Girsanov Theorem for Multifractional Brownian Processes

TL;DR

This paper introduces a new multifractional derivative operator based on variable order calculus, enabling analysis of stochastic differential equations driven by non-stationary multifractional Brownian motion with applications in stochastic and functional analysis.

Contribution

It develops a novel multifractional derivative operator as the inverse of the integral, extending fractional calculus to variable orders and applying it to stochastic differential equations.

Findings

Constructed a multifractional derivative operator solving Abel's integral equation.

Applied the operator to establish strong solutions for SDEs with multifractional Brownian motion.

Extended previous work on fractional calculus and stochastic analysis to variable order contexts.

Abstract

In this article we will present a new perspective on the variable order fractional calculus, which allows for differentiation and integration to a variable order, i.e. one differentiates (or integrates) a function along the path of a regularity function. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the last 20 years. We develop a multifractional derivative operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional derivative operator, we are able to analyze a variety of new problems, both in the field of stochastic analysis and in fractional and functional analysis, ranging from regularization properties of noise to solutions to multifractional differential equations. In this paper, we will focus on application of the derivative operator to the construction of strong solutions to stochastic differential equations where the drift coefficient is merely of linear growth, and the driving noise is given by a non-stationary multifractional Brownian motion with a Hurst parameter as a function of time. The Hurst functions we study will take values in a bounded subset of (0,1/2). The application of multifractional calculus to SDE's is based on a generalization of the works of D. Nualart and Y. Ouknine from 2002.

Equations

$$R_H(s,t)=E[{\tilde{B}}_s^H{\tilde{B}}_t^H]=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s{|}^{2H}\right),$$

$${\tilde{B}}_t^H={\int }_0^t{K}_H(t,s)d{B}_s,$$

$$I_{0+}^{\alpha }f(x)=\frac{1}{\Gamma (\alpha (x))}{\int }_0^x(x-t{)}^{\alpha (x)-1}f(t)dt,$$

$$D_{0+}^{\alpha }f(x)=\frac{1}{\Gamma \left(1-\alpha (x)\right)}\frac{d}{dx}{\int }_0^x\frac{f(t)}{(x-t{)}^{\alpha (x)}}dt.$$

$$D_{0+}^{\alpha }{I}_{0+}^{\alpha }=I+K,$$

$$X_t^x=x+{\int }_0^{t}b\left(s,X_s^x\right)ds+B_t^h;X_0^x=x\in \mathbb{R},$$

$$\parallel f{\parallel }_{\beta }=\left|f(0)\right|+\underset{s\ne t\in [0,T]}{\sup }\frac{\left|f(t)-f(s)\right|}{{\left|t-s\right|}^{\beta }}<\infty .$$

$$\parallel f{\parallel }_{L^p\left(V\right)}:={\left({\int }_V|f{|}^{p}d\mu \right)}^{\frac{1}{p}}.$$

$${△}^{(m)}[a,b]=\left\{(s_1,..,s_m)|a\le s_1<...<s_m\le b\right\}.$$

$$\parallel f{\parallel }_{\alpha \left(\cdot \right);[0,T]}:=|f(0)|+\underset{x,y\in [0,T]}{\sup }\frac{|f(x)-f(y)|}{|x-y{|}^{\max \left(\alpha (x),\alpha (y)\right)}}<\infty$$

$$|f(x)|\le |f(x)-f(0)|+f(0)\le \parallel f{\parallel }_{\alpha \left(\cdot \right);[0,T]}{x}^{\alpha (x)},$$

$$\underset{|h|\le 1;x+h\in [0,T]}{\sup }\frac{|f(x+h)-f(x)|}{|h{|}^{\alpha (x)}}\simeq \underset{x,y\in [0,T]}{\sup }\frac{|f(x)-f(y)|}{|x-y{|}^{\max \left(\alpha (x),\alpha (y)\right)}}.$$

$$[0,T]=\bigcup _{k=0}^{n-1}[T_k,T_{k+1}].$$

$$\begin{array}[]{cc}\beta_{k}=&\inf_{t\in[T_{k},T_{k+1}]}\alpha(t)\\ \hat{\beta}_{k}=&\sup_{t\in[T_{k},T_{k+1}]}\alpha(t)\end{array}$$

$${\mathcal{C}}^{{\beta }_k}\left(\left[T_k,T_{k+1}\right]\right)\supset {\mathcal{C}}^{\alpha (\cdot )}\left(\left[T_k,T_{k+1}\right]\right)\supset {\mathcal{C}}^{{\hat{\beta }}_k}\left(\left[T_k,T_{k+1}\right]\right).$$

$$\left(I_{c+}^{\alpha }f\right)(x)=\frac{1}{\Gamma \left(\alpha (x)\right)}{\int }_c^x(x-y{)}^{\alpha (x)-1}f(y)dy.$$

$$g(t)=\frac{1}{\Gamma \left(\alpha (t)\right)}{\int }_a^t\frac{f(s)}{{\left(t-s\right)}^{1-\alpha (t)}}ds.$$

$${\int }_a^x\frac{\Gamma \left(\alpha (t)\right)g(t)}{{\left(x-t\right)}^{\alpha (t)}}dt={\int }_a^{x}f(s){\int }_0^1{\tau }^{\alpha (s+\tau (x-s))-1}{\left(1-\tau \right)}^{-\alpha (s+\tau (x-s))}d\tau ds.$$

$$\frac{d}{dx}\left({\int }_a^x\frac{\Gamma \left(\alpha (t)\right)g(t)}{{\left(x-t\right)}^{\alpha (t)}}dt\right)$$

$$=f(x)B(\alpha (x),1-\alpha (x))+{\int }_a^{x}f(s){\int }_0^1\frac{d}{dx}\left({\tau }^{\alpha (s+\tau (x-s))-1}{\left(1-\tau \right)}^{-\alpha (s+\tau (x-s))}\right)d\tau ds,$$

$$B(\alpha (x),1-\alpha (x))f(x)=\frac{d}{dx}\left({\int }_a^x\frac{\Gamma \left(\alpha (t)\right)g(t)}{{\left(x-t\right)}^{\alpha (t)}}dt\right)$$

$$-{\int }_a^{x}f(s){\int }_0^1\frac{d}{dx}\left({\tau }^{\alpha (s+\tau (x-s))-1}{\left(1-\tau \right)}^{-\alpha (s+\tau (x-s))}\right)d\tau ds.$$

$$f(x)=G_a(g)(x)+{\int }_a^{x}f(s)F(s,x)ds.$$

$$f(x)=H(x)+{\int }_0^{x}f(s)F(s,x)ds,$$

$$G_0\left(g\right)(x)=\frac{1}{B(\alpha (x),1-\alpha (x))}\frac{d}{dx}\left({\int }_0^x\frac{\Gamma \left(\alpha (t)\right)g(t)}{{\left(x-t\right)}^{\alpha (t)}}dt\right),$$

$$\frac{d}{dx}\left({\tau }^{\alpha (s+\tau (x-s))-1}{\left(1-\tau \right)}^{-\alpha (s+\tau (x-s))}\right)$$

$$=\ln (\frac{\tau }{1-\tau })\frac{{\tau }^{\alpha (s+\tau (x-s))}}{{\left(1-\tau \right)}^{\alpha (s+\tau (x-s))}}{\alpha }^{\prime }(s+\tau (x-s)),$$

$$U(s,x;\tau ):={\alpha }^{\prime }(s+\tau (x-s))\times \left(\ln (\tau )-\ln \left(1-\tau \right)\right){\left(\frac{\tau }{1-\tau }\right)}^{\alpha (s+\tau (x-s))},and$$

$$F(s,x)={\int }_0^{1}U(s,x;\tau )d\tau .$$

$$\parallel F{\parallel }_{\infty ;{△}^{(2)}\left([0,T]\right)}:=\underset{(s,x)\in {△}^{(2)}([0,T])}{\sup }\left|{\int }_0^{1}U(s,x;\tau )d\tau \right|\le C(\alpha )\parallel \alpha {\parallel }_{C^1\left([0,T]\right)}<\infty ,$$

