Modern Stochastics: Theory and Applications

A σ-additive mapping $\mu :\mathcal{B}\to {\mathsf{L}_{0}}$ is called stochastic measure (SM).

(Lemma 3.1 in [16]) Let ${\phi _{l}}:\hspace{2.5pt}\mathbb{R}\to \mathbb{R},\hspace{2.5pt}l\ge 1$, be measurable functions such that $\tilde{\phi }(x)={\textstyle\sum _{l=1}^{\infty }}|{\phi _{l}}(x)|$ is integrable w.r.t. μ on $\mathbb{R}$. Then

(Lemma 3 in [17] or Lemma 3.3 in [18]) Let Z be an arbitrary set, and function $q(z,s):Z\times [0,T]\to \mathbb{R}$ be such that all paths $q(z,\cdot )$ are continuous on $[0,T]$. Denote

Then the random function has a version such that for all $\beta \gt 0$, $\omega \in \Omega $, $z\in Z$ where ${\Delta _{kn}^{(T)}}\subset {\Delta _{{k^{\prime }}(n-1)}^{(T)}}$.

The domain S belongs to a class ${A^{m+\beta }}$ (${A^{m}}$) in ${\mathbb{R}^{d}}$ (${\mathbb{R}^{d+1}}$) if for every point P of $\bar{S}$ there exists a sphere with center P and a function χ, which belongs to a class ${A^{m+\beta }}$ (${A^{m}}$), such that for certain $i\le d$ inside the sphere.

There exists $\beta \in (0,1)$ such that $\bar{S}$ belongs to a class ${A^{1+\beta }}$ in ${\mathbb{R}^{d+1}}$.

Function ${u_{0}}:\bar{B}\times \Omega \to \mathbb{R}$ is measurable and bounded for each fixed $\omega \in \Omega $.

Function $f(s,y,z):[0,T]\times \bar{B}\times \mathbb{R}\to \mathbb{R}$ is measurable, bounded and for some constants ${L_{f}}\gt 0$, $\beta (f)\gt 0$ and all $s\in [0,T]$, ${y_{1}}\hspace{0.1667em},{y_{2}}\in \bar{B}$, ${z_{1}}\hspace{0.1667em},{z_{2}}\in \mathbb{R}$.

Function $\sigma (s,y):[0,T]\times \bar{B}\to \mathbb{R}$ is measurable, bounded and for some constants ${L_{\sigma }}\gt 0$, $1\gt \beta (\sigma )\gt 1/2$ and all ${s_{1}},{s_{2}}\in [0,T]$, ${y_{1}}\hspace{0.1667em},{y_{2}}\in \bar{B}$.

Stochastic measure μ has bounded paths: for random constant ${C_{\mu }}(\omega )$ and all $t\in [0,T]$.

Stochastic measure μ has Hölder continuous paths: for random constant $C(\omega )$, deterministic constant $\beta (\mu )$ and all ${s_{1}},\hspace{0.1667em}{s_{2}}\in [0,T]$.

Let Assumptions 1, 2, 4, 5 hold. Then, for arbitrary set ${B^{\prime }}$, $d(\partial B,\bar{{B^{\prime }}})\gt 0$, the random process has a version of a kind (3), which is Hölder continuous with the exponent ${\gamma _{1}}$ on ${B^{\prime }}$ for all $t\in [0,T]$, ${\gamma _{1}}\lt \beta (\sigma )$.

