Almost Sure

19 July 10

Properties of Feller Processes

In the previous post, the concept of Feller processes was introduced. These are Markov processes whose transition function {\{P_t\}_{t\ge0}} satisfies certain continuity conditions. Many of the standard processes we study satisfy the Feller property, such as standard Brownian motion, Poisson processes, Bessel processes and Lévy processes as well as solutions to many stochastic differential equations. It was shown that all Feller processes admit a cadlag modification. In this post I state and prove some of the other useful properties satisfied by such processes, including the strong Markov property, quasi-left-continuity and right-continuity of the filtration. I also describe the basic properties of the infinitesimal generators. The results in this post are all fairly standard and can be found, for example, in Revuz and Yor (Continuous Martingales and Brownian Motion).

As always, we work with respect to a filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}. Throughout this post we consider Feller processes X and transition functions {\{P_t\}_{t\ge0}} defined on the lccb (locally compact with a countable base) space E which, taken together with its Borel sigma-algebra, defines a measurable space {(E,\mathcal{E})}.

Recall that the law of a homogeneous Markov process X is described by a transition function {\{P_t\}_{t\ge0}} on some measurable space {(E,\mathcal{E})}. This specifies that the distribution of {X_t} conditional on the history up until an earlier time {s<t} is given by the measure {P_{t-s}(X_s,\cdot)}. Equivalently,

\displaystyle  {\mathbb E}[f(X_t)\mid\mathcal{F}_s]=P_{t-s}f(X_s)

for any bounded and measurable function {f\colon E\rightarrow{\mathbb R}}. The strong Markov property generalizes this idea to arbitrary stopping times.

Definition 1 Let X be an adapted process and {\{P_t\}_{t\ge 0}} be a transition function.

Then, X satisfies the strong Markov property if, for each stopping time {\tau}, conditioned on {\tau<\infty} the process {\{X_{\tau+t}\}_{t\ge0}} is Markov with the given transition function and with respect to the filtration {\{\mathcal{F}_{\tau+t}\}_{t\ge0}}.

As we see in a moment, Feller processes satisfy the strong Markov property. First, as an example, consider a standard Brownian motion B, and let {\tau} be the first time at which it hits a fixed level {K>0}. The reflection principle states that the process {\tilde B} defined to be equal to B up until time {\tau} and reflected about K afterwards, is also a standard Brownian motion. More precisely,

\displaystyle  \tilde B_t=\begin{cases} B_t,&\textrm{if }t\le\tau,\\ 2K-B_t,&\textrm{if }t\ge\tau, \end{cases}

is a Brownian motion. This useful idea can be used to determine the distribution of the maximum {B^*_t=\max_{s\le t}B_s}. If {B^*_t\ge K} then either the process itself ends up above K or it hits K and then drops below this level by time t, in which case {\tilde B_t>K}. So, by the reflection principle,

\displaystyle  {\mathbb P}(B^*_t\ge K)={\mathbb P}(B_t\ge K)+{\mathbb P}(\tilde B_t> K)=2{\mathbb P}(B_t\ge K).

In fact, the reflection principle is a consequence of the strong Markov property, as follows. First, by the strong Markov property, the process {W_t\equiv B_{\tau+t}-B_\tau} is a Brownian motion independently of the stopped process {B^\tau_t=B_{t\wedge\tau}}. As the law of standard Brownian motion is symmetric, this has the same distribution as {\tilde B_{\tau+t}-\tilde B_\tau=-W_t}. So, {B_t=B^\tau_t+W_{(t-\tau)_+}} has the same joint distribution as {\tilde B_t=B^\tau_t-W_{(t-\tau)_+}}.

Let us now write out explicitly what the strong Markov property means. If {\tau} is a stopping time then, conditioning on {\{\tau<\infty\}}, the Markov property applied to {X_{\tau+t}} under the filtration {\mathcal{F}_{\tau+t}} is equivalent to

\displaystyle  {\mathbb E}\left[1_{\{\tau<\infty\}}f(X_{\tau+t})\mid\mathcal{F}_{\tau+s}\right]=1_{\{\tau<\infty\}}P_{t-s}f(X_{\tau+s})

(1)

(almost surely) for all times {s<t}. Simplifying this a bit gives the following.

Lemma 2 A process X is strong Markov with transition function {\{P_t\}_{t\ge0}} on state space E if and only if, for every finite stopping time {\tau}, every {t>0}, and bounded measurable {f\colon E\rightarrow{\mathbb R}},

\displaystyle  {\mathbb E}[f(X_{\tau+t})\mid\mathcal{F}_\tau]=P_tf(X_\tau).