Full text

1. 1 Introduction

2. 1.1 Notation and preliminaries

3. 2 multifractional Calculus

4. 2.1 Remark on multifractional Calculus

5. 3 Riemann-Liouville multifractional Brownian Motion

6. 4 Girsanov theorem and Existence of Strong Solutions

7. 4.1 Girsanov’s theorem

8. 4.2 Existence of weak solutions

9. 4.3 Uniqueness in law and path-wise uniqueness.

10. 4.4 Existence of strong solutions

11. 5 Conclusion

1. Introduction

Fractional calculus is a well studied subject in the field of functional analysis, and has been adopted in the field of stochastic analysis as a tool for analysing stochastic processes which exhibit some memory of its own past, called fractional Brownian motion (fBm), denoted as $\tilde{B}_{\cdot}^{H}:[0,T]\rightarrow\mathbb{R}$ for some $H\in(0,1)$. The process is characterized by its co-variance function

[TABLE]

and the process is often represented as a type of fractional integral with respect to a Brownian motion, in the sense that

[TABLE]

where $K_{H}$ is a singular and square integrable Volterra kernel and $B_{\cdot}$ is a Brownian motion. This type of process has been studied in connection with various physical phenomena, such as weather and geology, but also in modelling of internet traffic and finance.

In the 1990’s, a generalization of the fractional Brownian motion was proposed by letting the Hurst parameter $H$ of the process $B_{\cdot}^{H}$ to be dependent on time (e.g. [12]). The process was named multifractional Brownian motion (or mBm for short), referring to the fact that the fractional parameter $H$ was a function depending on time taking values between [math] and $1$. There has later been a series of articles on this type of processes, relating to local time, local and global Hölder continuity of the process, and other applications ( see [9, 10, 16, 6, 13] to name a few). However, there has not been, to the best of our knowledge, any articles on solving stochastic differential equations driven by this type of process. The mBm is a non-stationary stochastic process which makes the it more intricate to handle in differential equations, and require sufficient tools from fractional analysis in the sense of multifractional calculus (or fractional calculus of variable order). This type of fractional calculus was originally proposed by S. Samko ([14] ) and others in the beginning of the 1990’s, and generalizes the classical Riemann Liouville fractional integral by considering

[TABLE]

where $f\in L^{1}\left(\mathbb{R}\right)$ and $\alpha:\mathbb{R}\rightarrow[c,d]\subset(0,1)$. In the same way, the authors also proposed to generalize the fractional derivative i.e.

[TABLE]

However, by generalizing the fractional derivative in this way, the authors found that the derivative is no longer the inverse of the integral operator, but one rather finds that

[TABLE]

where $I$ is the identity, and under some conditions $K$ is a compact operator. Some of the research then focused on understanding the properties of the operator $K$, but fractional calculus of variable order has been scarcely treated in the literature, so far.

It is well known in stochastic analysis that one can construct weak solutions to SDE’s by Girsanov’s theorem, see for example [8] in the case when the SDE is driven by a fBm. To be able to develop a Girsanov’s theorem to SDE’s driven by multifractional noise, we need to have an inverse operator relating to the multifractional integral.

We will in this paper construct the multifractional derivative of a function $f\in L^{p}\left([0,T]\right)$ as the inverse of the multifractional integral, and see that this multifractional derivative operator is well defined on a certain class of functions. We will then apply this operator to the construction of a Girsanov theorem for Riemann-Liouville multifractional Brownian motions, and finally we show existence of strong solutions to certain SDE’s.

The SDE we will investigate is given by

[TABLE]

where $B^{h}$ is a Riemann Liouville multifractional Brownian motion with $regularity$ $function$ $h:[0,T]\rightarrow[a,b]\subset(0,\frac{1}{2})$. Actually, it turns out that we only need $t\mapsto\int_{0}^{t}b\left(s,X_{s}^{x}\right)ds$ to be locally $\beta$ - Hölder continuous for some well chosen $\beta$ to construct weak solutions, but we will need $b$ to be of linear growth to get strong solutions by the comparison theorem.

1.1. Notation and preliminaries

We will make use of a space of Hölder continuous functions defined as functions $f:[0,T]\rightarrow V$ for some Banach space $V$ (we will mostly use $V=\mathbb{R}^{d}$ in this article), which is such that the Hölder norm $\parallel f\parallel_{\beta}$ defined by

[TABLE]

We denote this space by $\mathcal{C}^{\beta}\left([0,T];V\right)$. In addition, for $a\in[0,T]$ let us define $\mathcal{C}_{a}^{\beta}\left([0,T];V\right)$ to be the subspace of $\mathcal{C}^{\beta}\left([0,T];V\right)$ such that for $f\in\mathcal{C}_{a}^{\beta}\left([0,T];V\right)$, $f(a)=0$. In particular, we will be interested in $\mathcal{C}_{0}^{\beta}\left([0,T];V\right)$. We use the standard notation for $L^{p}\left(V;\mu\right)$ spaces, where $V$ is a Banach space and $\mu$ is a measure on the Borel sets of $V$ (usually what measure we use is clear from the situation)i.e.,

[TABLE]

Furthermore let $\triangle^{(m)}\left[a,b\right]$ denote the $m$-simplex. That is, define $\triangle^{(m)}[a,b]$ to be given by

[TABLE]

We will use the notion of variable order exponent spaces, which has in recent years become increasingly popular in potential analysis, see for example [2, 3] for two new books on the subject. We will in this paper be particularly concerned with the variable exponent Hölder space, as we are looking to differentiate functions to a variable order. We will therefore follow the definitions of such a space introduced in [5], and give some preliminary properties.

Definition 1**.**

Let $\alpha$ be a $C^{1}$ regularity function with values in $[a,b]\subset(0,1)$, as described above. We define the space of locally Hölder continuous functions $f:[0,T]\rightarrow\mathbb{R}$ by the norm

[TABLE]

We denote this space by $\mathcal{C}^{\alpha\left(\cdot\right)}\left([0,T];\mathbb{R}\right)$. Moreover denote by $\mathcal{C}_{0}^{\alpha(\cdot)}\left([0,T];\mathbb{R}\right)$ the space of locally Hölder continuous functions which start in $0,$ i.e. for $f\in\mathcal{C}_{0}^{\alpha(\cdot)}\left([0,T];\mathbb{R}\right)$ then $f(0)=0$.