Let Here $z=(t,{x_{1}},{x_{2}})$. The function (13) is continuous in $[0,T]$ as a function of s, as follows from We give the brief proof of (14). Fix $\varepsilon \gt 0$. Then for all $0\le r\lt t$ We can choose r such that $C|t-r{|^{\beta (\sigma )}}\le \varepsilon /2$. On the other hand, as follows from [7, Chapter 3, Sec. 7, Definition]. Therefore, there exists $\delta \gt 0$ which may depend on t and x such that for all $s\gt t-\delta $, , and the convergence (14) holds. Therefore, we can apply Lemma 2 for q, which is defined by (13). At first, we estimate ${\omega _{2,[0,t]}}(q,r)$. Consider the difference ${I_{1}}$ is estimated in the same way as ${A_{2}}(s,h)$ in [3], where we estimate the derivatives using (8). More precisely, we get Therefore, we obtain that Denote Now we apply the definition in [7, Chapter 3, Sec. 7] to obtain the properties of v: Now consider (16) as a boundary value problem for each fixed s. Theorem 11 in [8, Sec. 1] implies that it has unique solution; consequently, v does not depend on τ. Therefore, We can construct the extension of a function $\sigma (s,y)$, which is bounded and Hölder continuous in $[0,T]\times {\mathbb{R}^{d}}$ with the same exponent. This follows, for example, from [7, Chapter 3, Theorem 2, p. 60]. Now we note that $v(t,x,s)={v^{(1)}}(t,x,s)-{v^{(2)}}(t,x,s)$, where ${v^{(1)}}$ is a solution of the Cauchy problem in $[0,T]\times {\mathbb{R}^{d}}$, and ${v^{(2)}}$ is a solution of a boundary value problem in $[0,T]\times B$. We represent ${I_{2}}$ in a form ${I_{21}}-{I_{22}}$, where According to [8, Sec. 4, Theorem 2], ${v^{(1)}}$ can be represented in the form where Therefore, in the notations of [3]. Recall the estimates for ${A_{1}}(s,h)$ from the mentioned article: Therefore, On the other hand, estimating ${A_{1}}(s,h)$ in a similar way to estimation [3.54] in [18], we get where for the i-th summand we used the substitutions From (4) and (18) it follows that Now we estimate ${I_{22}}$. Fix $\alpha \in (0,1)$. As the functions $v(t,x,s)$ and ${v^{(1)}}(t,x,s)$ are bounded in $\bar{Q}$ uniformly on $t,\hspace{0.1667em}x,\hspace{0.1667em}s$, the same holds for ${v^{(2)}}(t,x,s)$. Let us prove it, for example, for v: It is possible to take the domains ${B^{\prime\prime }}$ and ${B^{\prime\prime\prime }}$ such that ${\bar{B}^{\prime\prime }}\subset B$, ${\bar{B}^{\prime\prime\prime }}\subset {B^{\prime\prime }}$, $\bar{{B^{\prime }}}\subset {B^{\prime\prime\prime }}$ and $\partial {B^{\prime\prime }},\hspace{0.1667em}\partial {B^{\prime\prime\prime }}\in {A^{3}}$ (see Definition 2).