(2)

Proof: Applying (1) with s=0 gives (2). To prove the converse, suppose that (2) holds and pick any stopping time {\tau} and times {0\le s<t}. Then, {\sigma\equiv T\wedge\tau+s} is a finite stopping time for any constant {T\in{\mathbb R}_+} and, by (2),

\displaystyle  {\mathbb E}\left[f(X_{\sigma+t-s})\mid\mathcal{F}_\sigma\right]=P_{t-s}f(X_{\sigma}).

Then, using the fact that {\mathcal{F}_{\tau+s}} and {\mathcal{F}_{\tau\wedge T+s}} agree on the set {\{\tau\le T\}},

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle 1_{\{\tau\le T\}}{\mathbb E}\left[1_{\{\tau<\infty\}}f(X_{\tau+t})\mid\mathcal{F}_{\tau+s}\right] &\displaystyle =1_{\{\tau\le T\}}{\mathbb E}\left[f(X_{\sigma+t-s})\mid\mathcal{F}_{\sigma}\right]\smallskip\\ &\displaystyle=1_{\{\tau\le T\}}P_{t-s}f(X_{\sigma}). \end{array}

Letting T increase to infinity gives (1). \Box

We now prove that Feller processes are strong Markov.

Theorem 3 Every cadlag Feller process satisfies the strong Markov property.

Proof: Suppose that X is a cadlag Feller process with transition function {\{P_t\}_{t\ge0}}. We just need to show that (2) holds for a bounded stopping time {\tau} and fixed time {t>0}. First, the case where {\tau} takes values in a countable set {S\subset{\mathbb R}_+} is easily handled, and does not even require the Feller property. Just the Markov property is needed. Letting Z be a bounded {\mathcal{F}_\tau}-measurable random variable, {1_{\{\tau=s\}}Z} is {\mathcal{F}_s}-measurable for all fixed times s, giving

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb E}\left[f(X_{\tau+t})Z\right] &\displaystyle=\sum_{s\in S}{\mathbb E}\left[f(X_{s+t})1_{\{\tau=s\}}Z\right]\smallskip\\ &\displaystyle=\sum_{s\in S}{\mathbb E}\left[P_tf(X_s)1_{\{\tau=s\}}Z\right] ={\mathbb E}\left[P_tf(X_\tau)Z\right]. \end{array}

Next let {\tau} be any stopping time and, for each nonnegative integer n, let {\tau_n} be the first time after {\tau} which is a multiple of 1/n. That is {\tau_n=\inf\{k/n\colon k\in{\mathbb N},k/n>\tau\}}, which is a stopping time taking values in a countable subset of {{\mathbb R}_+}. So,

\displaystyle  {\mathbb E}\left[f(X_{\tau_n+t})Z\right]={\mathbb E}\left[P_tf(X_{\tau_n})Z\right]

for bounded {\mathcal{F}_\tau}-measurable Z. From the definition of the Feller property, if {f\in C_0(E)} then {P_tf(X_s)} is jointly continuous in t and right-continuous in s. So, taking limits

\displaystyle  {\mathbb E}[f(X_{\tau+t})Z]=\lim_{n\rightarrow\infty}{\mathbb E}[f(X_{\tau_n+t})Z]=\lim_{n\rightarrow\infty}{\mathbb E}[P_tf(X_{\tau_n})Z]={\mathbb E}[P_tf(X_\tau)Z].

This extends to all bounded measurable {f\colon E\rightarrow{\mathbb R}} by the functional monotone class theorem, giving (2) as required. \Box

In the study of continuous-time stochastic processes it is common to assume that the underlying filtration satisfies the `usual conditions’. That is, it is complete and right-continuous. On the other hand, in these notes, I have not done this. Completeness of the filtration has been used throughout, so that the debut theorem for right-continuous processes holds and so that we can take cadlag versions of processes. However, I have not been assuming right-continuity of the filtration. For Feller processes, as it turns out, right-continuity is automatic in any case. That is, the complete filtration generated by a Feller process is right-continuous.

Theorem 4 Let X be a Feller process and

\displaystyle  \mathcal{F}^0_t\equiv\sigma\left(\{X_s\colon s\le t\}\right)

be its natural filtration. Suppose that {\mathcal{F}_t} is the sigma algebra generated by {\mathcal{F}^0_t} together with the zero probability sets in {\mathcal{F}}.

Then, {\{\mathcal{F}_t\}_{t\ge0}} is a right-continuous filtration,

\displaystyle  \mathcal{F}_t=\mathcal{F}_{t+}\equiv\bigcap_{s>t}\mathcal{F}_s.

