Perturbed uncertain differential equations and perturbed reflected canonical process

1.   Introduction

There now exists a considerable body of literature devoted to the study of 'perturbed' versions of familiar stochastic and deterministic equations. An example is Carmona et al. ^[1] and Norris et al. ^[2] investigated the following doubly perturbed Brownian motion

We study 'perturbed canonical process', that can be, loosely speaking, described as follows: they behave exactly as a canonical process has stationary and independent increments except when they hit their past maximum or/and minimum where they get an extra 'push'. Many researchers have devoted themselves to studying the perturbed process (see ^[3,4,5,6]). Following them, Doney and Zhang ^[7] studied the following singly perturbed Skorohod equations

The authors proved the existence and uniqueness of the solution for (1.2) when the coefficients $b$, $\sigma$ is the global Lipschitz.

On the other hand, to describe the evolution of the uncertain phenomenon, Liu ^[8] proposed uncertain process and designed a Liu process ^[9]. Meanwhile, Liu ^[9] introduced uncertain calculus to handle the integral and differential with respect to an uncertain process. Uncertain differential equations driven by a Liu process, which were first proposed by Liu ^[8], have attracted the increasing attentions due to the wide applications in in many fields such as finance (^[10]), optimal control (^[11,12]), differential game (^[13]), population growth (^[14]), heat conduction (^[15]), string vibration (^[16]), spring vibration (^[17]), and epidemic spread (^[18]).

As far as we known, there is no result on the perturbed uncertain differential equations. Motivated by the need of the applications and in connection with the above discussions, it is worthwhile to develop some techniques and methods to explore the perturbed uncertain differential equations. To this end, in this paper, we will investigate the following perturbed uncertain differential equations,

where $C$ is a canonical process starting from $0$, $\alpha < 1$ is a real constant, $\sigma(x)$, $b(x)$ be Lipschitz continuous function on $R$. By the reflection principle and a successive approximation method, we obtain the existence and uniqueness of the solution to the considered equations.

Our other main aim is to deal with the analogous question for a general diffusion. Specifically we study the equation

where $\sigma$ is a Lipschitz continuous function on $R$, $\alpha < 1$ is a real constant, $x\geq0$, and $L_{t}$ denotes a local time at zero of $X$. Since the cases $x = 0$ and $x > 0$ are quite different, we will treat them separately. Finally we exploit our result on the uncertain differential equation together with Picard iteration to establish existence and uniqueness of a solution to (1.4).

The rest of the paper is organized as follows. Some preliminary concepts of uncertainty theory are recalled in Section 2. The method to solve perturbed uncertain differential equations is presented in Section 3. An existence and uniqueness theorem for perturbed reflected canonical process is proved in Section 4. At last, a brief summary is given in Section 5.

2.   Preliminaries

In this section, we will introduce some foundational concepts and properties of uncertainty theory, which will be used throughout this paper.

Theorem 2.1. (^[19]) An uncertain process $C_{t}$ is said to be a canonical process if

(ⅰ) $C_{0} = 0$ and almost all sample paths are Lipschitz continuous;

(ⅱ) $C_{t}$ has stationary and independent increments;

(ⅲ) every increment $C_{s+t}-C_{s}$ is a normal uncertain variable with expected value $0$ and vartiance $t^{2}$.

It is clear that a canonical process $C_{t}$ is a stationary independent increment process with normal uncertainary distribution

and inverse uncertainty distribution

Theorem 2.2. (^[20]) Let $C_{t}$ be a canonical process. Then its expected value is

and variance is

In other words, Liu process $C_{t}$ is a normal uncertain process with expected value $0$ and variance $t^{2}$, i.e., $C_{t}\sim\mathcal{N}(0, t)$.

Theorem 2.3. (^[21]) Set $W_{0} = \{f\in C([0, \infty)\rightarrow R); f(0) = 0\}$ and $W^{+} = \{f\in C([0, \infty)\rightarrow R); f(t)\geq0 \; for \; all \; t\geq0\}$. Given $f\in W_{0}$ and $0\leq \alpha < 1$, there exist unique $g\in W^{+}$ and $h\in W^{+}$ such that

(i) $g(t) = f(t)+\alpha\max_{0\leq s\leq t}g(s)+h(t)$;

(ii) $h(0) = 0$ and $t\rightarrow h(t))$ is non-decreasing;

(iii) $\int^{t}_{0}\chi\{g(s) = 0\}dh(s) = h(t)$.

$(g, h)$ is called a solution to the perturbed Skorohod equation for the function $f$.

Lemma 2.1. (^[22]) Suppose that $C_{t}$ is a canonical process, and $X_{t}$ is an integrable uncertain process on $[a, b]$ with respect to $t$. Then the inequality

holds, where $K(\gamma)$ is the Lipschitz constant of the sample path $X_{t}(\gamma)$.

