UDC 517.934

APPLICATION OF THE THEORY OF ADDITIVE FUNCTIONALS TO STOCHASTIC DIFFERENTIAL EQUATIONS

E. I. ZHELEZNOV

For many questions in the theory of stochastic differential equations it is desirable to know the probability distribution for the trajectories of a moving point. In some cases the theory of additive functionals gives quite satisfactory answers. The aim of this article is to demonstrate the method of additive functionals on the simplest differential equations. In what follows we shall consider the differential equation

\[ \dot{x}=v[y(t)]x, \tag{1} \]

where \(x\) is a real number, \(y(t)\) is a Markov process, homogeneous in time. Denote by \(\varphi(\lambda,t,y)\) the conditional characteristic function for the random variable

\[ z_t=\int_0^t v[y(t)]\,dt, \tag{2} \]

i.e.,

\[ \varphi(\lambda,t,y)=M[\exp(i\lambda z_t)\mid y(0)=y]. \tag{3} \]

The following theorem of E. B. Dynkin holds: [1]

The characteristic function \(\varphi(\lambda,t,y)\) is the unique solution of the differential equation

\[ \frac{d\varphi}{dt}=A\varphi+i\lambda v\varphi, \tag{4} \]

satisfying the initial condition

\[ \lim_{t\to 0}\varphi(\lambda,t,y)=1 \]

and the boundedness condition \(\varphi\exp(-\|v\|t)\) as \(t\to\infty\) (here \(A\) is the infinitesimal operator of the semigroup).

§ 1. Consider the differential equation

\[ \dot{x}=v[y(t)]x, \]

whose general solution has the form

\[ x(t)=x_0\exp\int_0^t v[y(t)]\,dt. \tag{1.1} \]

Let \(\Phi(t,z)\) be the distribution function of the random variable (2). If \(F(x_0,x,t)\) is the distribution function for the random variable \(x(t)\), then we obtain the relation

\[ F(x_0,x,t)= \begin{cases} \Phi\left(t,\ln\dfrac{x}{x_0}\right), & x_0>0,\quad x>0,\\[6pt] 0, & x_0>0,\quad x<0,\\[6pt] 1, & x_0<0,\quad x>0,\\[6pt] 1-\Phi\left(t,\ln\dfrac{x}{x_0}\right), & x_0<0,\quad x<0. \end{cases} \tag{1.2} \]

If, further, \(y(t)\) is a Wiener process, then equation (4) has the form

\[ \frac{d\varphi}{dt}=\frac{1}{2}\varphi''(y)+i\lambda v\varphi . \tag{1.3} \]

Passing to the Laplace transform

\[ \Psi(\lambda,p,y)=\int_0^\infty \varphi(\lambda,t,y)\exp(-pt), \]

we obtain

\[ \frac{1}{2}\frac{d^2\Psi}{dy^2}=[p-i\lambda v(y)]\Psi-1 . \tag{1.4} \]

§ 2. If \(v(y)=\dfrac{1}{2}[1+\operatorname{sgn} y]\), then we obtain the well-known result from probability theory concerning the mathematical expectation of the time spent by a particle on one of the half-axes. Thus, in the case \(y\leq 0\),

\[ \varphi(\lambda,t,y)=\frac{2}{\pi}\exp(i\lambda t)\int_0^{y/\sqrt{2t}}\exp(-u^2)\,du+ \frac{1}{\pi}\int_0^t \frac{\exp(i\lambda\tau-y^2/2\tau)}{\sqrt{\tau(t-\tau)}}\,d\tau . \tag{2.1} \]

If \(y=0\), then we have the well-known arcsine law

\[ \Phi(t,z)= \begin{cases} 0, & z<0,\\[4pt] \dfrac{2}{\pi}\arcsin \sqrt{\dfrac{z}{t}}, & 0\leq z<t,\\[8pt] 1, & t<z. \end{cases} \tag{2.2} \]

For (2.2) see [2,3]. From (2.2) it follows that

\[ P[x(t)<x\mid y(0)=0]= \begin{cases} 0, & x<x_0,\quad x_0>0,\\[6pt] \dfrac{2}{\pi}\arcsin \sqrt{\dfrac{\ln x_0^{-1}x}{t}}, & 0<x_0\leq x\leq x_0\exp t,\\[10pt] 1, & 0<x_0\exp t<x. \end{cases} \tag{2.3} \]

§ 3. In this section we consider the case of a time-homogeneous Markov process with two states \(a_1\) and \(a_2\). Let \(P(t)=(p_{ij}(t))\) be the matrix of transition probabilities, where

\[ p_{ij}(t)=P[y(t)=a_j\mid y(0)=a_i]. \tag{3.1} \]

In the theory of Markov processes [4,5], under sufficiently broad assumptions, the existence of the limits

\[ q_i=\lim_{h\downarrow 0}\frac{1-p_{ii}(h)}{h} \]

is proved.