Proof: Suppose that X is Feller with state space E and transition function {\{P_t\}_{t\ge0}}. The aim is to prove the identity

\displaystyle  {\mathbb E}[Z\mid\mathcal{F}_{t+}]={\mathbb E}[Z\mid\mathcal{F}_t]

(3)

(almost surely) for every bounded {\mathcal{F}_\infty}-measurable random variable Z. As {\mathcal{F}_\infty} is the sigma-algebra generated by {\{X_t\colon t\in{\mathbb R}_+\}} (up to zero probability sets), it is enough to prove the result for Z of the form

\displaystyle  Z=f_1(X_{t_1})\cdots f_n(X_{t_n})

(4)

for times {t_1<\cdots<t_n} and functions {f_1,\ldots,f_n\in C_0(E)}. The functional monotone class theorem then extends this to all bounded and {\mathcal{F}_\infty}-measurable Z.

First, consider {Z=f(X_T)} for some time {T>t} and {f\in C_0(E)}. As the Feller process X has a cadlag modification, {P_{T-t}f(X_t)} will be right-continuous in probability w.r.t. t. So, the Markov property gives

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb E}[f(X_T)\mid\mathcal{F}_{t+}] &\displaystyle=\lim_{n\rightarrow\infty}{\mathbb E}[f(X_T)\mid\mathcal{F}_{t_n}]\smallskip\\ &\displaystyle=\lim_{n\rightarrow\infty}P_{T-t_n}f(X_{t_n})\smallskip\\ &\displaystyle=P_{T-t}f(X_t)={\mathbb E}[f(X_T)\mid\mathcal{F}_t] \end{array}

as required.

Extending the result to all Z of the form (4) is just an application of induction on n. So, suppose that the (3) holds for n replaced by n-1. If {t_n\le t} then Z is {\mathcal{F}_t}-measurable, and the result is trivial. If {t_n>t\ge t_{n-1}} then setting {Z^\prime=f(X_{t_1})\cdots f(X_{t_{n-1}})} gives

\displaystyle  {\mathbb E}[Z\mid\mathcal{F}_{t+}]=Z^\prime{\mathbb E}[f(X_{t_n})\mid\mathcal{F}_{t+}]=Z^\prime{\mathbb E}[f(X_{t_n})\mid\mathcal{F}_t]={\mathbb E}[Z\mid\mathcal{F}_t].

Finally, if {t_{n-1}\ge t} then setting {\tilde f_{n-1}(x)=f_{n-1}(x)P_{t_n-t_{n-1}}f_n(x)} and

\displaystyle  Z^\prime=f_1(X_{t_1})\cdots f_{n-2}(X_{t_{n-2}})\tilde f_{n-1}(X_{t_{n-1}}),

the Markov property gives {Z^\prime={\mathbb E}[Z\mid\mathcal{F}_{t_{n-1}}]}. Also, the induction hypothesis gives (3) with {Z^\prime} in place of Z. So,

\displaystyle  {\mathbb E}[Z\mid\mathcal{F}_{t+}]={\mathbb E}[Z^\prime\mid\mathcal{F}_{t+}]={\mathbb E}[Z^\prime\mid\mathcal{F}_t]={\mathbb E}[Z\mid\mathcal{F}_t]

as required. \Box

One consequence of the right-continuity of the filtration is the following zero-one law. Consider, for example, a standard Brownian motion B. It can be shown {B_t} hits zero infinitely often for t in any neighborhood of 0, with probability one. Events such as this are {\mathcal{F}_t}-measurable for each {t>0} and, therefore, {\mathcal{F}_{0+}} measurable. The following result states that all such events have probability zero or one. So, in fact, any Feller process with fixed initial state {X_0=x} must either equal x infinitely often in any neighbourhood of 0 with probability one or, again with probability one, there must be a nonempty interval {(0,\tau)} on which it does not equal x.

Corollary 5 (Blumenthal’s zero-one law) Let X be a Feller process with fixed initial value {X_0=x}. Letting {\mathcal{F}^0_t} be its natural filtration, every {A\in\mathcal{F}^0_{0+}} satisfies {{\mathbb P}(A)=0} or {{\mathbb P}(A)=1}.

Proof: As {X_0} is fixed at x, the sigma-algebra {\mathcal{F}^0_0} it generates contains only sets with probability zero or one. If {\mathcal{F}_t} is the sigma-algebra generated by {\mathcal{F}^0_t} together with the zero probability sets then, by Theorem 4, {\mathcal{F}^0_{0+}\subseteq\mathcal{F}_{0+}=\mathcal{F}_0}. However, {\mathcal{F}_0} is generated by sets of zero probability. \Box

Recall that, for a Poisson process X, its jump times are totally inaccessible. That is, for any predictable stopping time {\tau}, it is almost surely continuous at time {\tau}, so {{\mathbb P}(X_{\tau-}=X_{\tau}=1)}. This property holds for all Feller processes, and is referred to as quasi-left-continuity. By definition, a stopping time {\tau>0} is predictable if there exists a sequence {\tau_n<\tau} of stopping times increasing to {\tau}. Then, {X_{\tau_n}\rightarrow X_{\tau-}} as {n\rightarrow\infty}. For arbitrary jointly measurable processes, quasi-left-continuity is defined as follows.

