A remark on conditions that a diffusion in the natural scale is a martingale

TL;DR

This paper investigates conditions under which a diffusion process in the natural scale behaves as a martingale, providing simple and analytic proofs for existing results.

Contribution

It offers straightforward, analytic proofs for known conditions that ensure a diffusion in natural scale is a martingale.

Findings

Identifies conditions for martingale property in diffusions

Provides simplified proofs for existing results

Enhances understanding of diffusion behavior in natural scale

Abstract

We consider a diffusion processes $\{ X_t \}$ on an interval in the natural scale. Some results are known under which $\{ X_t \}$ is a martingale, and we give simple and analytic proofs for them.

Equations

$${\tau }_a$$

$$X_t^{\tau }$$

$${\int }_{[r,l_{+})}xm(dx)=\infty ,r\in (l_{-},\infty )$$

$${\int }_{[r,l_{+})}xm(dx)=\infty \text{mbox}and{\int }_{(l_{-},r]}|x|m(dx)=\infty ,r\in (-\infty ,\infty ).$$

$$\mathcal{L}f:=\frac{d^2}{dmdx}f$$

$$\underset{z\to \infty }{\lim }{f}_{-}^{\prime }(z)=\infty$$

$$\underset{z\to \infty }{\lim }{f}_{-}^{\prime }(z)=\infty \text{mbox}and\underset{z\to -\infty }{\lim }{f}_{+}^{\prime }(z)=-\infty .$$

$$\underset{t\to \infty }{\lim }{E}_x[X_{t\wedge {\tau }_{-}}]=l_{-}$$

$$f_{-}^z(x)=f_{-}(x)-\frac{f_{-}(z)}{f_{+}(z)}{f}_{+}(x),f_{+}^z(x)=f_{+}(x).$$

$$f_{-}^{\prime }(x)$$

$$\underset{z\to \infty }{\lim }z{\int }_0^{\infty }{e}^{-\lambda t}{P}_x({\tau }_z<t)dt$$

$${\int }_0^{\infty }{e}^{-\lambda t}\underset{z\to \infty }{\mathrm{liminf}}z{P}_x({\tau }_z<t)dt=0.$$

$$\underset{n\to \infty }{\lim }{z}_n{P}_x({\tau }_{z_n}<t)=0.$$

$$x\le E_x[X_{t\wedge {\tau }_{z_n}}]=z_n{P}_x({\tau }_{z_n}<t)+E_x[X_t;{\tau }_{z_n}\ge t].$$

$$G(x,y,\lambda )$$

$$h$$

$$\int^{\infty}_{l_{-}}G(x,y,\lambda)(y-l_{-})m(dy)=E_{x}\Big{[}\int^{\infty}_{0}e^{-\lambda t}(X_{t\wedge\tau_{-}}-l_{-})dt\Big{]}.$$

$$f_{+}(x)={\alpha }_{+}-{\int }_x^{\infty }{f}_{+}^{\prime }(y)dy.$$

$$f_{+}^{\prime }(x)$$

$$f_{+}(x)$$

$$\lambda {\int }_x^{\infty }y{f}_{+}(y)m(dy)$$

$$f_{-}^{\prime }(y)$$

$$f_{-}(x)$$

$$\lambda {\int }_{l_{-}}^{x}y{f}_{-}(y)m(dy)$$

$${\int }_0^{\infty }{e}^{-\lambda t}{E}_x[X_{t\wedge {\tau }_{-}}-l_{-}]dt$$

$$\lim_{\lambda\to 0}\Big{(}x-l_{-}-\frac{\alpha_{+}f_{-}(x)}{h}\Big{)}=0.$$

$$\varphi (0)=1,{\varphi }^{\prime }(0)=0$$

$$\psi (0)=0,{\psi }^{\prime }(0)=1.$$

$$f_{+}(x)=\varphi (x)-\left(\underset{x\to \infty }{\lim }\frac{\varphi (x)}{\psi (x)}\right)\psi (x),f_{-}(x)=\varphi (x)-\left(\underset{x\to l_{-}}{\lim }\frac{\varphi (x)}{\psi (x)}\right)\psi (x).$$

$$\varphi (x)=1+\sum _{n=1}^{\infty }{\lambda }^n{\varphi }_n(x),{\varphi }_0(x)=1,{\varphi }_n(x)={\int }_0^x(x-y){\varphi }_{n-1}(y)m(dy)$$