Remark 2*.*

Notice that in $\mathcal{C}_{0}^{\alpha(\cdot)}\left([0,T];\mathbb{R}\right)$ , all functions $f$ satisfy $|f(x)|\leq Cx^{\alpha(x)}.$ Indeed, just write

[TABLE]

where we have used that $\max(\alpha\left(x\right),\alpha\left(y\right))\geq\alpha\left(x\right).$ Further, we have the following equivalence

[TABLE]

To show what we mean by the local regular property, we may divide the interval $[0,T]$ into $n\geq 2$ intervals, by defining $T_{0}=0$, $T_{1}=\frac{T}{n}$ and $T_{k}=\frac{kT}{n}$, then

[TABLE]

Furthermore, we may define two sequences $\{\beta_{k}\}$ and $\{\hat{\beta}_{k}\}$ of $\sup$ and $\inf$ values of $\alpha$ restricted to each interval $[T_{k},T_{k+1}]$, i.e

[TABLE]

Then we have the following inclusions

[TABLE]

By letting $n$ be a large number, we have by the continuity of $\alpha$ that $\hat{\beta}_{k}-\beta_{k}=\varepsilon$ gets small. Therefore the space $\mathcal{C}^{\alpha(\cdot)}\left(\left[T_{k},T_{k+1}\right]\right)$ is similar to the spaces $\mathcal{C}^{\beta_{k}}\left(\left[T_{k},T_{k+1}\right]\right)$ and $\mathcal{C}^{\hat{\beta}_{k}}\left(\left[T_{k},T_{k+1}\right]\right)$ when the $T_{k+1}-T_{k}$ is small, but on large intervals the spaces may be different depending on the chosen $\alpha$.

We will often write $f_{*}:=\inf_{t\in[0,T]}f(t)$ and $f^{*}:=\sup_{t\in[0,T]}f(t)$.

2. multifractional Calculus

In this section we will give meaning to the multifractional calculus. Many references in fractional analysis refer to this concept as fractional calculus of variable order, as the idea is to let the order of integration (or differentiation) be dependent on time (or possibly space, but for our application, time will be sufficient), see for example [14, 15] and the references therein. We will use the word multifractional calculus for the concept of fractional calculus of variable order, as this is coherent with the notion of multifractional stochastic processes.

Definition 3**.**

($Multifractional$ $Riemann$-$Liouville$ $integrals$) For $0<c<d$, assume $f\in L^{1}\left([c,d]\right),$ and $\alpha:[c,d]\rightarrow[a,b]\subset(0,\infty)$ is a differentiable function. We define the left multifractional Riemann Liouville integral operator $I_{c+}^{\alpha}$ by

[TABLE]

And define the space $I_{c+}^{\alpha}L^{p}\left([0,T]\right)$ as the image of $L^{p}\left([0,T]\right)$ under the operator $I_{c+}^{\alpha}.$

By the definition of the space $I_{c+}^{\alpha}L^{p}\left([0,T]\right),$ we have that for all $g\in I_{c+}^{\alpha}L^{p}\left([0,T]\right)$, $g(c)=0.$ Indeed, since $g=I_{c+}^{\alpha}f,$ we must have $I_{c+}^{\alpha}f(c)=0$ regardless of the function $\alpha$. This property will become very important later, and we will give a more thorough discussion of the properties of the multifractional integral and derivative at the end of this section.

We want to define the multifractional derivative of a function $g\in I_{c+}^{\alpha}L^{p}\left([0,T]\right)$ as the inverse operation of $I_{c+}^{\alpha}$, such that if $g\in I_{c+}^{\alpha}L^{p}\left([0,T]\right)$ then there exists a unique $f\in L^{p}$ which satisfies $g=I_{c+}^{\alpha}f$ and we define the fractional derivative $D_{c+}^{\alpha}g=f$. In contrast to this methodology used in [14], we believe that to be able to construct a coherent fractional calculus, one must choose to generalize either the derivative or the integral operator, and the other operator must be found through the definition of the first. Therefore, we will use a method similar to that solving Abel’s integral equation, often used to motivate the definition of the fractional derivative in the case of constant regularity function. However, by generalizing the integral equation, the calculations become a little more complicated.

We will start to give formal motivation for how we obtain the derivative operator corresponding to the multifractional integral. First, for an element $g\in I_{a+}^{\alpha}L^{p}\left([0,T]\right)$, we know there exists a $f\in L^{p}\left([0,T]\right)$ such that

[TABLE]

By some simple manipulation of the equation above, and using Fubini’s theorem, we obtain the equation,

[TABLE]

If $\alpha$ is constant, the last integral is simply the Beta function. However, the fact that $\alpha$ is a function complicates the expression. In the end, we are interested in solving the above equation by obtaining $f(x)$, therefore, we may try to differentiate w.r.t $x$ on both sides of the equation. We find that

[TABLE]

[TABLE]

and re-ordering the terms, we get

[TABLE]

[TABLE]

In this sense, the multifractional derivative $D_{a+}^{\alpha}$ of a function $g\in I_{a+}^{\alpha}L^{p}\left([0,T]\right)$ is given by $f$ which is a solution to the ordinary differential equation (ODE) above. Writing the above equation more compactly, we have

[TABLE]

We will use the rest of this section to prove that the ODE that we obtained above, actually is well defined, and that the multifractional derivative can be defined as the solution to that ODE. In the above motivation we looked at the multifractional derivative as the inverse operator to the multifractional integral with initial value of integration at a point $a$. However, we will mostly be interested in looking at the inverse operator of the integral starting at $a=0$, and therefore the construction of the multifractional will be focused on this particular case, although it is straight forward to generalize this operator to any $a\in[0,T]$. We will give a suitable definition of the multifractional derivative in the following steps:

1. (1)

The function $F:\triangle^{(2)}\left([0,T]\right)\rightarrow\mathbb{R}$ in equation (2.2) is well defined and bounded on $\triangle^{(2)}\left([0,T]\right)$ as long as $\alpha$ is $C^{1}$. 2. (2)

We show that if $H\in L^{p}\left([0,T]\right),$ then the ODE

[TABLE]

has a unique solution $f$ in $L^{p}\left([0,T]\right).$ 3. (3)

The functional $G_{0}$ defined by

[TABLE]

is such that the mapping $x\mapsto G_{0}\left(g\right)(x)\in L^{p}\left([0,T]\right)$ for all functions $g\in\mathcal{C}^{\alpha\left(\cdot\right)+\epsilon}\left([0,T]\right)$, where $p>1$ and some small $\epsilon>0$ depending on $\alpha$. 4. (4)

Then the ODE as motivated above has a unique solution in $L^{p}\left([0,T]\right)$ for all $g\in\mathcal{C}^{\alpha\left(\cdot\right)+\epsilon}\left([0,T]\right)$, and we will define the multifractional derivative $D_{0+}^{\alpha}g=f$ as the solution to the equation (2.2).

First, we will look at the function $F:\triangle^{(2)}\left([0,T]\right)\rightarrow\mathbb{R}$ in equation (2.2). Notice that the derivative of $\tau^{\alpha(s+\tau(x-s))-1}\left(1-\tau\right)^{-\alpha(s+\tau(x-s))}$ is explicitly given by

[TABLE]

[TABLE]

and we arrive at a lemma which gives estimates on this derivative.