Definition 6 A process X taking values in a topological space E is quasi-left-continuous if and only if, for each finite stopping time {\tau>0} and sequence of stopping times {\tau_n<\tau} increasing to {\tau},

\displaystyle  X_{\tau_n}\rightarrow X_\tau

almost surely as {n\rightarrow\infty}.

We now prove that Feller processes satisfy this property.

Theorem 7 Every cadlag Feller process is quasi-left-continuous.

Proof: Let X be a cadlag Feller process on state space E and with transition function {\{P_t\}_{t\ge0}}. Then choose a finite predictable stopping time {\tau>0} and a sequence of stopping times {\tau_n<\tau} increasing to {\tau}. The aim is to prove the identity

\displaystyle  {\mathbb E}[u(X_{\tau-},X_{\tau})]={\mathbb E}[u(X_{\tau-},X_{\tau-})]

(5)

for bounded measurable functions {u\colon E\times E\rightarrow{\mathbb R}}. Then, taking {u(x,y)=1_{\{x=y\}}} gives {{\mathbb P}(X_{\tau-}=X_{\tau})={\mathbb P}(X_\tau=X_\tau)=1} as required.

As the Borel sigma-algebra on an lccb space E is generated by functions in {C_0(E)}, it is sufficient to prove (5) for {u(x,y)=f(x)g(y)} for functions {f,g\in C_0(E)}, as the functional monotone class theorem then extends (5) to all bounded and measurable u. Given any such functions f, g, the continuity of {P_tg(x)} in both x and t together with the strong Markov property (2) applied at the stopping times {\tau_n} gives,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb E}[f(X_{\tau-})g(X_{\tau+t-})]&\displaystyle=\lim_{n\rightarrow\infty}{\mathbb E}[f(X_{\tau_n})g(X_{\tau_n+t})]\smallskip\\ &\displaystyle=\lim_{n\rightarrow\infty}{\mathbb E}[f(X_{\tau_n})P_tg(X_{\tau_n})]\smallskip\\ &\displaystyle={\mathbb E}[f(X_{\tau-})P_tg(X_{\tau-})] \end{array}

for any {t>0}. Taking the limit as {t\rightarrow0} gives

\displaystyle  {\mathbb E}[f(X_{\tau-})g(X_{\tau})]={\mathbb E}[f(X_{\tau-})g(X_{\tau-})],

as required. \Box

Another property of homogeneous Poisson processes is that the times between jumps are exponentially distributed. This also carries through to arbitrary Feller processes in the following form; the time spent at any fixed level x, if it is not zero or infinite, has the exponential distribution. In fact, the following result almost holds for all right-continuous Markov processes. The Feller property is only used to show that {\tau} is a stopping time (using right-continuity of the filtration) and that X is discontinuous at {\tau} whenever {0<\tau<\infty} (using the strong Markov property).

Theorem 8 Let X be a cadlag Feller process with fixed initial value {X_0=x}, and set

\displaystyle  \tau=\inf\left\{t\ge 0\colon X_t\not=x\right\}.

If the underlying filtration is complete then {\tau} is a stopping time. Furthermore, one of the following properties holds.

  1. {\tau=0} almost surely.
  2. {\tau=\infty} almost surely.
  3. There is a {\lambda>0} such that {\tau} has the {{\rm Exp}(\lambda)} distribution. In this case, {X_\tau\not=x} with probability one, so X is discontinuous at {\tau}.

Proof: Without loss of generality we can assume that the underlying filtration is the one generated by X together with the zero probability sets, since the filtration can always be replaced by this without altering the conclusion of the theorem. By right-continuity of X,

\displaystyle  \left\{\tau<t\right\}=\bigcup_{\substack{s\in{\mathbb Q}_+,\\ s<t}}\left\{X_s\not=x\right\}\in\mathcal{F}_t.

Theorem 4 says that the filtration is right continuous and, by the following, {\tau} is a stopping time.

\displaystyle  \left\{\tau\le t\right\}=\bigcap_{s>t}\left\{\tau<s\right\}\in\bigcap_{s>t}\mathcal{F}_s=\mathcal{F}_{t+}=\mathcal{F}_t.

Now define the function {f\colon{\mathbb R}_+\rightarrow[0,1]} by {f(t)={\mathbb P}(\tau>t)}. For any {s,t>0}, conditioning on the set {\{\tau>s\}\in\mathcal{F}_s} gives {X_s=x=X_0} and, by the Markov property, the process {X_{s+\cdot}} has the same distribution as {X_\cdot}. Therefore, {\tau-s} conditioned on {\{\tau>s\}} has the same distribution as {\tau},

\displaystyle  f(s+t)={\mathbb E}[1_{\{\tau> s\}}{\mathbb P}(\tau-s>t\mid\mathcal{F}_s)]={\mathbb E}[1_{\{\tau>s\}}f(t)]=f(s)f(t).