$$\psi (x)=x+\sum _{n=1}^{\infty }{\lambda }^n{\psi }_n(x),{\psi }_0(x)=x,{\psi }_n(x)={\int }_0^x(x-y){\psi }_{n-1}(x)m(dy)$$

$$\underset{\lambda \to 0}{\lim }\varphi (x)=1,\underset{\lambda \to 0}{\lim }{\varphi }^{\prime }(x)=0,\underset{\lambda \to 0}{\lim }\psi (x)=x,\underset{\lambda \to 0}{\lim }{\psi }^{\prime }(x)=1.$$

$$\underset{\lambda \to 0}{\lim }\left(-\underset{x\to {\text{l}}_{-}}{\lim }\frac{\psi (x)}{\varphi (x)}\right)=\underset{\lambda \to 0}{\lim }\left({\int }_{l_{-}}^0\frac{1}{(\varphi (x){)}^2}dx\right)={\int }_{l_{-}}^{0}dx=-l_{-}$$

$$\underset{\lambda \to 0}{\lim }{f}_{-}(x)=1-\frac{x}{l_{-}},\underset{\lambda \to 0}{\lim }{f}_{-}^{\prime }(x)=-\frac{1}{l_{-}}.$$

Full text

Mathematics Subject Classification (2010): 60J60, 60G44

1 Introduction

Let $-\infty\leq l_{-}<l_{+}\leq\infty$ and let $m$ be a Borel measure with $\mbox{ supp }m=(l_{-},l_{+})$. We denote by \Bigl{\{}\{X_{t}\}_{t\geq 0},\{P_{x}\}_{x\in(l_{-},l_{+})}\Bigr{\}} the minimal diffusion process on $(l_{-},l_{+})$ with the speed measure $m$ and the scale function $s(x)=x$. It is well known that a local martingale $\{X_{t}\}$ is a martingale if and only if $\{X_{T}:T\mbox{ is a stopping time with }\;T\leq t\}$ is uniformly integrable for any $t\geq 0$. Here our aim is to have more explicit condition for the one-dimensional diffusions in the natural scale. If $|l_{\pm}|<\infty$, $\{X_{t}\}$ is bounded so that it is a martingale. If $l_{-}=-\infty$, $l_{+}<\infty$, this can be reduced to the case of $l_{-}<\infty$, $l_{+}=\infty$ by replacing $X_{t}$ by $-X_{t}$. Hence it suffices to consider the following two cases.

Case I : $-\infty<l_{-}$, $l_{+}=+\infty$,  Case II : $l_{-}=-\infty$, $l_{+}=+\infty$.

Let $P(l_{-},l_{+})$ be the set of Borel measures on $(l_{-},l_{+})$, and for $\mu\in P(l_{-},l_{+})$ let $P_{\mu}(\cdot):=\int_{(l_{-},l_{+})}P_{x}(\cdot)\mu(dx)$. According to Lemma 4.1 ([1], Lemma 2), $\{X_{t}^{\tau}\}$ is a $P_{\mu}$-martingale for some $\mu\in P(l_{-},l_{+})$ with $\int_{(l_{-},l_{+})}|x|\mu(dx)<\infty$ if and only if $\{X_{t}^{\tau}\}$ is $P_{x}$-martingale for any $x\in(l_{-},l_{+})$. We further set

[TABLE]

Kotani [1] showed the following theorem.

Theorem 1.1

[1]* $\{X^{\tau}_{t}\}$ is a $P_{x}$-martingale for any $x\in(l_{-},l_{+})$ if and only if

Case I :*

[TABLE]

Case II :

[TABLE]

By Feller’s criterion, $P_{x}(\tau_{\sharp}=\infty)=1$ if $|l_{\sharp}|=\infty$, $\sharp=\pm\infty$. Thus Theorem 1.1 implies that $\{X^{\tau}_{t}\}$ is a martingale if and only if the boundaries at infinity are natural. Hulley, Platen [2] derived another condition. Let

[TABLE]

be the generator of $\{X_{t}\}$ and for $\lambda>0$ let $f_{-}$ (resp. $f_{+}$) be the positive increasing (resp. positive decreasing) solution to the equation ${\cal L}f=\lambda f$, which are unique up to constants unless the boundary is regular.