Lemma 4**.**

Let $\alpha\in C^{1}\left(\left[0,T\right];(0,1)\right)$. Define

[TABLE]

[TABLE]

Then

[TABLE]

where $\parallel\alpha\parallel_{C^{1}\left([0,T]\right)}:=\sup_{t\in[0,T]}|\alpha(t)|+\sup_{t\in[0,T]}|\alpha^{\prime}(t)|.$

Proof.

Look at,

[TABLE]

[TABLE]

The fact that the last integral is finite follows by simple calculations, knowing that $\alpha(t)\in[c,d]\subset(0,1)$ for all $t\in[0,T].$ Indeed, we can write

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Now we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\alpha_{*}:=\inf_{t\in[0,T]}\alpha(t)$ and $\alpha^{*}:=\sup_{t\in[0,T]}\alpha(t)$. ∎

We now show that the solution to the differential equation given by

[TABLE]

is well-defined as an element of $L^{p}\left(\left[0,T\right]\right)$ when $H\in L^{p}\left([0,T]\right)$, and $F$ is given as above.

Lemma 5**.**

Let $H\in L^{p}\left(\left[0,T\right]\right)$ and $F$ be given as in Lemma 4. Then there exists a unique solution $f$ to the equation

[TABLE]

in $L^{p}\left(\left[0,T\right]\right)$.

Proof.

For simplicity, write $L^{p}=L^{p}\left([0,T]\right)$. We consider a usual Picard iteration, and define

[TABLE]

then for $x\in[0,T]$ we use Lemma 4 to see that

[TABLE]

[TABLE]

where we have used the Hölder inequality . By iteration, we obtain

[TABLE]

[TABLE]

and more generally that for $m>n$

[TABLE]

[TABLE]

Therefore, $\left\{f_{n}\right\}_{n\in\mathbb{N}}$ is Cauchy in $L^{p}$ and$\left\|f_{m}-f_{n}\right\|_{L^{p}}\rightarrow 0$ as $n,m\rightarrow\infty$ , and we denote $f^{\infty}$ to be its limit. Furthermore, we can choose a sub sequence $f_{n_{k}}$ converging almost surely and in $L^{p}$ to $f^{\infty}$. By applying dominated convergence theorem, $f^{\infty}$ solves the original ODE,

[TABLE]

and we set $f:=f^{\infty}.$ The uniqueness of the solution is immediate by the linearity of the equation, and can bee seen explicit from the representation in Theorem 9. ∎

The next lemma shows that the functional $G_{0}$ in equation (2.2) is an element of $L^{p}$ when acting on a certain class of functions.

Lemma 6**.**

Let $g\in\mathcal{C}_{0}^{\alpha(\cdot)+\epsilon}\left([0,T];\mathbb{R}\right)$ with $\alpha\in C^{1}\left([0,T],[a,b]\right)$ for $[a,b]\subset(0,1)$ and $\epsilon<1-\alpha^{*}$. Furthermore, assume that for some $p>1,$ the regularity function $\alpha$ satisfies the inequality

[TABLE]

Then the functional $G_{0}$ evaluated in $g$ defined by

[TABLE]

is an element of $L^{p}\left(\left[0,T\right]\right)$.

Proof.

Let us first look at $G_{0}(g)(x)$, but ignoring the factor with the Beta function as this is behaving well already, i.e, there exists a $\beta>0$ such that

[TABLE]

as $\alpha(x)\in[a,b]$ for all $x\in[0,T]$. Therefore, we need to prove that

[TABLE]

is an element of $L^{p}\left([0,T]\right).$ In particular, we will show that the integral above is in fact continuously differentiable on $(0,T],$ and then we show that it is an element in $L^{p}\left([0,T]\right)$. Expanding the integral by adding and subtracting the point $g(x)$, we can see that,

[TABLE]

Then one can show that

[TABLE]

[TABLE]

See the appendix in section Appendix for proof a of this relation, which is based on using the definition of the derivative. In this way, we can make sense of this derivative by proving that the above two terms are elements of $L^{p}\left(\left[0,T\right]\right)$ when $g\in\mathcal{C}^{\alpha\left(\cdot\right)+\epsilon}\left(\left[0,T\right]\right)$.

Writing the second term explicitly, we have by straight forward derivation that

[TABLE]

Notice that $\frac{d}{dx}K(x,t)$ is singular, but since $g\in\mathcal{C}_{0}^{\alpha(\cdot)+\epsilon}\left([0,T]\right)$, we get the estimate

[TABLE]

[TABLE]

The gamma function is well defined on any $[a,b]\subset(0,1)$, and $\alpha$ is $C^{1}$ and we know

[TABLE]

we have that the singularity is integrable, hence the above is well defined for all $x\in[0,T]$. The function $P_{1}$ is dependent on both $x$ and $\epsilon$, but also on $\alpha$, and we can see that

[TABLE]

Therefore, $P_{1}$ is $L^{p}$ integrable, for all $p$ as long as $\epsilon>0$.

We are left to prove that the first term is an element of $L^{p}$, i.e., we need to show that

[TABLE]

Write $M(x)=\int_{0}^{x}\frac{\Gamma\left(\alpha(t)\right)}{(x-t)^{\alpha(t)}}dt,$ and set $u=x-t,$ then by a variable change $M(x)=\int_{0}^{x}\frac{\Gamma\left(\alpha(x-u)\right)}{u^{\alpha(x-u)}}du,$ and by Leibniz integral rule, we have for all $x\in(0,T],$

[TABLE]

and

[TABLE]

The derivative inside the integral in $M_{2}$ can be calculated explicitly as follows,

[TABLE]

where $\psi_{0}$ is the digamma function. First we have that

[TABLE]

which is in $L^{p}\left([0,T]\right)$ if $\left(\alpha_{*}+\epsilon-\alpha(0)\right)\times p>-1.$

Secondly we look at

[TABLE]

[TABLE]

The first term is well defined since $\alpha$ is a differentiable function on a compact interval, and the gamma function is decreasing on some compact sub interval of $(0,1)$, therefore,

[TABLE]

where $\alpha^{*}=\sup_{x\in[0,T]}\alpha(x)$ . The above is well defined as an element of $L^{p}\left([0,T]\right)$ for any $p$. At last, for $g(x)M_{2,2}(x)$ we get a very similar estimate as for $M_{2,1}$ by using that the digamma function is bounded on a compact sub interval of $(0,1),$ and in fact for $x\in[a,b]\subset(0,1)$ , $|\psi_{0}\left(x\right)|\leq|\psi_{0}\left(a\right)|$. Therefore we have that

[TABLE]

for any $p$ satisfying the inequality $\left(\alpha_{*}+\epsilon-\alpha(0)\right)\times p>-1$.

Combining the results above, we obtain that the $G_{0}$ functional in our representation is well behaved on the space $\mathcal{C}_{0}^{\alpha(\cdot)+\epsilon}\left(\left[0,T\right]\right)$ and satisfies

[TABLE]

By the above lemma, we know that

[TABLE]

is in $L^{p}\left(\left[0,T\right]\right)$ for a some well chosen $p$. ∎

From the last result, we now have a concrete way to define a multifractional derivative which is the inverse operator of the generalized Riemann-Liouville integral.