3.   Perturbed diffusion processes

In this section, we assume that $\sigma(x)$, $b(x)$ be Lipschitz continuous function on $R$, i.e., there exists a constant $c$ such that

and linear growth condition

For $\alpha < 1$, consider the following uncertain differential equation:

Theorem 3.1. Assume that the random variable $X_{0}$ is independent of $L$. There exists a unique, continuous, $\mathscr{F}$-adpted solution $X_{t}$, $t\geq0$ to the uncertain differential (1.3) for any $T > 0$ if the coefficients $\sigma(X_{t})$ and $b(X_{t})$ satisfy the assumption (3.1)-(3.3) for some constants $c > 0$.

We construct the solution by iteration. Let

For $n\geq{0}$ define $X^{n+1}_{t}$ to be the unique, continuous, adapted solution to the following equation:

Such a solution exists and can be expressed explicitly as

This is a consequence of the reflection principle. We will show that $X^{n}$ converges uniformly on compact intervals almost surely. It following from (3.7) that

where we used the fact that $|\max_{0\leq v \leq s }f(v)-\max_{0\leq v \leq s }g(v)|\leq\max_{0\leq v \leq s }|f(v)-g(v)|$ holds for any two continuous functions $f$ and $g$. Thus,

For any sample $\gamma$, we define

We claim that

where $T$ is a constant. Indeed for $n = 0$, it follows from Lemma 2.1 that

Then

This confirms the claim for $n = 0$. Next we assume the claim is true for $n-1$. Then

Note that (3.13) and (3.14) are induced form Lemma 2.1 and the inductive assumption, respectively. This proves the claim. Therefore,

holds for all $n\geq0$. It follows from Weierstrass' criterion that, for each sample $\gamma$,

Thus $X^{k}_{t}(\gamma)$ converges uniformly in $t\in[0, T]$. We denote the limit by

Then

Therefore $X_{t}$ is the solution of (1.3) for all $t\geq0$ since $T$ is arbitrary.

Next, we will prove that the solution of uncertain differential (1.3) is unique. Assume that both of $X_{t}$ and $X_{t}^{*}$ are solutions of (1.3) with the same initial value $X_{0}$. Then

Arguing as above, there is a constant $C$ such that

Then for each $\gamma\in\Gamma$, we have

It follows from Gronwall inequality that

for any $\gamma$. Hence $X_{t} = X_{t}^{*}$, the solution is unique. The theorem is proved.

4.   Perturbed reflected diffusions

Let $\sigma$ be as in Section 2. For $x\geq0$, consider the uncertain differential equation:

Definition 4.1 We say that $(X_{t}, L_{t}, t\geq0)$ is a solution to (1.4) if

(ⅰ) $X_{0} = x$, $X_{t}\geq0$ for $t\geq0$;

(ⅱ) $X_{t}$, $L_{t}$ are adapted to the filtration of $C$;

(ⅲ) $L_{t}$ is non-decreasing with $L_{0} = 0$ and

(ⅳ) $(X_{t}, L_{t}, t\geq0)$ satisfies (4.1) almost surely for every $t > 0$.

The cases $x = 0$ and $x > 0$ are quite different. We will treat them separately.

Theorem 4.1. Assume $\alpha < 1$ and $\sigma$ is Lipschitz. If $x > 0$, there exists a unique solution $(X_{t}, L_{t}, t\geq0)$ to (1.4).

Proof. We construct the solution iteratively in a similar way to (3.11). Define $Y^{0}_{t}$ to be the unique solution to the equation:

It is known from Section 2 that such a solution exists. Set $T_{1} = \inf \{t\geq0;Y^{0}_{t} = 0\}$. Then $T_{1} > 0$ a.s. as $x > 0$. Define

Put $C_{t}^{1} = C_{t+T_{1}}-C_{T_{1}}$ for $t\geq0$. It is well known that $C^{1}_{t}$, $t\geq0$ is a normal uncertain variable with expected value $0$ and variance $t^{2}$. Consider the uncertain differential equation with reflecting boundary:

The definition of a solution to this equation is the same as Definition 4.1 with $x = 0$ and $\alpha = 0$. It is known that a unique solution $(Z^{1}_{t}, L^{1}_{t})$ to the (4.4) exists, see e.g. ^[11] or ^[15]. In general, suppose that $(X_{t}, L_{t})$ has been defined for $0\leq t\leq T_{2n-1}$. We can construct $(X_{t}, L_{t})$ for $T_{2n-1}\leq t\leq T_{2n+1}$ as follows. Let $Z^{2n-1}_{t}$ be the solution to the equation:

where $C^{2n-1}_{t} = C_{t+T_{2n-1}}$. Put $T_{2n} = \inf\{t > T_{2n-1};Z^{^{2n-1}_{t-T_{2n-1}}} = \max_{0\leq s\leq T_{2n-1}}X_{s}\}$ and set

Let $Y^{2n}_{t}$ denote the solution to equation:

where $C^{2n}_{t} = C_{t+T_{2n}}-C_{T_{2n}}$. Set $T_{2n+1} = \inf\{t > T_{2n}, Y_{t-T_{2n} = 0}\}$ and