Theorem 1.2

[2]* $\{X^{\tau}_{t}\}$ is a $P_{x}$-martingale for any $x\in(l_{-},l_{+})$ if and only if

Case I :*

[TABLE]

Case II :

[TABLE]

Gushchin, Urusov, and Zervos [3] derived a condition that $\{X^{\tau}_{t}\}$ is a submartingale or a supermartingale.

Theorem 1.3

[3]* $\{X^{\tau}_{t}\}$ is a $P_{x}$-submartingale if and only if $\int_{r}^{\infty}xm(dx)=\infty$, $r\in(l_{-},l_{+})$.*

By [2] Proposition 3.16, 3.17, this condition is equivalent to $\lim_{t\to\infty}f^{\prime}_{-}(t)=\infty$. Together with Theorem 1.3 we thus have

Theorem 1.4

$\{X^{\tau}_{t}\}$* is a $P_{x}$-submartingale if and only if $\lim_{t\to\infty}f^{\prime}_{-}(t)=\infty$.*

Moreover in [3], they further derived a condition in Case I such that $\{X^{\tau}_{t}\}$ is a strict $P_{x}$ supermartingale, that is, $\{X^{\tau}_{t}\}$ is a $P_{x}$-supermartingale but is not a $P_{x}$-martingale.

Theorem 1.5

[3]* Let $-\infty<l_{-}$, $l_{+}=\infty$. Then $\{X^{\tau_{-}}_{t}\}$ is a strict $P_{x}$-supermartingale if and only if*

[TABLE]

for any $x\in(l_{-},l_{+})$.

We believe that Theorem 1.5 is also true for $l_{-}=-\infty$. The goal of this paper is :

(1) To give a simple analytic proof of Theorem 1.4 without using the results in [2]. We note that the proofs of Proposition 3.16, 3.17 in [2] is more or less probabilistic using Tanaka’s formula.

(2) To give a simple analytic proof of Theorem 1.5 ; the original proof of that in [3] is done by embedding $\{X_{t}\}$ into the geometric Brownian motion on the torus.

The rest of this paper is organized as follows. In Section 2(resp. Section 3), we give a proof of Theorem 1.4 (resp. Theorem 1.5). In Appendix, we prepare some tools for these proofs.

2 A proof of Theorem

1.4

In Case I, the statement follows from Theorem 1.2, for $\{X^{\tau_{-}}_{t}\}$ is always a $P_{x}$-supermartingale being bounded from below. Henceforth we consider Case II.

Suppose $\{X_{t}\}$ is a $P_{x}$-submartingale and let $z<x$. Then $\{X^{\tau_{z}}_{t}\}$ is bounded from blow so that it is a $P_{x}$-martingale. For $\lambda>0$, let $f^{z}_{-}$ (resp. $f^{z}_{+}$) be the positive increasing (resp. positive decreasing) solution to the equation ${\cal L}f=\lambda f$ such that $f^{z}_{-}(z)=0$. Then we have

[TABLE]

Since $f_{+}^{\prime}$ is increasing, we have

[TABLE]

Applying Theorem 1.2 to $\{X^{\tau_{z}}_{t}\}$ yields $\lim_{t\to\infty}{f_{-}^{z}}^{\prime}(t)=\infty$ and thus $\lim_{t\to\infty}f^{\prime}_{-}(t)=\infty$.

Conversely, suppose $\lim_{t\to\infty}f^{\prime}_{-}(t)=\infty$ and let $z<x$. Then

[TABLE]

where we used Lemma 4.3 and l’Hospital’s rule. By Fatou’s lemma,

[TABLE]

Hence $\liminf_{z\to\infty}zP_{x}(\tau_{z}<t)=0$ so that we can find a sequence $\{z_{n}\}\subset(x,\infty)$ with $\lim_{n\to\infty}z_{n}=\infty$ such that

[TABLE]

On the other hand $\{X^{\tau_{z_{n}}}_{t}\}$ is a $P_{x}$-submartingale being bounded from above and

[TABLE]

Since $\lim_{n\to\infty}P_{x}(\tau_{z_{n}}\geq t)=1$, $x\leq E_{x}[X_{t}]$. Markov property implies $\{X_{t}\}$ is a $P_{x}$-submartingale.