Corollary 7**.**

Under the assumptions from Lemma 6, define the multifractional derivative $D_{0+}^{\alpha}g$ of a function $g\in\mathcal{C}^{\alpha(\cdot)+\epsilon}\left([0,T]\right)$ to be the solution of the differential equation

[TABLE]

where $G_{0}$ (dependent on $g$) and $F$ is given as in Lemma 6 and 4.

Remark 8*.*

The derivative operator that we have defined above is indeed the inverse operator of the multifractional integral. To see this, assume for a moment that for a function $g\in\mathcal{C}^{\alpha(\cdot)+\epsilon}$, we have the derivative

[TABLE]

then we can look at equation (2.2), which we now know is well posed, and we can go backwards in the derivation method, which is found in the beginning of section 2, to find that

[TABLE]

and hence $g\in I_{0+}^{\alpha}L^{p}\left(\left[0,T\right]\right)$.

By simple iterations of the above ODE, we can get an explicit representation of this multifractional derivative.

Theorem 9**.**

The multifractional derivative can be represented in $L^{p}\left([0,T]\right)$ as the infinite sequence of integrals

[TABLE]

for any $g\in\mathcal{C}_{0}^{\alpha(\cdot)+\epsilon}\left([0,T];\mathbb{R}\right)$ with $\alpha\in C^{1}\left([0,T],[a,b]\right)$ for $[a,b]\subset(0,1)$ and $\epsilon<1-\alpha^{*}$. Furthermore, assume that for some $p>1,$ the regularity function $\alpha$ satisfies the inequality

[TABLE]

Then we have the estimate

[TABLE]

where $x\mapsto C(T,\epsilon,\alpha;x)$ is an $L^{p}$ function for any $p$ satisfying the inequality and hence $D_{0+}^{\alpha}g\in L^{p}\left([0,T]\right).$

Proof.

The representation is immediate as a consequence of Definition 7. We know from Lemma 6 and 4 that

[TABLE]

and

[TABLE]

for any $p$ satisfying the inequality $(\alpha_{*}+\epsilon-\alpha(0)\times p)>-1.$

We therefore obtain

[TABLE]

[TABLE]

[TABLE]

We can then use Cauchy’s formula for repeated integration given by

[TABLE]

Using this, we get the estimate on the representation

[TABLE]

[TABLE]

[TABLE]

which completes the proof. ∎

Remark 10*.*

With the last Theorem 9, we can see that the multifractional derivative operator is well defined on any local Hölder continuous space of order $\alpha(\cdot)+\epsilon$ for some small $\epsilon$, as long as $p$ and $\epsilon$ satisfies $\left(\alpha_{*}+\epsilon-\alpha(0)\right)\times p>-1$. Therefore, we have the relation

[TABLE]

2.1. Remark on multifractional Calculus

Most of the articles we have found on the multifractional calculus or fractional calculus of variable order has been related to applications in physics. In an article by Hartley and Lorenzo [17] the authors suggest many applications of multifractional calculus in physics. Particular examples are given when physical phenomena is modeled by a fractional differential equation, i.e.

[TABLE]

but the fractional order parameter $\alpha$ is dependent on a variable, which again is dependent on time. An example could be that $\alpha$ was estimated on the basis of temperature, but temperature changes in time. Therefore, they suggest that by using multifractional differential operators, one can overcome this problem, as one would be able to give a different differential order $\alpha$ to different times. The multifractional calculus enables us to construct more accurate differential equations for processes where the local time regularity of the process is depending on time. Of course, this can also be generalized to multifractional differential operators in space, where the spatial variables of a system has time dependent local regularity.

Although the multifractional calculus seems like a suitable tool for differential equations described above, the soul concept of multifractional or even fractional calculus can be very difficult to grasp, and use in practice. When considering fractional calculus we usually consider an operator behaving well (as inverse, etc. ) with respect to a fractional integral. This makes the operator dependent on the initial point of the integral, and it is not linear with respect to this initial point, in the sense that $I_{a+}^{\alpha}I_{b+}^{\alpha}\neq I_{(a+b)+}^{\alpha}$. Furthermore, we can not fractionally differentiate a constant (different than [math]), as it is not contained in $I_{a+}^{\alpha}L^{p}$. There is therefore a strong dependence of the derivative operator on the choice of the integral (as we have seen it is actually completely determined by the chosen integral). The intuition of the framework is therefore far from that of the regular calculus of Newton and Leibniz, and a more rigorous understanding of the properties of multifractional calculus is needed to consider differential equations and partial differential equations with such operators.

3. Riemann-Liouville multifractional Brownian Motion

In this section we give some preliminary results relating to the multifractional Brownian motion (mBm) with Riemann Liouville and present a Girsanov transform for this process. The multifractional Brownian motion (fBm) was first proposed in the 1990’s by Peltier and Lèvy-Vehèl in [12] and independently by Benassi, Jaffard, Roux in [4]. The process is non-stationary and on very small time steps it behaves like a fractional Brownian motion. However, by letting the Hurst parameter in the fractional Brownian motion be a function of time, the Hölder regularity of the process is depending on time, and therefore it makes more sense to talk about local regularities rather than global. The process was initially proposed as a generalization with respect to the fBm representation given by Mandelbrot and Van-Ness, that is, the mBm was defined by

[TABLE]

where $\left\{B_{t}\right\}_{t\in[0,T]}$ is a real valued Brownian motion, and $h:[0,T]\rightarrow(0,1)$ is a continuous function. Notice in the above representation that $\tilde{B}_{t}^{(1),h}$ is always measurable with respect to the filtration $\tilde{\mathcal{F}}_{0}$ (generated by the Brownian motion), as the stochastic process only “contributes” from $-\infty$ to $0.$ Therefore, we can think of $\tilde{B}_{t}^{(2),h}$ as the only part which contributes to the stochasticity of $\tilde{B}_{t}^{h}$ when $t>0$. The reason why one also considers the process $\tilde{B}_{t}^{(1),h}$ when analyzing regular fractional Brownian motions (in the case $h(t)=H$) is to ensure stationarity of the process. However, when we are considering the generalization $\tilde{B}_{t}^{h}$ above, when $h$ is not constant, we do not get stationary of the process even though we consider the a representation as the one above. We are therefore inclined to choose $\tilde{B}_{t}^{(2),h}$ to be the multifractional noise we consider in this article due to its very simplistic nature. This multifractional process is often called in the literature the $Riemann$-$Liouville$ multifractional Brownian motion, inspired by the original definition of the fractional Brownian motion defined by Lèvy in the 1940’s. The Riemann-Liouville multifractional Brownian motion was first analyzed by S. C. Lim in [11], and is well suited to the use of multifractional calculus, constructed above, in the analysis of differential equations driven by this process. In this section we will recite some of the basic properties of the Riemann-Liouville multifractional Brownian motion from [11], and then we will use the next section to apply this process with respect to the Girsanov theorem.

We begin by giving a proper definition of what we mean by a Riemann-Liouville multifractional Brownian motion.

