Solution to a stochastic differential equation
In probability theory and statistics , diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion process is stochastic in nature and hence is 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 .
Mathematical definition [ edit ] 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)} b i ( x , t ) {\displaystyle b^{i}(x,t)} P a ; b ξ , τ {\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }} τ ≥ 0 {\displaystyle \tau \geq 0} ξ ∈ R d {\displaystyle \xi \in \mathbb {R} ^{d}} Ω = C ( [ 0 , ∞ ) , R d ) {\displaystyle \Omega =C([0,\infty ),\mathbb {R} ^{d})} σ {\displaystyle \sigma }
1. (Initial Condition) The process starts at ξ {\displaystyle \xi } 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 ))} 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} P a ; b ξ , τ {\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }} t ≥ τ {\displaystyle t\geq \tau } M t [ f ] = 0 {\displaystyle M_{t}^{[f]}=0} t ≤ τ {\displaystyle t\leq \tau }
This family P a ; b ξ , τ {\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }} L a ; b {\displaystyle {\mathcal {L}}_{a;b}}
SDE Construction and Infinitesimal Generator [ edit ] It is clear that if we have an L a ; b {\displaystyle {\mathcal {L}}_{a;b}} ( X t ) t ≥ 0 {\displaystyle (X_{t})_{t\geq 0}} ( Ω , F , F t , P a ; b ξ , τ ) {\displaystyle (\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\mathbb {P} _{a;b}^{\xi ,\tau })} X t {\displaystyle X_{t}} 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} 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)} σ i j ( x , t ) {\displaystyle \sigma ^{ij}(x,t)} b i ( x , t ) {\displaystyle b^{i}(x,t)} X t {\displaystyle X_{t}} X τ = ξ {\displaystyle X_{\tau }=\xi } f ∈ C 2 , 1 ( R d × [ τ , ∞ ) ) {\displaystyle f\in C^{2,1}(\mathbb {R} ^{d}\times [\tau ,\infty ))} 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}.} 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},} X t {\displaystyle X_{t}} P a ; b ξ , τ {\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }} Ω = C ( [ 0 , ∞ ) , R d ) {\displaystyle \Omega =C([0,\infty ),\mathbb {R} ^{d})} σ , b {\displaystyle \sigma \!,\!b} L a ; b + ∂ ∂ s {\displaystyle L_{a;b}+{\tfrac {\partial }{\partial s}}} A {\displaystyle {\mathcal {A}}} X t {\displaystyle X_{t}} f ( x , t ) ∈ C 2 ( R d × R + ) {\displaystyle f(\mathbf {x} ,t)\in C^{2}(\mathbb {R} ^{d}\times \mathbb {R} ^{+})} A {\displaystyle {\mathcal {A}}} 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}}.}
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![{\displaystyle \mathbb {P} _{a;b}^{\xi ,\tau }[\psi \in \Omega :\psi (t)=\xi {\text{ for }}0\leq t\leq \tau ]=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faf754d3063f77a2dec5771f1943eb2753a3bb97)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0bc8fc9c6d0f27980665c6593e31c9c774b84e)
![{\displaystyle M_{t}^{[f]}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/022e232602687ed374b8b869a38f57958fa71eb1)