3 A proof of Theorem 1.5

Without losing generality, we may suppose $l_{-}<0$. For $\lambda>0$, let $f_{-}$ (resp. $f_{+}$) be the positive increasing (resp. positive decreasing) solution to the equation ${\cal L}f=\lambda f$ such that $f_{-}(l_{-})=0$. Let $G$ be Green’s function of ${\cal L}$ :

[TABLE]

Then we have

[TABLE]

Let $\alpha_{+}:=\lim_{t\to\infty}f_{+}(t)$. Then $f_{+}^{\prime}\in L^{1}(a,\infty)$ for $a\in(l_{-},\infty)$ and

[TABLE]

Therefore $\lim_{x\to\infty}f^{\prime}_{+}(x)=0$. The equation ${\cal L}f_{+}=\lambda f_{+}$ yields

[TABLE]

so that we have

[TABLE]

Similarly,

[TABLE]

Substituting them into (3.2) yields

[TABLE]

We note that (3.3) and Lemma 4.1 also proves Theorem 1.1 in Case I.

Suppose $\{X^{\tau_{-}}_{t}\}$ is a strict $P_{x}$-supermartingale. The discussion above implies $\alpha_{+}>0$. We shall show below that

[TABLE]

Let $\phi$, $\psi$ be the solution to ${\cal L}f=\lambda f$ with the initial condition

[TABLE]

Then $f_{\pm}$ satisfy

[TABLE]

$\psi$, $\psi$ can be composed by the method of successive approximation :

[TABLE]

which is convergent locally uniformly w.r.t. $\lambda$ [4] which yields

[TABLE]

Moreover

[TABLE]

implies

[TABLE]

On the other hand, by $\alpha_{+}>0$ and by Lemma 4.2, we have $\int^{\infty}_{r}xm(dx)<\infty$, $r\in(l_{-},\infty)$ so that we can find $g$ satisfying

[TABLE]

by successive approximation. Using $\alpha_{+}>0$, $\lim_{t\to\infty}f^{\prime}_{+}(t)=0$, $\lim_{t\to\infty}g(t)=1$ and $\lim_{t\to\infty}g^{\prime}(t)=0$, we have

[TABLE]

which implies $f_{+}(x)=Cg(x)$ for some positive constant $C$. Because $\lim_{\lambda\to 0}g(x)=1$, $\lim_{\lambda\to 0}g^{\prime}(x)=0$,

[TABLE]

Therefore

[TABLE]

proving (3.4). Since $X_{t\wedge\tau_{-}}$ is a supermartingale, $f(t):=E_{x}[X_{t\wedge\tau_{-}}-l_{-}]\in C^{1}[0,\infty)$ is monotone decreasing which shows that $\lim_{t\to\infty}f(t)$ exists and $f^{\prime}\in L^{1}(0,\infty)$. Thus by (3.3) and Lemma 4.4

[TABLE]

Conversely, suppose that $\displaystyle\lim_{t\to\infty}E_{x}[X_{t\wedge\tau_{-}}-l_{-}]=0$. Then

[TABLE]

which implies $\alpha_{+}>0$ since otherwise it would contradict to (3.3), (3.4). Therefore $\{X^{\tau_{-}}_{t}\}$ is not a martingale.

4 Appendix

Lemma 4.1

**(Lemma 2 in [1])

**Suppose $\{X_{t\wedge\tau_{-}}\}$ is a $P_{\mu}$-martingale for some \mu\in P\big{(}l_{-},\infty\big{)}. Then for any $t\geq 0$, $x\in(l_{-},\infty)$,

[TABLE]

Conversely, if (4.1) is valid, then $\{X_{t\wedge\tau_{-}}\}$ is a $P_{\mu}$-martingale for any \mu\in P\big{(}l_{-},\infty\big{)} with $\int_{l_{-}}^{\infty}|x|\mu(dx)<\infty$.

Lemma 4.2

Let $\lambda>0$ and let $f_{+}$ be the positive decreasing solution to ${\cal L}f=\lambda f$ with $\displaystyle\alpha_{+}:=\lim_{x\to\infty}f_{+}(x)$. Then the following three conditions are equivalent.

[TABLE]

Lemma 4.3

Let $f_{\pm}$ be the ones defined in the proof of Theorem 1.5. Then

[TABLE]

Lemma 4.4

*Suppose $f\in C^{1}[0,\infty)$ and $f^{\prime}\in L^{1}(0,\infty)$. Then

(1) $\lim_{t\to\infty}f(t)$ exists, and

(2) $\lim_{t\to\infty}f(t)=\lambda\lim_{\lambda\downarrow 0}\int_{0}^{\infty}e^{-\lambda t}f(t)dt.$*