The only solutions to this functional equation for a right-continuous and decreasing function {f\colon{\mathbb R}_+\rightarrow[0,1]} are,

  1. {f(t) = 0}, in which case {\tau=0} almost surely.
  2. {f(t)=1}, in which case {\tau=\infty} almost surely.
  3. {f(t)=\exp(-\lambda t)} for some constant {\lambda>0}. So, {\tau} has the exponential distribution of rate {\lambda}. Now define the process {\tilde X_t=X_{\tau+t}}. By definition of {\tau}, {\tilde\tau=\inf\{t\ge0\colon\tilde X_t\not=0\}} is almost surely zero. However, by the strong Markov property, conditioned on the event {\{\tilde X_0=x\}} the random variable {\tilde\tau} has the same distribution as {\tau} and is therefore almost surely positive. So,

    \displaystyle  {\mathbb P}(X_\tau=x)={\mathbb P}(\tilde X_0=x)\le{\mathbb P}(\tilde\tau>0)=0.

\Box

Infinitesimal Generators

Except in certain simple cases it is often not possible to explicitly write out the transition function describing a Feller process. Instead, the infinitesimal generator is used. This approximately describes the transition kernel {P_t} for small times t, and can be viewed as the derivative of {P_t} at time 0, {A=dP_t/dt\vert_{t=0}}. As the transition function is likely not to be differentiable in any strong sense, the generator is only defined on some subset of {C_0}.

Definition 9 Let {\{P_t\}_{t\ge0}} be a Feller transition function on the lccb space E. Then, {f\in C_0(E)} is said to be in the domain {\mathcal{D}_A} of the infinitesimal generator if the limit

\displaystyle  Af=\lim_{t\rightarrow0}\frac1t(P_tf-f)

(6)

exists under the uniform topology on {C_0(E)}.

The operator {A\colon\mathcal{D}_A\rightarrow C_0(E)} is called the infinitesimal generator of the semigroup {\{P_t\}_{t\ge 0}}.

Equation (6) can alternatively be written as

\displaystyle  P_tf = f + tAf + o(t)

(7)

where {o(t)} denotes a term vanishing faster than t as {t\rightarrow0}. So, the generator A gives the first-order approximation to {P_t} for small t.

Restricted to {\mathcal{D}_A}, the operator {P_t} is differentiable with derivative given by {AP_t=P_tA}. Equation (8) is a version of the Kolmogorov backward equation.

Theorem 10 Let {f\in\mathcal{D}_A}. Then, {P_tf\in\mathcal{D}_A} for all {t\ge0}. Furthermore, {t\mapsto P_tf} is differentiable (w.r.t. the norm topology on {C_0(E)}) and

\displaystyle  \frac{d}{dt}P_tf = AP_tf=P_tAf.

(8)

Proof: By definition, {(P_hf-f)/h} tends to {Af} in {C_0(E)} as {h\rightarrow0}, over {h>0}. Applying the bounded linear operator {P_t} to this gives

\displaystyle  \frac1h(P_{t+h}f-P_tf)=\frac1hP_t(P_hf-f)\rightarrow P_tAf

as {h\rightarrow 0} over {h>0}. This shows that {t\mapsto P_tf} has the right-hand derivative {P_tAf}. We need to extend this to show that it is actually differentiable. By the Feller property, {P_sAf} is continuous in s. In fact, all functions with a continuous right-hand derivative are differentiable. We can integrate to get

\displaystyle  \frac1h(P_{t+h}f-P_tf)=\frac1h\int_t^{t+h}P_sAf\,ds\rightarrow P_tAf

as {h\rightarrow 0}. Therefore, {P_tf} is indeed differentiable. Next,

\displaystyle  \frac1h(P_hP_tf-P_tf)=\frac1h(P_{t+h}f-P_tf)\rightarrow P_tAf

as {h\rightarrow0}, over {h>0}. By definition this means that {P_tf\in\mathcal{D}_A} and {AP_tf=P_tAf}. \Box

Recall from the previous post that, associated with the transition function {P_t}, there is a resolvent {R_\lambda} defined for any {\lambda>0} as

\displaystyle  R_\lambda f(x)=\int_0^\infty e^{-\lambda t}P_tf(x)\,dt.

This is a transition kernel on E and, if {P_t} is Feller, then {R_\lambda f\in C_0(E)} for all {f\in C_0}. Restricted to {C_0(E)}, {R_\lambda\colon C_0(E)\rightarrow C_0(E)} is a continuous linear map satisfying {\Vert R_\lambda\Vert=1/\lambda} and {\Vert\lambda R_\lambda f-f\Vert\rightarrow 0} as {\lambda\rightarrow\infty}. Resolvents and infinitesimal generators are closely related.