Denote ${S^{\prime\prime }}=[0,T]\times \partial {B^{\prime\prime }}$, ${B^{\prime\prime }_{0}}=\{0\}\times {B^{\prime\prime }}$. It is obvious that ${v^{(2)}}\in C([0,T]\times \bar{{B^{\prime\prime }}})$. As $[0,T]\times \bar{{B^{\prime\prime }}}$ is a compact, there exist polynomials ${\Psi _{m}}$ such that ${\Psi _{m}}\to {v^{(2)}}$ in $C([0,T]\times \bar{{B^{\prime\prime }}})$. Therefore, there exists a sequence ${\psi _{m}}(t,x)={\Psi _{m}}(t,x)-{\Psi _{m}}(0,x)$ such that ${\psi _{m}}\in {C^{3}}({S^{\prime\prime }}\cup {B^{\prime\prime }_{0}})$, ${\psi _{m}}=0$ on $\bar{{B^{\prime\prime }_{0}}}$ and ${\psi _{m}}\to {v^{(2)}}$ on $C({S^{\prime\prime }}\cup {B^{\prime\prime }_{0}})$. Let ${v_{m}^{(2)}}$ be a solution of the boundary value problem on $[0,T]\times {B^{\prime\prime }}$. Theorem 7 in [7, Chap. III, Sec. 3] implies that ${v_{m}^{(2)}}\in {C_{2+\alpha }}([0,T]\times {\bar{B}^{\prime\prime }_{0}})$. Therefore, we can apply Theorem 4 in [7, Chap. IV, Sec. 7] for the functions ${v_{m}^{(2)}}-{v_{n}^{(2)}}$, where ${B^{\prime\prime }}$, ${B^{\prime\prime\prime }}$ and $(0,T)\times {B^{\prime\prime }}$ are the sets R, ${R_{0}}$ and D in the formulation of the theorem, respectively. Using, in addition, the maximum principle, we obtain and sequence $\{{v_{m}^{(2)}}:m\ge 1\}$ converges in $\| \cdot {\| _{0,2+\alpha }^{{R_{0}},D}}$ to a limit function ${\tilde{v}^{(2)}}$; for example, ${M_{0,0}^{{R_{0}},D}}[{v_{m}^{(2)}}-{\tilde{v}^{(2)}}]\to 0,\hspace{0.1667em}m\to 0$. On the other hand, according to [7, Chap. III, Sec. 6, Corollary of Theorem 15], sequence $\{{v_{m}^{(2)}}:m\ge 1\}$ converges uniformly to ${v^{(2)}}$ on $[0,T]\times \bar{{B^{\prime\prime }}}$. Therefore, ${\tilde{v}^{(2)}}={v^{(2)}}$ and where constants K and ${K_{1}}$ depend only on a, α, ${B^{\prime\prime\prime }}$ and ${B^{\prime\prime }}$. This implies the inequality We can choose γ in (15), λ in (19), α in (20) such that Estimating $q(z,s)$ in the same way as ${I_{2}}$ for $s\lt t$ and using Hölder continuity of σ for $t\le s\le T$, we obtain that for each ${\tilde{\gamma }_{1}}\lt \beta (\sigma )$, Now we proceed to the estimating of ${\omega _{2,[0,T]}}(q,r)$. We obtain that where Triangle inequality for the norm $\| \cdot {\| _{{L_{2}}}}$ together with (21) implies that where we take ${\tilde{\gamma }_{1}}\in ({\gamma _{1}},\beta (\sigma ))$. On the other hand, the difference $q(z,t)-q(z,s)$ can be rewritten in the following way: Remark that The same estimates can be applied for $|{v^{(1)}}(0,{x_{2}},t)-{v^{(1)}}(t-s,{x_{2}},s)|$. That leads to the inequality For the second summand in (23) we can use the same estimates as in (20) and obtain that Here we also applied the fact ${v^{(2)}}(0,{x_{i}},t)={v^{(2)}}(0,{x_{i}},s)=0$. Eqs. (24) and (25) imply that Together with (22), (26) leads to the estimate where ${\theta _{2}}\gt 1/2$. In conclusion, As a result, for a sufficiently small ε. The only fact left to prove is that where $C(\omega )$ does not depend on t. Assume that for each n, $t\in {\Delta _{{k_{n}}n}^{(T)}}$; then by Assumption 5  □

Let Assumptions 1, 2, 4, 6 hold. Then the random process has a version of a kind (3), which is Hölder continuous on $[\delta ,T]$ with the exponent ${\gamma _{2}}$ for all $x\in B$, $T\gt \delta \gt 0$, ${\gamma _{2}}\lt \beta (\mu )$, ${\gamma _{2}}\lt \beta (\sigma )/(4-2\beta (\sigma ))$. If $x\in {B^{\prime }}$, where ${\bar{B}^{\prime }}\subset B$, we can choose Hölder constant that depends only on σ, μ, ${\gamma _{2}}$, δ and ${B^{\prime }}$.

