{\tau }^{t}{\bigl (}L_{a;b}+{\tfrac {\partial }{\partial s}}{\bigr )}f(\psi (s),s),円ds}${\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),円ds} is a local martingale under ${\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }}${\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }} for ${\displaystyle t\geq \tau }${\displaystyle t\geq \tau }, with ${\displaystyle M_{t}^{[f]}=0}${\displaystyle M_{t}^{[f]}=0} for ${\displaystyle t\leq \tau }${\displaystyle t\leq \tau }.

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

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

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

• Markov processes

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