Theorem 11 Let {\{P_t\}} be a Feller transition function and {\lambda>0}. Then, {\mathcal{D}_A} is equal to the image of the resolvent {R_\lambda} on {C_0(E)} and, considering {R_\lambda} as a map from {C_0(E)} to {\mathcal{D}_A},

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle R_\lambda = (\lambda-A)^{-1},\smallskip\\ &\displaystyle A=\lambda-R_\lambda^{-1}. \end{array}

(9)

Proof: The second of equations (9) is just a rearrangement of the first. We can apply {P_h} to {R_\lambda f} for any {h>0} and {f\in C_0(E)},

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle P_hR_\lambda f&\displaystyle=\int_0^\infty e^{-\lambda t}P_{t+h}f\,dt\smallskip\\ &\displaystyle=e^{\lambda h}\int_h^\infty e^{-\lambda s}P_sf\,ds\smallskip\\ &\displaystyle=e^{\lambda h}R_\lambda f-e^{\lambda h}\int_0^he^{-\lambda s}P_sf\,ds \end{array}

By the Feller property, {P_sf\rightarrow f} as {s\rightarrow0}, giving

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle P_hR_\lambda f&\displaystyle=e^{\lambda h}R_\lambda f-he^{\lambda h}f+o(h)\smallskip\\ &\displaystyle=R_\lambda f+h(\lambda R_\lambda f-f)+o(h). \end{array}

So, by equation (7), {R_\lambda f\in\mathcal{D}_A} and {AR_\lambda f=\lambda R_\lambda f-f}. Rearranging gives

\displaystyle  (\lambda-A)R_\lambda = 1.

(10)

Now, choosing any {f\in\mathcal{D}_A}, equation (8) and integration by parts gives,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle R_\lambda Af&\displaystyle=\int_0^\infty e^{-\lambda t}P_tAf\,dt\smallskip\\ &\displaystyle= \int_0^\infty e^{-\lambda t}\frac{d}{dt}P_tf\,dt\smallskip\\ &\displaystyle=-f+\lambda\int_0^\infty e^{-\lambda t}P_tf\,dt\smallskip\\ &\displaystyle= \lambda R_\lambda f-f. \end{array}

Rearranging gives {R_\lambda(\lambda-A)f=f}, so {R_\lambda(\lambda-A)=1}. Together with (10), this gives equation (9) as required. \Box

Using the resolvent simplifies the proofs of some of the properties of the generator. In particular, its domain is dense in {C_0(E)} and A is closed. In the following lemma, the topology given by the uniform norm on {C_0(E)} is used.

Lemma 12 The domain {\mathcal{D}_A} is a dense subset of {C_0(E)}, and A is a closed operator. That is, if {f_n\in\mathcal{D}_A} converge to {f\in C_0(E)} such that {A f_n\rightarrow g\in C_0(E)}, then {f\in\mathcal{D}_A} and {Af=g}.

Proof: Setting {g_n=A f_n}, Theorem 11 gives {f_n=\lambda R_\lambda f_n-R_\lambda g_n}. Taking the limit {n\rightarrow\infty} then, as {R_\lambda} is a bounded operator, this gives {f=\lambda R_\lambda f-R_\lambda g}. Again by Theorem 11, this says that f is in {\mathcal{D}_A} and {Af=g}. \Box

A Feller transition function is uniquely determined by its generator. However, this result not nearly as useful as it sounds. This is because, in all but a small number of special cases, we do not know what the domain {\mathcal{D}_A} is. Usually, we just have A defined on some dense subspace of {C_0(E)}, such as the twice continuously differentiable functions, and this is not enough to apply the following result.

Lemma 13 Let {\{P_t\}_{t\ge0}} and {\{Q_t\}_{t\ge0}} be Feller transition functions on an lccb space E with generators A and B respectively.

Suppose that the domains of the generators satisfy {\mathcal{D}_A\subseteq\mathcal{D}_B} and, restricted to {\mathcal{D}_A}, {B=A}. Then, {P_t=Q_t} for all t.

Proof: Denote the resolvents of {\{P_t\}} and {\{Q_t\}} by {R_\lambda} and {S_\lambda} respectively. Theorem 11 gives

\displaystyle  R_\lambda f=(\lambda-A)^{-1}f=(\lambda-B)^{-1}f=S_\lambda f

for all {f\in C_0(E)}. So, the resolvents are identical. However, resolvents are just Laplace transforms of the transition function and, by invertibility of Laplace transforms, {P_tf(x)=Q_tf(x)} for almost every t. By continuity in t, {P_t=Q_t} as required. \Box

An alternative proof of Lemma 13 follows from applying (8) for any {f\in\mathcal{D}_A},

\displaystyle  \frac{d}{ds}Q_sP_{t-s}f= Q_sBP_{t-s}f-Q_sAP_{t-s}f=Q_s(A-B)P_{t-s}f=0.