Definition 11**.**

Let $\left\{B_{t}\right\}_{t\in[0,T]}$ be a one dimensional Brownian motion on a filtered probability space $\left(\Omega,\mathcal{F},P\right)$, and let $h:[0,T]\rightarrow[a,b]\subset(0,1)$ be a $C^{1}$ function. We define the Riemann Liouville multifractional Brownian motion $(RLmBm)$ $\left\{B_{t}^{h}\right\}_{t\in[0,T]}$ by

[TABLE]

where $\Gamma$ is the Gamma function. The function $h$ is called the regularity function of $B_{\cdot}^{h}$ .

With this definition, it is useful to know the co-variance function of the process that we are interested in. Although this function is slightly more complicated than the co-variance of a regular fBm, or even more complicated than that proposed for mBm’s in article [1], the co-variance obtained here is explicit, and suits our purposes well. The below proposition can also be found in [11], and the proof is a simple exercise in standard integration.

Proposition 12**.**

(Co-variance of RLmBm) let the RLmBm be defined as above. Then for $0\leq s\leq t\leq T$, we have

[TABLE]

where $F$ is the hyper-geometric function.

Proof.

We know that

[TABLE]

[TABLE]

∎

When discussing fractional and multifractional Brownian motions, their Hölder regularities seem to be of great interest. We will therefore give some short comments on the regularity of the Riemann-Liouville mBm as this is an uncommon representation.

Proposition 13**.**

Let $\left\{B_{t}^{h}\right\}_{t\in[0,T]}$ be a RLmBm process, with $h\in C^{1}\left(\left[0,T\right]\right)$ with values in some compact subset of $(0,1)$. Then we have global regularity

[TABLE]

and local regularity

[TABLE]

Proof.

Both the claims are thoroughly proved in the seminal paper [12] by Peltier and Vehï¿œl. This article uses the representation $\tilde{B}_{t}^{h}$ as mentioned above, but do split their arguments in to the case of $\tilde{B}^{(1),h}$ and $\tilde{B}^{(2),h}$, such that the proof is very simple to read. ∎

For a longer discussion on the properties of the multifractional Brownian motion, we refer to [1, 4, 6, 16, 10, 9, 11, 7].

4. Girsanov theorem and Existence of Strong Solutions

In this section we will apply the multifractional calculus that we developed in Section 2 to analyze differential equations driven by multifractional noise. By multifractional noise (or multifractional Brownian motion), we will from here on out use the Riemann-Liouville multifractional Brownian motion, as discussed earlier. We do this by the same techniques used in the article by Nualart and Ouknine [8], but we must generalize those arguments to allow for multifractional noise, using the multifractional derivative operator. We start to prove a Girsanov theorem for multifractional Brownian motion, and then we use this to construct weak solutions. We then check for uniqueness in distribution, and find that this again implies path-wise uniqueness. To construct strong solutions, the authors of [8] use approximation techniques to show that when $b$ is of linear growth, we can find an appropriate sequence of functions which is converging to $b$ such that the solution of the corresponding SDE is strong for any of the functions in the approximating sequence. The corresponding solutions can then be shown to converge to a stochastic process, which the authors show is a strong solution to the SDE. The whole argument is based on the behavior of the drift function $b$ , and not on the multifractionality of the noise. Therefore, the existence of strong solutions in our case follows by construction of weak solutions and application of the results relating to strong solutions given by Proposition 6, and 7 and Theorem 8 in [8]. However, as we want this article to be self contained, we include the propositions and proofs which is given in [8], and we will make it very clear what we have used from this article.

4.1. Girsanov’s theorem

As before, let us denote by $B_{t}^{h}$ the multifractional Brownian motion with its natural filtration $\left\{\mathcal{F}_{t};t\in[0,T]\right\}$. We would like to show through a Girsanov theorem, that a perturbation of $B^{h}$ with a specific type of function will give us a mBm under another new probability measure. Similar to the derivation found in [8] in case of fBm, we consider the following perturbation

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

The multifractional derivative $D_{0+}^{h+\frac{1}{2}}\left(\int_{0}^{\cdot}u_{s}ds\right)$ in the last equation is well defined as an element in $L^{2}\left(\left[0,T\right]\right)$ as long as $\int_{0}^{\cdot}u_{s}ds\in I_{0+}^{h+\frac{1}{2}}L^{2}\left([0,T]\right).$

Theorem 14**.**

(Girsanov theorem for mBm) Let $\left\{B_{t}^{h}\right\}_{t\in[0,T]}$ be a Riemann-Liouville multifractional Brownian motion with regularity function $h:[0,T]\rightarrow\left[a,b\right]\subset(0,\frac{1}{2})$ such that $h\in C^{1}$. Assume that

$(i)$* $\int_{0}^{\cdot}u_{s}ds\in\mathcal{C}^{h(\cdot)+\frac{1}{2}+\epsilon}\left([0,T]\right)\subseteq I_{0+}^{h+\frac{1}{2}}L^{2}\left([0,T]\right)$ a.s.*

$(ii)$* $E\left[Z_{T}\right]=1$ for*

[TABLE]

Then the stochastic process

[TABLE]

is an RLmBm under the measure $\tilde{P}$ defined by

[TABLE]

Proof.

The proof follows from the derivations above the theorem, and then applying the regular Girsanov theorem to $\tilde{B}_{t}=B_{t}+\int_{0}^{t}D_{0+}^{h+\frac{1}{2}}\left(\int_{0}^{\cdot}u_{s}ds\right)\left(r\right)dr,$ showing that $\tilde{B}_{t}$ is a BM under $\tilde{P}.$ ∎

Remark 15*.*

The reason that we only look at $h:[0,T]\rightarrow\left[a,b\right]\subset(0,\frac{1}{2})$ is that the multifractional derivative is only constructed for $\alpha:\left[0,T\right]\rightarrow\left[c,d\right]\subset(0,1)$, and as we need to look at $\alpha=h+\frac{1}{2}$ to consider the RLmBm, we need to restrict $0<h(t)<\frac{1}{2}$ for all $t\in[0,T]$. We are currently working on a way to consider the multifractional derivative for any function $\alpha$, i.e. such that we can differentiate to variable order with largest value above $1$ and smallest value less than 1.

4.2. Existence of weak solutions

Theorem 16**.**

Let $b$ be integrable and of linear growth, i.e. $|b(t,x)|\leq\tilde{C}(1+|x|)$ for all $(t,x)\in\left[0,T\right]\times\mathbb{R}$. Then there exists a weak solution $\left\{X_{t}^{x}\right\}_{t\in[0,T]}$ to equation

[TABLE]

where $B^{h}$ is given as in Definition 11 .

Proof.

Set $u_{s}=b\left(s,B_{s}^{h}+x\right),$ and $\tilde{B}_{t}^{h}=B_{t}^{h}-\int_{0}^{t}u_{s}ds$, and

[TABLE]

We need to check that the process $\int_{0}^{\cdot}u_{s}ds$ satisfies condition $(i)$ and $(ii)$ in theorem 14. First we will show $(i)$ , i.e. $\int_{0}^{\cdot}u_{s}ds\in I_{0+}^{h+\frac{1}{2}}L^{2}\left(\left[0,T\right]\right)$, which is equivalent to showing $v\in L^{2}\left(\left[0,T\right]\right)$.** **By theorem 9, we know that

[TABLE]

where $t\mapsto C(T,\epsilon,h+\frac{1}{2};t)\in L^{2}\left(\left[0,T\right]\right)$ since** $\left(h_{*}+\epsilon-h(0)\right)\times 2>-1$** for some small $\epsilon>0.$ Moreover, by the linear growth of $b$ we have the following estimate

[TABLE]

and we know by Fernique that $|B^{h}|_{\infty}$ has finite exponential moments of all orders, and therefore we have that

[TABLE]

Next we look at $(ii)$, where we need to have that $u$ is adapted and satisfies the Novikov condition. It is sufficient to prove that there exists a $\lambda>0$ such that

[TABLE]

The adaptedness of $v$ follows, as $D_{0+}^{h+\frac{1}{2}}$ is a deterministic operator. By equation (4.2) and Fernique’s theorem, the above condition is satisfied, and the existence of a weak solution follows from the Girsanov Theorem 14. ∎

Let us continue by checking for uniqueness of the solution.