In this case the infinitesimal operator will be represented by the matrix

\[ A=\begin{pmatrix} -q_1 & q_1\\ q_2 & -q_2 \end{pmatrix}. \tag{3.2} \]

E. B. Dynkin’s equation (4) in this case will have the following form:

\[ \begin{aligned} \dot{\varphi}_1&=(-q_1+i\lambda v_1)\varphi_1+q_1\varphi_2,\\ \dot{\varphi}_2&=q_2\varphi_1+(-q_2+i\lambda v_2)\varphi_2, \end{aligned} \tag{3.3} \]

where the notation

\[ \varphi_i(\lambda,t)=\varphi(\lambda,t,a_i),\qquad v_i=v(a_i). \]

The roots of the characteristic equation of system (3.3) will be equal to

\[ r_{1,2}=-\frac{1}{2}\left[-(q_1+q_2)+i\lambda(v_1+v_2)\pm \sqrt{(q_1+q_2)^2+2i(v_1-v_2)(q_2-q_1)\lambda}\right]. \]

§ 4. Consider the case \(q_1=0,\ q_2=q>0\). System (3.3) will have the form

\[ \begin{aligned} \dot\varphi_1&=i\lambda v_1\varphi_1,\\ \dot\varphi_2&=q_2\varphi_1+(-q_2+i\lambda v_2)\varphi_2 . \end{aligned} \tag{4.1} \]

It is easy to see that the solution of system (4.1) can be written in the following form:

\[ \varphi_1(\lambda,t)=\exp(i\lambda v_1 t), \]

\[ \varphi_2(\lambda,t)= \frac{1}{q+i(v_1-v_2)\lambda} \left[i(v_1-v_2)\lambda \exp\{-[q-i\lambda v_2]t\} +q\exp(i\lambda v_1t)\right]. \tag{4.2} \]

From these conditional characteristic functions we can find the corresponding conditional distribution functions by the inversion formula

\[ \Phi_i(t,z_2)-\Phi_i(t,z_1)= \frac{1}{2\pi i}\int_{-\infty}^{+\infty} \frac{\exp(-i\lambda z_1)-\exp(-i\lambda z_2)}{\lambda}\, \varphi_i\,d\lambda, \tag{4.3} \]

where the integral is understood in the sense of the principal value, and the function \(\Phi_i(t,z)\) is normalized, i.e., its values at discontinuity points are taken to be equal to

\[ \frac{1}{2}\left[\Phi_i(t,z-0)+\Phi_i(t,z+0)\right]. \]

Using the theory of residues, we obtain the following:

\[ \int_{-\infty}^{+\infty} \frac{\exp(-i\lambda x)}{\lambda}\,\varphi_2\,d\lambda = \pi i \begin{cases} \displaystyle 2\exp\frac{q(x-v_1t)}{v_1-v_2}, & x<v_2t,\quad v_1>v_2,\\[1.2ex] \displaystyle -2\exp\frac{q(x-v_1t)}{v_1-v_2}, & x>v_2t,\quad v_1<v_2,\\[1.2ex] 0, & \text{in all other cases} \end{cases} \]

\[ -\pi i \begin{cases} 1, & x>v_1t,\quad v_1>v_2,\\ \displaystyle -1+2\exp\!\left[q(x-v_1t)(v_1-v_2)^{-1}\right], & x<v_1t,\quad v_1>v_2,\\[1.2ex] \displaystyle 1-2\exp\!\left[q(x-v_1t)(v_1-v_2)^{-1}\right], & x>v_1t,\quad v_1<v_2,\\[1.2ex] -1, & x<v_1t,\quad v_1<v_2. \end{cases} \tag{4.4} \]

Case 4.A: \(v_1>v_2\). From the preceding equality (4.4) it follows easily that

\[ \int_{-\infty}^{+\infty}\frac{\varphi_2}{\lambda}\,d\lambda = \pi i \begin{cases} 1, & v_2>0,\\ \displaystyle 1-2\exp\!\left[-qv_1t\,(v_1-v_2)^{-1}\right], & v_2<0<v_1,\\[1.2ex] -1, & v_1<0. \end{cases} \tag{4.5} \]