So, {P_tf=Q_0P_{t-0}f=Q_tP_{t-t}f=Q_tf}.

Finally, the generator of a Feller process can be rewritten in terms of a martingale problem. The martingale approach, introduced by Stroock and Varadhan, is a very useful method in the theory of stochastic differential equations and Markov processes. For example, it is instrumental in the famous Stroock-Varadhan uniqueness theorem which shows that stochastic differential equations of the form

\displaystyle  dX^i_t=\sum_{j=1}^na_{ij}(X_t)\,dB^j+b_i(X_t)\,dt

satisfy uniqueness in law for the n-dimensional process X. This is under the condition that {a_{ij}\colon{\mathbb R}^n\rightarrow{\mathbb R}} are continuous functions such that {\{a_{ij}(x)\}_{ij}} is a nonsingular nxn matrix for any x, and {b_i\colon{\mathbb R}^n\rightarrow{\mathbb R}} are bounded and measurable.

The martingale description of the generator is as follows.

Lemma 14 Let X be a cadlag Feller process with transition function {\{P_t\}} and generator A. Then, for {f\in\mathcal{D}_A}, the process

\displaystyle  M_t\equiv f(X_t)-\int_0^t Af(X_s)\,ds

is a martingale.

Proof: Choose times {s<t} and let Z be a bounded {\mathcal{F}_s}-measurable random variable. Taking expectations of {Z(M_t-M_s)} and applying the Markov property gives,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb E}\left[Z(M_t-M_s)\right]&\displaystyle={\mathbb E}\left[Z(f(X_t)-f(X_s))\right]-\int_s^t{\mathbb E}[ZAf(X_u)]\,du\smallskip\\ &\displaystyle={\mathbb E}\left[Z(P_{t-s}f(X_t)-f(X_s))\right]-\int_0^{t-s}{\mathbb E}[ZP_uAf(X_s)]\,du\smallskip\\ &\displaystyle={\mathbb E}\left[Z\left(P_{t-s}f(X_s)-f(X_s)-\int_0^{t-s}\frac{d}{du}P_uf(X_s)\,ds\right)\right]\smallskip\\ &\displaystyle=0 \end{array}

as required. Here, (8) has been used to express {P_uAf(X_s)} as a derivative. \Box

Lemma 14 almost gives a complete description of the generator associated with a Feller process X, but not quite. To strengthen this result to give an alternative characterization of the generator, we must consider running the Markov process X from each possible starting position {x\in E}.

In the following theorem, we let {\Omega\subseteq E^{{\mathbb R}_+}} be the set of cadlag functions {\omega\colon{\mathbb R}_+\rightarrow E} with coordinate process {X_t(\omega)\equiv\omega(t)}. Then, {\mathcal{F}^0_t} is the sigma-algebra generated by {\{X_s\colon s\le t\}} and {\mathcal{F}^0=\mathcal{F}^0_\infty}. For each {x\in E}, Corollary 4 of the previous post implies that there is a unique probability measure {{\mathbb P}_x} on {(\Omega,\mathcal{F}^0)} under which X is a Feller process with the given transition function and {{\mathbb P}_x(X_0=x)=1}. Then, {(\Omega,\mathcal{F}^0,\{\mathcal{F}^0_t\}_{t\ge0},{\mathbb P}_x)} is a filtered probability space.

Theorem 15 Let {\{P_t\}} be a Feller transition function and {f,g\in C_0(E)}. Then, {f\in\mathcal{D}_A} and {Af=g} if and only the process

\displaystyle  M_t\equiv f(X_t)-\int_0^t g(X_s)\,ds

is a martingale under the measure {{\mathbb P}_x} for all {x\in E}.

Proof: If {Af=g} then Lemma 14 says that M is a martingale. Conversely, suppose that M is a martingale under the measure {{\mathbb P}_x}. Then,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle P_tf(x)&\displaystyle={\mathbb E}_x[f(X_t)]\smallskip\\ &\displaystyle=f(x)+{\mathbb E}\left[M_t-M_0+\int_0^t g(X_s)\,ds\right]\smallskip\\ &\displaystyle=f(x)+\int_0^tP_sg(x)\,ds \end{array}

However, the Feller property says that {P_sg\rightarrow g} as {s\rightarrow 0}, giving

\displaystyle  P_tf=f+tg+o(t).