By this procedure, we obtain a sequence of increasing stopping times $T_{n}$, $n\geq 0$. Set $T = \lim_{n\rightarrow \infty T_{n}}$. Then $T$ is again a stopping time, and $(X_{t}, L_{t})$ is a well defined continuous process for all $0\leq t < T$. We will show that $(X_{t}, L_{t}, t < T)$ satisfies (1.4) in the sense of Definition 4.1. To achieve this, it is sufficient to prove that $(X_{t}, L_{t})$ satisfies (1.4) for $T_{2n}\leq t\leq T_{2n+1}$ and $n = 0, 1...$. We will do this by induction. It is obvious that $(X_{t}, L_{t})$ is a solution to (1.4) for $0\leq t < T_{1}$. If $T_{1}\leq t\leq T_{2}$, it follows that

since $max_{0\leq s\leq T_{1}}X_{s} = max_{0\leq s\leq t}X_{s}$ for $T_{1}\leq t\leq T_{2}$, and $X_{T_{1}} = 0$.

Furthermore, if $T_{1}\leq t\leq T_{2}$, we see that

Thus we have showed that $(X_{t}, L_{t})$ is a solution to (4.1) for $0\leq t\leq T_{2}$. Suppose that $(X_{t}, L_{t})$ satisfies (4.1) for $0\leq t\leq T_{2n}$. If $T_{2n}\leq t\leq T_{2n+1}$, it follows that

where we have used the fact that $X_{T_{2n}} = \max_{0\leq s\leq T_{2n}}X_{s}$ and $Y^{2n}_{0} = X_{T_{2n}}$ from their definitions. Since $X_{t}\neq0$ for $T_{2n}\leq t < T_{2n+1}$, we also have

So $(X_{t}, L_{t})$ satisfies (4.1) also for $T_{2n}\leq t < T_{2n+1}$. Repeating similar arguments as for (4.10), we also can show that $(X_{t}, L_{t})$ satisfies (4.1) for $T_{2n+1}\leq t < T_{2n+2}$.

Finally we show that $T = \infty$ a.s. By the construction of $X$, we can have that

Suppose $T < \infty$ with positive probability. Letting $n\rightarrow \infty$ in (4.13), we get $0 = \max_{0\leq s\leq T}X_{s}$ which contradicts the fact that $X_{0} = (1-\alpha)^{-1}x > 0$. The proof of existence is complete.

On the other hand, it is easily seen that the solution is unique since it is unique on each interval $[T_{n}, T_{n+1}]$.

Theorem 4.2. Assume $x = 0$. If $0\leq \alpha < \frac{1}{2}$, then there exists a unique solution $(X_{t}, T_{t}, t\geq0)$ to (1.4).

Proof. We will use the Picard iteration method. Define $X^{0}_{t}\equiv0$ and $(X^{n+1}_{t}, L^{n+1}_{t})$ to be the unique solution to the equation:

The existence and uniqueness of this solution follow from Section 3. Observe that by the reflection principle,

Now (4.14) and (4.15) imply that

Consequently,

Let $\beta = \sup|X^{1}_{s}-X^{0}_{s}|$. Then

holds for all $n\geq1$. It follows from Weierstrass' criterion that, for each sample $\gamma$,

Thus $X^{n}_{s}$ converges uniformly to a continuous, adapted process $X$ on $[0, T]$ almost surely. It is also seen that $M_{n}(t): = \int^{t}_{0}\sigma(X^{n}_{s})dC_{s}$ converges uniformly on $[0, T]$ to $M_{n}(t): = \int^{t}_{0}\sigma(X^{n}_{s})dC_{s}$ almost surly. Thus, by (4.14), we see that $L^{n}_{t}$ converges uniformly to a continuous non-decreasing process $L$ on $[0, T]$ almost surly. Letting $n\rightarrow \infty$ in (4.14) gives

Then

Therefore, $X_{t}$ is the solution of (1.3) for all $t\geq0$ since $T$ is arbitrary.

To show that $(X_{t}, L_{t})$ is a solution to (1.4), we need to prove

This will follow if we can show that for any $f\in C_{0}(0, \infty)$

Indeed,

Next we show the uniqueness. Let $(X_{t}^{1}, L_{t}^{1})$, $(X_{t}^{2}, L_{t}^{2})$ be two solutions to (1.4). Using the similar arguments as above, it can be shown that

By Gronwall's inequality, it follows that $X^{1} = X^{2}$, and hence $L^{1} = L^{2}$.

5.   Conclusions

In this paper, a new type of differential equations within the framework of uncertainty theory was discussed for the first time. First of all, we was first to provide an existence and uniqueness theorem under Lipschitz condition and linear growth condition. And then, as an application, we establish the existence and uniqueness of some perturbed reflected canonical process. In the future work, we will try to explore the stability for this type of perturbed uncertain differential equations.

Acknowledgement

This research is partially supported by the NNSF of China (No.11901058).

Conflict and interest

The authors declare that they have no conflict of interest.