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Infinitesimal generator (stochastic processes)
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<h2 id="definition">Definition</h2> <h3>General case</h3> <p>For a <a href="/facts/Feller_process/tBLxmCLS">Feller process</a> ( X t ) t ≥ 0 {\displaystyle (X_{t})_{t\geq 0}} with Feller semigroup T = ( T t ) t ≥ 0 {\displaystyle T=(T_{t})_{t\geq 0}} and state space E {\displaystyle E} we define the generator<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> ( A , D ( A ) ) {\displaystyle (A,D(A))} by D ( A ) = { f ∈ C 0 ( E ) : lim t ↓ 0 T t f − f t exists as uniform limit } , {\displaystyle D(A)=\left\{f\in C_{0}(E):\lim _{t\downarrow 0}{\frac {T_{t}f-f}{t}}{\text{ exists as uniform limit}}\right\},} A f = lim t ↓ 0 T t f − f t , for any f ∈ D ( A ) . {\displaystyle Af=\lim _{t\downarrow 0}{\frac {T_{t}f-f}{t}},~~{\text{ for any }}f\in D(A).} Here C 0 ( E ) {\displaystyle C_{0}(E)} denotes the Banach space of continuous functions on E {\displaystyle E} vanishing at infinity, equipped with the supremum norm, and T t f ( x ) = E x f ( X t ) = E ( f ( X t ) | X 0 = x ) {\displaystyle T_{t}f(x)=\mathbb {E} ^{x}f(X_{t})=\mathbb {E} (f(X_{t})|X_{0}=x)} . In general, it is not easy to describe the domain of the Feller generator. However, the Feller generator is always closed and densely defined. If X {\displaystyle X} is R d {\displaystyle \mathbb {R} ^{d}} -valued and D ( A ) {\displaystyle D(A)} contains the <a href="/facts/Test_functions/yfaJ4mUp">test functions</a> (compactly supported smooth functions) then<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> A f ( x ) = − c ( x ) f ( x ) + l ( x ) ⋅ ∇ f ( x ) + 1 2 div Q ( x ) ∇ f ( x ) + ∫ R d ∖ { 0 } ( f ( x + y ) − f ( x ) − ∇ f ( x ) ⋅ y χ ( | y | ) ) N ( x , d y ) , {\displaystyle Af(x)=-c(x)f(x)+l(x)\cdot \nabla f(x)+{\frac {1}{2}}{\text{div}}Q(x)\nabla f(x)+\int _{\mathbb {R} ^{d}\setminus {\{0\}}}\left(f(x+y)-f(x)-\nabla f(x)\cdot y\chi (|y|)\right)N(x,dy),} where c ( x ) ≥ 0 {\displaystyle c(x)\geq 0} , and ( l ( x ) , Q ( x ) , N ( x , ⋅ ) ) {\displaystyle (l(x),Q(x),N(x,\cdot ))} is a <a href="/facts/L%C3%A9vy_process/m9A3YQW8">Lévy triplet</a> for fixed x ∈ R d {\displaystyle x\in \mathbb {R} ^{d}} . </p> <h3>Lévy processes</h3> <p>The generator of Lévy semigroup is of the form A f ( x ) = l ⋅ ∇ f ( x ) + 1 2 div Q ∇ f ( x ) + ∫ R d ∖ { 0 } ( f ( x + y ) − f ( x ) − ∇ f ( x ) ⋅ y χ ( | y | ) ) ν ( d y ) {\displaystyle Af(x)=l\cdot \nabla f(x)+{\frac {1}{2}}{\text{div}}Q\nabla f(x)+\int _{\mathbb {R} ^{d}\setminus {\{0\}}}\left(f(x+y)-f(x)-\nabla f(x)\cdot y\chi (|y|)\right)\nu (dy)} where l ∈ R d , Q ∈ R d × d {\displaystyle l\in \mathbb {R} ^{d},Q\in \mathbb {R} ^{d\times d}} is positive semidefinite and ν {\displaystyle \nu } is a Lévy measure satisfying ∫ R d ∖ { 0 } min ( | y | 2 , 1 ) ν ( d y ) < ∞ {\displaystyle \int _{\mathbb {R} ^{d}\setminus \{0\}}\min(|y|^{2},1)\nu (dy)<\infty } and 0 ≤ 1 − χ ( s ) ≤ κ min ( s , 1 ) {\displaystyle 0\leq 1-\chi (s)\leq \kappa \min(s,1)} for some κ > 0 {\displaystyle \kappa >0} with s χ ( s ) {\displaystyle s\chi (s)} is bounded. If we define ψ ( ξ ) = ψ ( 0 ) − i l ⋅ ξ + 1 2 ξ ⋅ Q ξ + ∫ R d ∖ { 0 } ( 1 − e i y ⋅ ξ + i ξ ⋅ y χ ( | y | ) ) ν ( d y ) {\displaystyle \psi (\xi )=\psi (0)-il\cdot \xi +{\frac {1}{2}}\xi \cdot Q\xi +\int _{\mathbb {R} ^{d}\setminus \{0\}}(1-e^{iy\cdot \xi }+i\xi \cdot y\chi (|y|))\nu (dy)} for ψ ( 0 ) ≥ 0 {\displaystyle \psi (0)\geq 0} then the generator can be written as A f ( x ) = − ∫ e i x ⋅ ξ ψ ( ξ ) f ^ ( ξ ) d ξ {\displaystyle Af(x)=-\int e^{ix\cdot \xi }\psi (\xi ){\hat {f}}(\xi )d\xi } where f ^ {\displaystyle {\hat {f}}} denotes the Fourier transform. So the generator of a Lévy process (or semigroup) is a Fourier multiplier operator with symbol − ψ {\displaystyle -\psi } . </p> <h3>Stochastic differential equations driven by Lévy processes</h3> <p>Let L {\textstyle L} be a Lévy process with symbol ψ {\displaystyle \psi } (see above). Let Φ {\displaystyle \Phi } be locally Lipschitz and bounded. The solution of the SDE d X t = Φ ( X t − ) d L t {\displaystyle dX_{t}=\Phi (X_{t-})dL_{t}} exists for each deterministic initial condition x ∈ R d {\displaystyle x\in \mathbb {R} ^{d}} and yields a Feller process with symbol q ( x , ξ ) = ψ ( Φ ⊤ ( x ) ξ ) . {\displaystyle q(x,\xi )=\psi (\Phi ^{\top }(x)\xi ).} </p><p>Note that in general, the solution of an SDE driven by a Feller process which is not Lévy might fail to be Feller or even Markovian. </p><p>As a simple example consider d X t = l ( X t ) d t + σ ( X t ) d W t {\textstyle dX_{t}=l(X_{t})dt+\sigma (X_{t})dW_{t}} with a Brownian motion driving noise. If we assume l , σ {\displaystyle l,\sigma } are Lipschitz and of linear growth, then for each deterministic initial condition there exists a unique solution, which is Feller with symbol q ( x , ξ ) = − i l ( x ) ⋅ ξ + 1 2 ξ Q ( x ) ξ . {\displaystyle q(x,\xi )=-il(x)\cdot \xi +{\frac {1}{2}}\xi Q(x)\xi .} </p> <h2 id="mean-first-passage-time">Mean first passage time</h2> <p>The mean first passage time T 1 {\displaystyle T_{1}} satisfies A T 1 = − 1 {\displaystyle {\mathcal {A}}T_{1}=-1} . This can be used to calculate, for example, the time it takes for a Brownian motion particle in a box to hit the boundary of the box, or the time it takes for a Brownian motion particle in a potential well to escape the well. Under certain assumptions, the escape time satisfies the <a href="/facts/Arrhenius_equation/TdAeo3Lf">Arrhenius equation</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="generators-of-some-common-processes">Generators of some common processes</h2> <p>For finite-state continuous time Markov chains the generator may be expressed as a <a href="/facts/Transition_rate_matrix/6m9Tjrby">transition rate matrix</a>. </p><p>The general n-dimensional diffusion process d X t = μ ( X t , t ) d t + σ ( X t , t ) d W t {\displaystyle dX_{t}=\mu (X_{t},t)\,dt+\sigma (X_{t},t)\,dW_{t}} has generator A f = ( ∇ f ) T μ + t r ( ( ∇ 2 f ) D ) {\displaystyle {\mathcal {A}}f=(\nabla f)^{T}\mu +tr((\nabla ^{2}f)D)} where D = 1 2 σ σ T {\displaystyle D={\frac {1}{2}}\sigma \sigma ^{T}} is the diffusion matrix, ∇ 2 f {\displaystyle \nabla ^{2}f} is the <a href="/facts/Hessian_matrix/RguMNr3m">Hessian</a> of the function f {\displaystyle f} , and t r {\displaystyle tr} is the <a href="/facts/Trace_(linear_algebra)/dchFFvHt">matrix trace</a>. Its adjoint operator is<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> A ∗ f = − ∑ i ∂ i ( f μ i ) + ∑ i j ∂ i j ( f D i j ) {\displaystyle {\mathcal {A}}^{*}f=-\sum _{i}\partial _{i}(f\mu _{i})+\sum _{ij}\partial _{ij}(fD_{ij})} The following are commonly used special cases for the general n-dimensional diffusion process. </p> <ul><li>Standard Brownian motion on R n {\displaystyle \mathbb {R} ^{n}} , which satisfies the stochastic differential equation d X t = d B t {\displaystyle dX_{t}=dB_{t}} , has generator 1 2 Δ {\textstyle {1 \over {2}}\Delta } , where Δ {\displaystyle \Delta } denotes the <a href="/facts/Laplace_operator/kov9u3PT">Laplace operator</a>.