4.3. Uniqueness in law and path-wise uniqueness.

We will prove uniqueness in law and path wise uniqueness in the same manner as was done in [8] section 3.3. The technique used is very similar, although we have different bounds on our differential equations, corresponding to the multifractality of our equation.

Theorem 17**.**

Let $\left(X,B^{h}\right)$ be a weak solution to differential equation (4.1) defined on the corresponding filtered probability space $\left(\Omega,\mathcal{F},P\right)$. Then the weak solution is unique in law, and moreover is unique almost surely.

Proof.

Start again with

[TABLE]

and let a new probability measure $\tilde{P}$ be defined by the Radon-Nikodym derivative

[TABLE]

where $B$ denotes a regular Brownian motion. Although this Radon-Nikodym derivative is similar as before, notice that we now have the solution of the differential equation inside the function $b$. We will show that this new process $\tilde{u}$ still satisfies condition $(i)$ and $(ii)$ of Girsanov Theorem 14. Since we know that ,

[TABLE]

[TABLE]

we have by Grï¿œnwall’s inequality that

[TABLE]

Moreover, by the estimates on the multifractional derivative from Theorem 9, we know

[TABLE]

and since

[TABLE]

we obtain the estimate

[TABLE]

where $t\mapsto C\left(T,\epsilon,h+\frac{1}{2};t\right)\in L^{2}\left(\left[0,T\right]\right)$. Combining the above estimates, and again using Ferniques theorem, we have that Novikov’s condition is satisfied and $E\left[\frac{d\tilde{P}}{dP}\right]=1$. The classical Girsanov theorem then states that the process

[TABLE]

is an $\left\{\mathcal{F}_{t}\right\}$- Brownian motion under $\tilde{P}$, and we can then write

[TABLE]

where $K_{h}(t,s)=\frac{1}{\Gamma(h_{t}+\frac{1}{2})}(t-s)^{h_{t}-\frac{1}{2}}$. Now we have that $X_{t}-x$ is a $\left\{\mathcal{F}_{t}\right\}$-multifractional Brownian motion with respect to $\tilde{P}$ . Therefore, we must have that $X_{t}-x$ and $\tilde{B}_{t}^{h}$ has the same distribution under the probability $P$. We can show this by considering a measurable functional $\varphi$ on $C\left([0,T]\right)$, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

For the almost surely uniqueness, assume there exists two weak solutions $X^{1}$ and $X^{2}$ on the same probability space $\left(\Omega,\mathcal{F},\tilde{P}\right)$, then $\max\left(X^{1},X^{2}\right)$ and $\min\left(X^{1},X^{2}\right)$ are again two solutions, and according to the above proof, have the same distribution. We can rewrite $\max(x,y)=\frac{1}{2}\left(x+y+|x-y|\right)$ and $\min(x,y)=\frac{1}{2}\left(x+y-|x-y|\right),$ and therefore $E\left[\max\left(X^{1},X^{2}\right)\right]=E\left[\min\left(X^{1},X^{2}\right)\right]$ imply that

[TABLE]

but of course this is only true if $X^{1}=X^{2}$ a.s. ∎

Remark 18*.*

Although, in this paper we limit our selves to the study of SDE’s with drift coefficient of linear growth, one can easily obtain weak existence of time-singular Volterra equations of the form

[TABLE]

where $V$ is a singular and square integrable deterministic Volterra kernel, and the drift $b$ is still of linear growth. As we have seen in Theorem 17, we only need to prove that

[TABLE]

But this can be verified in the case that $|V(t,s)|\leq C|t-s|^{h+\frac{1}{2}+\epsilon-1},$as we then obtain $|\int_{0}^{t}V(t,s)ds|\leq C|t-s|^{h+\frac{1}{2}+\epsilon}.$ We are currently writing a paper studying such singular Volterra equations, with merely linear growth conditions on the drift, and their relation to stochastic fractional differential equations.

4.4. Existence of strong solutions

The following three theorems can be found in [8] given by Proposition 6, and 7 and Theorem 8, but is cited here with modifications to accommodate the multifractional Brownian motion.

Proposition 19**.**

Let $X$ denote a weak solution constructed above, but where the drift $b$ is a uniformly bounded function. Fix a constant $\rho>\sup_{t\in[0,T]}h_{t}+1$. For any measurable and non-negative function $g$ with $g:[0,T]\times\mathbb{R}\rightarrow\mathbb{R},$ there exists a constant $K$ depending on $\parallel b\parallel_{\infty}$, $\rho,$ and $T$ such that

[TABLE]

Proof.

For compact notation, Let the co variance in proposition 12 be written as,

[TABLE]

Let $Z=\frac{d\tilde{P}}{dP}$ be the Radon-Nikodym derivative that we constructed in Theorem 17. Then by the Hölder inequality for some $\frac{1}{\alpha}+\frac{1}{\beta}=1,$ we have the estimate

[TABLE]

[TABLE]

The expectation of $Z^{-\alpha}$ is explicitly given for any $\alpha>1$ by

[TABLE]

[TABLE]

obtained by arguments given in Theorem 17. Next we look at the other term,

[TABLE]

[TABLE]

[TABLE]

where we applied the Hölder inequality again, with $\gamma=\frac{\gamma^{\prime}}{\gamma^{\prime}-1}\Leftrightarrow\frac{1}{\gamma}+\frac{1}{\gamma^{\prime}}=1$ with $\sup_{t\in[0,T]}h_{t}+1<\gamma$. Explicit calculations then yield

[TABLE]

and the result follows. ∎

The next proposition will show that if we are able to find a sequence of bounded and measurable functions $\left\{b_{n}\right\}_{n\in\mathbb{N}}$ converging to the drift function $b$ in equation (4.1) almost surely, and the corresponding solutions $X^{(n)}$ to equation (4.1) constructed with $b_{n}$ converges to some process $X$. Then $X$ is a solution to the original differential equation (4.1). When we have proved this result, we will show that there actually exists such a sequence of functions $\left\{b_{n}\right\}_{n\in\mathbb{N}}$, when $b$ is of a certain class, and therefore, by the path wise uniqueness property of the solution, we obtain that the solution to equation (4.1) is a strong solution.

Proposition 20**.**

Let $\left\{b_{n}\right\}_{n\in\mathbb{N}}$ be a sequence of bounded and measurable functions on $[0,T]\times\mathbb{R}$ bounded uniformly by a constant $C$ such that

[TABLE]

Also, assume that the corresponding solutions $X_{t}^{(n)}$ of equation (4.1) given by

[TABLE]

converge to a process $X_{t}$ for all $t\in[0,T].$ Then $X_{\cdot}$ is a solution to equation 4.1.