By equation (7), this says that {f\in\mathcal{D}_A} and {Af=g}. \Box

For example, suppose that {X=(X^1,\ldots,X^n)} is an n-dimensional process satisfying the stochastic differential equation

\displaystyle  dX^i_t=\sum_{j=1}^ma_{ij}(X_t)\,dB^j_t+b_i(X_t)\,dt

for an m-dimensional Brownian motion B and measurable functions {a_{ij},b_i\colon{\mathbb R}^n\rightarrow{\mathbb R}}. Defining the function {A\colon{\mathbb R}^n\rightarrow{\mathbb R}^{n^2}} by {A(x)=a(x)a(x)^{\rm T}}, consider the differential operator

\displaystyle  Lf\equiv\frac12\sum_{ij}A_{ij}\frac{\partial^2f}{\partial x_i\partial x_j}+\sum_ib_i\frac{\partial f}{\partial x_i}.

For any twice-continuously differentiable function {f\colon{\mathbb R}^n\rightarrow{\mathbb R}}, Ito’s lemma gives

\displaystyle  f(X_t)=f(X_0)+\sum_i\int_0^t \frac{\partial f}{\partial x_i}(X_s)\,dB^i_s+\int_0^tLf(X_s)\,ds.

So, {f(X_t)-\int_0^tLf(X_s)\,ds} is a local martingale. If it is known that solutions to the SDE are in fact Feller processes, such as is the case when the coefficients are Lipschitz continuous, then Theorem 15 says that any twice continuously differentiable {f\in C_0({\mathbb R}^n)} such that {Lf} vanishes at infinity is in the domain of the generator, and the generator agrees with L on such functions.

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6 Comments »

  1. Hey George,
    just stumbled upon your blog, very cool stuff! I’d like to read a bit more on SDEs permitting a solution which is a Feller process. This can become very nasty I think, so I am interested in some (hopefully) easy to check criteria. I have seen that even simple SDEs, driven by Levy Processes are not Feller. There is a counterexample in the phd thesis of Schnurr (check apendix B): http://deposit.ddb.de/cgi-bin/dokserv?idn=995080879&dok_var=d1&dok_ext=pdf&filename=995080879.pdf

    I have some Levy driving process, so not sure if the solution is Feller (which would be nice). Any references? Thanks.

    Comment by pierre — 14 July 11 @ 9:34 PM | Reply

    • Hi.

      I don’t have any references to hand (but do remember that there are theorems giving sufficient conditions for a solution to an SDE to be Feller). You need to check several conditions (i) Uniquess in law (ii) Weak existence of solutions (iii) Continuity with respect to initial conditions (iv) Time homogenity and (v) E_x[f(X_t)]\to0 as \Vert x\Vert\to\infty for bounded functions f of compact support.

      Conditions (i) and (ii) are things that you need to check in any case. Also, (iii) is the kind of result that would normally be proven along with uniqueness in law, and will hold for continuous coefficients. Property (iv) should be clear when it holds, and holds for an SDE with no time dependence in the coefficients and driven by independent Lévy processes. Properties (i)-(iii) will be satisfied for locally Lipschitz coefficents as long as the solution doesn’t blow up. So, (v) can be the tricky one. You need to show that the probability of being within some compact region at a fixed time t vanishes as the initial state goes to infinity. Thinking about it, if you have an SDE

      \displaystyle dX = \sum_{k=1}^n a_k(X_-)\,dZ^k

      for a Lévy process Z = (Z1,…,Zn) then I think that (v) will be satisfied as long as ak has sub-linear growth. This is because the size of jump in Z required for X to jump back to near the origin grows to infinity as X becomes large. The counterexample you mention, (B.3) in the linked paper, has ak growing linearly in X. For continuous driving terms (Brownian motion + drift) linear growth is enough for (v) to be satisfied.

      I don’t have time to be looking for references now, but, if I come across any references I’ll post here.

      Comment by George Lowther — 18 July 11 @ 11:53 PM | Reply

  2. Very nice blog indeed.

    Something I have been wondering about for a while is under which conditions a Feller process that is strongly Markov would also have a reflection principle. More generally speaking, under which conditions would a (multi-dimensional) process that is strongly Markov also posses a reflection principle? I haven´t come across any references that treats this problem.

    Comment by Steven — 13 March 12 @ 6:28 PM | Reply

  3. Hello,
    I have a question that I would like to ask you. If we have a property A for Feller processes with negative jumps then we can say that the property A holds for Levy processes with negative jumps, right?

    Is this true? Feller processes contain Levy processes?

    I hope to receive your answer soon!
    Thank you!

    Comment by Son — 27 March 12 @ 10:33 AM | Reply

  4. [...] http://almostsure.wordpress.com/2010/07/19/properties-of-feller-processes/ Share this:TwitterFacebookLike this:LikeBe the first to like this post. This entry was posted in Probability Theory and tagged Brownian Motion, discrete time, dyadic approximation, Feller process, Feller property, Markov chain, Markov process, markov property, Strong Markov Property by dominicyeo. Bookmark the permalink. [...]

    Pingback by Feller Processes and the Strong Markov Property | Eventually Almost Everywhere — 6 April 12 @ 1:13 PM | Reply


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