For simplicity of the discussion, we assume \(z>0\), and write the inversion formula (4.3) in the form

\[ \Phi_2(t,z)-\Phi_2(t,0)= \frac{1}{2\pi i}\int_{-\infty}^{+\infty} \frac{1-\exp(-i\lambda z)}{\lambda}\, \varphi_2(\lambda,t)\,d\lambda, \]

since

\[ \int_{-\infty}^{+\infty}\frac{\varphi_2}{\lambda}\,d\lambda = \pi i \begin{cases} 1, & v_2>0,\\ \displaystyle 1-2\exp\!\left[-qv_1t\,(v_1-v_2)^{-1}\right], & v_2<0<v_1,\\[1.2ex] -1, & v_1<0, \end{cases} \tag{4.6} \]

then it is easy to calculate

\[ \Phi_2(t,z)= \begin{cases} 0, & z<v_2t,\\ \exp\left[q(z-v_1t)(v_1-v_2)^{-1}\right], & v_2t<z<v_1t,\\ 1, & v_1t<z. \end{cases} \tag{4.7} \]

Next it is easy to verify that

\[ \Phi_1(t,z)= \begin{cases} 0, & z<v_1t,\\ 1, & z>v_1t. \end{cases} \tag{4.8} \]

If the initial probability distribution is

\[ p_i=P[y(0)=a_i], \]

then the unconditional distribution of the random variable \(z_t\) will be, for \(z>0\),

\[ \Phi(t,z)= \begin{cases} 0, & z<v_2t,\\ p_2\exp\left[q(z-v_1t)(v_1-v_2)^{-1}\right], & v_2t<z<v_1t,\\ 1, & v_1t<z. \end{cases} \tag{4.9} \]

Using relation (1.2) for the probability distribution of the trajectories of the differential equation (1), we obtain the following:

\[ P\{|x(t)|<x\}= \begin{cases} 0, & |x|<|x_0|\exp v_2t,\\ p_2\exp\left[q\bigl(\ln|x_0^{-1}x|-v_1t\bigr)(v_1-v_2)^{-1}\right], & |x|<|x_0|\exp v_1t,\quad |x|>|x_0|\exp v_2t,\\ 1, & |x|>|x_0|\exp v_1t. \end{cases} \tag{4.10} \]

Case 4.B: \(v_1<v_2\). Arguments analogous to the preceding case make it possible to write the following formula for the probability distribution of the trajectories of the differential equation (1):

\[ P\{|x(t)|<x\}= \begin{cases} 0, & |x|<|x_0|\exp v_1t,\\ 1-p_2\exp\dfrac{q\bigl(\ln|x_0^{-1}x|-v_1t\bigr)}{v_1-v_2}, & |x_0|\exp v_1t<|x|<|x_0|\exp v_2t,\\ 1, & |x_0|\exp v_2t<|x|. \end{cases} \tag{4.11} \]

§ 5. Here we consider the case \(q_1=q_2=q,\ v_1=-v_2=v\), where \(v_i=v(a_i)\). In this case the solution of E. B. Dynkin’s equation (4) will be the following pair of functions:

\[ \varphi_1(\lambda,t)=\exp(-qt) \left[ \frac{q+i\lambda v}{\sqrt{q^2-\lambda^2v^2}}\, \operatorname{sh} t\sqrt{q^2-\lambda^2v^2} +\operatorname{ch} t\sqrt{q^2-\lambda^2v^2} \right], \]

\[ \varphi_2(\lambda,t)=\exp(-qt) \left[ \frac{q-i\lambda v}{\sqrt{q^2-\lambda^2v^2}}\, \operatorname{sh} t\sqrt{q^2-\lambda^2v^2} +\operatorname{ch} t\sqrt{q^2-\lambda^2v^2} \right]. \tag{5.1} \]

The direct application of the inversion formula to these characteristic functions appears to be very difficult. Therefore, in order to find the corresponding distribution functions, we consider an indirect route. First of all, proceeding only from the form of the functions \(\varphi_k(\lambda,t)\), one can obtain interesting information about the behavior of the trajectories of the differential equation

\[ \dot{x}=v[y(t)]x \]

in the case of an equiprobable initial distribution, i.e., if

\[ P[y(0)=a_1]=P[y(0)=a_2]. \tag{5.2} \]

Theorem. If the initial distribution is such that (5.2) is satisfied, then the probability distribution for the trajectories has the property

\[ F(x_0,x,t)=1-F(x,x_0,t), \tag{5.3} \]

i.e., in other words:

\[ P[x(x_0,t)<x]=P[x(x,t)>x_0],\qquad x_0>0. \]

Proof. If the initial distribution is such that (5.2) is satisfied, then the materiality of the corresponding unconditional characteristic function is obvious, and this entails the property [6]

\[ \Phi(t,z)=1-\Phi(t,-z). \tag{5.4} \]

From (1.2) and (5.4), (5.3) follows.