Proof.

We need to show that

[TABLE]

Adding and subtracting $b_{n}\left(s,X_{s}\right)$ in the above expression, we can find

[TABLE]

[TABLE]

[TABLE]

Moreover, there exists a smooth function $\kappa:\mathbb{R}\rightarrow\mathbb{R}$ such that $\kappa(0)=1$ and $\kappa(z)\in[0,1]$ for all $|z|<1$ and $\kappa(z)=0$ for all $|z|\geq 1$. Fix $\epsilon>0$ and let $R$ be a constant such that

[TABLE]

By the Frechet-Kolmogorov theorem (a compactness criterion for $L^{2}\left(B\right)$ when $B$ is a bounded subset of $\mathbb{R}^{d}$), the sequence of functions $\left\{b_{k}\right\}_{k\in\mathbb{N}}$ is relatively compact in $L^{2}\left(\left[0,T\right]\times\left[-R,R\right]\right)$, and therefore we can find a finite set of smooth functions $\{H_{1},..,H_{N}\}\in L^{2}\left(\left[0,T\right]\times\left[-R,R\right]\right)$ such that for every $k$,

[TABLE]

for some $H_{i}\in\{H_{1},..,H_{N}\}$. Using this we can find

[TABLE]

[TABLE]

[TABLE]

We begin by looking at the components $I_{u}$, $u=1,2,3$ separately, and start with $u=1.$ Using the $\kappa$ function we defined earlier, we can see that

[TABLE]

[TABLE]

then by Proposition 19, we know that the first of the two elements above is bounded in the following way,

[TABLE]

[TABLE]

where $K_{1}$ depends on $\parallel b\parallel_{\infty}$ and $\sup_{i}\parallel H_{i}\parallel_{\infty}$ and $T$. We can then show for the second component,

[TABLE]

[TABLE]

where $K_{2}$ also depends on $\parallel b\parallel_{\infty}$ and $\sup_{i}\parallel H_{i}\parallel_{\infty}$. Set $K=\max K_{1},K_{2}$, by the properties of $H_{i}$ and $\kappa$ shown above in relation to the sequence of $\left\{b_{k}\right\},$ we have that,

[TABLE]

In the same way, we can show that for a constant $K$ similar to the one above,

[TABLE]

at last, of course $\lim_{n\rightarrow\infty}I_{2}(n)=0$ since $X^{(n)}\rightarrow X,$ and therefore,

[TABLE]

For $J_{1}$ we can use a very similar argument, as we can decompose it into

[TABLE]

[TABLE]

and use Proposition 19, just as above. ∎

The next theorem shows how we can combine Proposition 19 and Proposition 20 to obtain strong solutions when $b$ is of linear growth. Nualart and Ouknine do this by constructing a proper sequence of functions which is bounded and measurable such that this sequence converges to $b,$ and we can apply Propositions 19, and 20.

Theorem 21**.**

Let the drift $b$ in the SDE in equation (4.1) be of linear growth, i.e. $|b(t,x)|\leq C(1+|x|)$ for a.a. $(t,x)\in[0,T]\times\mathbb{R}$. Then there exists a unique strong solution to equation (4.1).

Proof.

The path-wise uniqueness is already obtained in Theorem 17. Therefore, the object of interest here is the strong existence. Define a function $b_{R}(t,x)=b(t,\left(x\vee-R\right)\wedge R),$ which by the linear growth condition we can see is bounded and measurable. Next, let $\varphi$ be a non-negative test function with compact support in $\mathbb{R}$ such that $\int_{\mathbb{R}}\varphi(y)dy=1$. For $j\in\mathbb{N}$ define the function

[TABLE]

which one can verify is Lipschitz in the second variable uniformly in $t$., Indeed,

[TABLE]

Furthermore, define the functions

[TABLE]

Both is again Lipschitz in the second variable uniformly in $t$, and we see that $b_{R,n,k}\downarrow b_{R,n}$ as $k\rightarrow\infty$ for a.a. $x\in\mathbb{R}$ and $t\in[0,T]$. Now, we can construct a unique solution $\tilde{X}_{R,n,k}$ from equation (4.1) with corresponding drift coefficient $b_{R,n,k}.$ By the comparison theorem for ordinary differential equations, the sequence $\tilde{X}_{R,n,k}$ is decreasing as $k$ grows. Therefore $\tilde{X}_{R,n,k}$ has a limit $\tilde{X}_{R,n}$. By the comparison theorem again, we have that $\tilde{X}_{R,n,k}$ and $\tilde{X}_{R,n}$ are bounded from above and below by $R$ and $-R$ respectively. Moreover, the solution $\tilde{X}_{R,n}$ is increasing as $n$ gets larger, and converge to a limit $X_{R}$. Now we can apply Proposition 20, and we obtain that there exists a unique strong solution. ∎

5. Conclusion

In this article we have constructed a new type of multifractional derivative operator acting as the inverse of the generalized fractional integral. We have then applied this derivative operator to construct strong solutions to SDE’s where the drift is of linear growth and the noise is of non-stationary and multifractional type. The applications of such equations may be found in a range of fields, including finance, physics and geology. For future work, we are currently working on a way to construct solutions to Volterra SDE’s with singular Volterra drift, and driven by self exciting multifractional noise. The methodology will be similar to the above, but we will need to generalize the last to sections to account for Volterra kernels of singular type. Furthermore, we believe that the multifractional differential operator may shed new light on both multifractional (partial) differential equations, with possibly random order differentiation, and are currently working on a project relating to such equations.

Appendix

In this section we will prove that the integral

[TABLE]

satisfies for all $x>0,$

[TABLE]

[TABLE]

The existence of the terms on the right hand side is shown in Lemma 6, where it is shown that the right hand side is elements of $L^{p}\left([0,T]\right)$, which is sufficient for our application. To see the above relation, we shall use the definition of the derivative. Start by adding and subtracting the point $g(x),$ and get

[TABLE]

We then look at the right hand side as increments between $x+h$ to $x$, and after simple manipulations get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The second part is given by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Combining $J_{1}$ and $J_{2}$ and we get and notice that the term $\left(g(x+h)-g(x)\right)\int_{0}^{x}K(x+h,t)dt$ cancels out and we obtain

[TABLE]

[TABLE]

[TABLE]

Let us first look at the last line above,

[TABLE]

we know,

[TABLE]

as $h\rightarrow 0$. Further, we see

[TABLE]

Moreover, set $t=x+\lambda h$, and then we have

[TABLE]

But the integral on the right hand side is bounded, i.e

[TABLE]

This imply that

[TABLE]

We are left to look at the limits of the integrals

[TABLE]

We will look at them separately, staring with the one one the left above. By dominated convergence, we have

[TABLE]

Secondly, it is straight forward to see that,

[TABLE]

The fact that $\frac{d}{dx}\int_{0}^{x}K(x,t)dt$ indeed exists is proved in Lemma 6 using the fact that $K(x,t)=\frac{\Gamma\left(\alpha(t)\right)}{\left(x-t\right)^{\alpha(t)}}$.