</li> <li>The two-dimensional process Y {\displaystyle Y} satisfying: d Y t = ( d t d B t ) {\displaystyle \mathrm {d} Y_{t}={\mathrm {d} t \choose \mathrm {d} B_{t}}} where B {\displaystyle B} is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator: A f ( t , x ) = ∂ f ∂ t ( t , x ) + 1 2 ∂ 2 f ∂ x 2 ( t , x ) {\displaystyle {\mathcal {A}}f(t,x)={\frac {\partial f}{\partial t}}(t,x)+{\frac {1}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x)} </li> <li>The <a href="/facts/Ornstein%E2%80%93Uhlenbeck_process/oFaDcovm">Ornstein–Uhlenbeck process</a> on R {\displaystyle \mathbb {R} } , which satisfies the stochastic differential equation d X t = θ ( μ − X t ) d t + σ d B t {\textstyle dX_{t}=\theta (\mu -X_{t})dt+\sigma dB_{t}} , has generator: A f ( x ) = θ ( μ − x ) f ′ ( x ) + σ 2 2 f ″ ( x ) {\displaystyle {\mathcal {A}}f(x)=\theta (\mu -x)f'(x)+{\frac {\sigma ^{2}}{2}}f''(x)} </li> <li>Similarly, the graph of the Ornstein–Uhlenbeck process has generator: A f ( t , x ) = ∂ f ∂ t ( t , x ) + θ ( μ − x ) ∂ f ∂ x ( t , x ) + σ 2 2 ∂ 2 f ∂ x 2 ( t , x ) {\displaystyle {\mathcal {A}}f(t,x)={\frac {\partial f}{\partial t}}(t,x)+\theta (\mu -x){\frac {\partial f}{\partial x}}(t,x)+{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x)} </li> <li>A <a href="/facts/Geometric_Brownian_motion/0Ve9dW6c">geometric Brownian motion</a> on R {\displaystyle \mathbb {R} } , which satisfies the stochastic differential equation d X t = r X t d t + α X t d B t {\textstyle dX_{t}=rX_{t}dt+\alpha X_{t}dB_{t}} , has generator: A f ( x ) = r x f ′ ( x ) + 1 2 α 2 x 2 f ″ ( x ) {\displaystyle {\mathcal {A}}f(x)=rxf'(x)+{\frac {1}{2}}\alpha ^{2}x^{2}f''(x)} </li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Dynkin%27s_formula/hffzVfUM">Dynkin's formula</a></li></ul> <ul><li>Calin, Ovidiu (2015). <i>An Informal Introduction to Stochastic Calculus with Applications</i>. Singapore: World Scientific Publishing. p. 315. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-981-4678-93-3. (See Chapter 9)</li> <li><a href="/facts/Bernt_%C3%98ksendal/Au4oSbnG">Øksendal, Bernt K.</a> (2003). <i>Stochastic Differential Equations: An Introduction with Applications</i>. Universitext (Sixth ed.). Berlin: Springer. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-14394-6">10.1007/978-3-642-14394-6</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-04758-1. (See Section 7.3)</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Böttcher, Björn; Schilling, René; Wang, Jian (2013). Lévy Matters III: Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Springer International Publishing. ISBN 978-3-319-02683-1. <a href="978-3-319-02683-1" target="_blank">978-3-319-02683-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Böttcher, Björn; Schilling, René; Wang, Jian (2013). Lévy Matters III: Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Springer International Publishing. ISBN 978-3-319-02683-1. <a href="978-3-319-02683-1" target="_blank">978-3-319-02683-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Böttcher, Björn; Schilling, René; Wang, Jian (2013). Lévy Matters III: Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Springer International Publishing. ISBN 978-3-319-02683-1. <a href="978-3-319-02683-1" target="_blank">978-3-319-02683-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"Lecture 10: Forward and Backward equations for SDEs" (PDF). cims.nyu.edu. <a href="https://cims.nyu.edu/~holmes/teaching/asa19/handout_Lecture10_2019.pdf" target="_blank">https://cims.nyu.edu/~holmes/teaching/asa19/handout_Lecture10_2019.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"Lecture 10: Forward and Backward equations for SDEs" (PDF). cims.nyu.edu. <a href="https://cims.nyu.edu/~holmes/teaching/asa19/handout_Lecture10_2019.pdf" target="_blank">https://cims.nyu.edu/~holmes/teaching/asa19/handout_Lecture10_2019.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>