Remark. The geometric meaning of this theorem is quite simple. Namely, if a trajectory starts from the point \(x_0>0\), then the probability of the inequality \(x(t)<x\) will be exactly the same as for the inequality \(x_1(t)>x_0\), where \(x_1(t)\) is the trajectory starting from the point \(x\).

§ 6. Let \(q_1=q_2=q,\ v_1=-v_2=v\). Denote by \(\psi_i(\lambda,p)\) the Laplace transform of the function \(\varphi_i(\lambda,t)\), i.e.

\[ \psi_i(\lambda,p)=p\int_0^\infty \varphi_i(\lambda,t)\exp(-pt)\,dt, \tag{6.1} \]

and, passing in system (3.3) to the Laplace transform and solving the obtained system, we shall have

\[ \psi_1(\lambda,p)= \frac{p(p+q)+p(q+i\lambda v)} {(p+q)^2+v^2(\lambda^2-m^2)}, \]

\[ \psi_2(\lambda,p)= \frac{p(p+q)+p(q-i\lambda v)} {(p+q)^2+v^2(\lambda^2-m^2)},\qquad m=\frac{q}{|v|}. \tag{6.2} \]

Compute the integral needed for what follows:

\[ J=\int_{-\infty}^{+\infty}\frac{\exp(-i\lambda x)}{\lambda}\,\varphi_1(\lambda,t)\,d\lambda. \]

For this purpose, to the subintegral function \(\varphi_1(\lambda,t)\) we apply the inversion formula for the Laplace transform

\[ J=\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty} \frac{\exp(pt)}{p} \int_{-\infty}^{+\infty} \frac{p(p+q)+p(q+i\lambda v)} {\lambda\left[(p+q)^2+v^2\lambda^2-p^2\right]} \exp(-i\lambda x)\,d\lambda\,dp. \tag{6.3} \]

The computations give the following result:

\[ J=\frac{\operatorname{sgn}x}{2} \begin{cases} 0, & t<|x|,\\[6pt] \displaystyle \exp\!\left(-\frac{q|x|}{2}\right) -\frac{qx}{2}\int_{|x|}^{t} \exp\!\left(-\frac{qy}{2}\right) \frac{J_1\!\left(-\frac{q}{2}\sqrt{y^2-x^2}\right)} {\sqrt{y^2-x^2}}\,dy, & |x|<t, \end{cases} \]

\[ \quad +\frac{1}{2} \begin{cases} 0, & t<|x|,\\[6pt] \displaystyle \exp\!\left(-\frac{qt}{2}\right) J_0\!\left(-\frac{q}{2}\sqrt{t^2-x^2}\right), & t>|x|, \end{cases} -\frac{1}{2}\operatorname{sgn}x. \tag{6.4} \]

Using the inversion formula (4.3) for characteristic functions, we obtain for \(z>0\)

\[ \Phi_1(t,z)=\frac{1}{2}-J. \tag{6.5} \]

From relation (1.2) we have

\[ P[x(t)<x\mid y(0)=a_1] = \frac{1}{2} - J\left(t,\ln\frac{x}{x_0}\right), \qquad x>x_0>0. \tag{6.6} \]

In an analogous manner one can obtain an expression also for

\[ P[x(t)<x\mid y(0)=a_2]. \]

References

1. Dynkin E. B. Dokl. Akad. Nauk SSSR, 104, No. 5, 691—694, 1955.
2. Skorokhod A. V. Random Processes with Independent Increments, 1964.
3. Gelfand I. M., Yaglom A. M. Uspekhi Mat. Nauk, 2 (67), 77—114, 1956.
4. Doob J. L. Stochastic Processes. Moscow, IL, 1956, p. 218.
5. Loeve M. Probability Theory. Moscow, IL, 1962, p. 615.
6. Khinchin A. Ya. Limit Laws for Sums of Independent Random Variables. ONTI, 1938, p. 11.

Received by the editors
September 20, 1965.

Ural Polytechnic Institute
named after S. M. Kirov