Let ${t_{1}}\le {t_{2}}$. We represent the difference of the integrals (27) in the form where We fix a domain ${B^{\prime }}$ such that $x\in {B^{\prime }}$, ${\bar{B}^{\prime }}\subset B$ and, in the notations of Lemma 3, obtain that where the constant C in the last inequality depends on ${B^{\prime }}$. We take ${k_{n1}}$ and ${k_{n2}}$ such that ${t_{1}}\in {\Delta _{{k_{n1}}n}^{(T)}}$ and ${t_{2}}\in {\Delta _{{k_{n2}}n}^{(T)}}$ and choose ${n_{0}}$ that satisfies the inequality For such ${n_{0}}$, ${k_{{n_{0}}1}}+1={k_{{n_{0}}2}}$ or ${k_{{n_{0}}1}}+2={k_{{n_{0}}2}}$, while for smaller n, ${k_{n1}}+1={k_{n2}}$ or ${k_{n1}}={k_{n2}}$. We can easily obtain by induction that for each $n\ge {n_{0}}$ The function $\bar{q}(z,s)$ was already defined on $[{t_{1}},{t_{2}}]$, let $\bar{q}(z,s)=\bar{q}(z,{t_{1}})$ for $s\lt {t_{1}}$ and $\bar{q}(z,s)=\bar{q}(z,{t_{2}})$ for $s\gt {t_{2}}$. Now we can use Lemma 2 to estimate integral ${J_{1}}$: For each n we can omit summands for $k\le {k_{n1}}$, as for such k, $\bar{q}(z,{d_{(k-1)n}^{(T)}})=\bar{q}(z,{t_{1}})=\bar{q}(z,{d_{(k-2)n}^{(T)}})$, and summands for $k\gt {k_{n2}}$, as for such k, ${\Delta _{kn}^{(T)}}\cap ({t_{1}},{t_{2}}]=\varnothing $: Now we estimate the sums ${S_{1}}$ and ${S_{2}}$, using (29). where we choose $\beta \gt 0$ such that such β exists as $1\gt \beta (\sigma )$. Therefore, In order to estimate ${J_{2}}$, we need to prove some properties of the function $\bar{Q}$. Firstly, notice that in the notations of Lemma 3 $\bar{Q}(z,s)=v({t_{2}}-s,x,s)-v({t_{1}}-s,x,s)$ and The difference $|{v^{(1)}}({t_{2}}-s,x,s)-{v^{(1)}}({t_{1}}-s,x,s)|$ was already estimated in [3], see formulas (13)–(15): This leads to the following estimates for $|\bar{Q}(z,s)|$: Eqs. (31) and (32) directly imply that Rewrite the difference $\bar{Q}(z,s+h)-\bar{Q}(z,s)$ in a form Using (9), we obtain that ${F_{2}}$ can be estimated similarly to (29): The estimates hold for ${F_{3}}$, too. That leads to the following analogue of formula (19) in [3]: The next inequality is proved with the help of (29): Raising (35) to the power λ and (34) to the power $1-\lambda $, where $\lambda \in (1/(2-\beta (\sigma )),1)$, we get that where Raising (37) to the power λ and (36) to the power $1-\lambda $, we obtain that We choose ${m_{0}}$ which satisfies a condition The function $\bar{Q}(z,s)$ was already defined on $[0,{t_{1}}]$, let $\bar{Q}(z,s)=\bar{Q}(z,{t_{1}})$ for $s\gt {t_{1}}$. Now function $\bar{Q}$ is continuous on $[0,{t_{2}}]$ and we can use Lemma 2: Using (33), we easily obtain that In order to estimate ${U_{3}}$, we use (38): Now we estimate ${U_{4}}$, applying (39): The estimates (42) and (43) hold for each $\beta \gt 0$. For each fixed ${\gamma _{2}}\lt \beta (\sigma )/(2(2-\beta (\sigma )))$ we take Choose β such that $\beta +2{\gamma _{2}}\lt \beta (\sigma )/(2-\beta (\sigma ))$; then $\lambda \gt 1/(2-\beta (\sigma ))$. Taking into consideration that $\beta (\sigma )/2\gt \beta (\sigma )/(2(2-\beta (\sigma )))\gt {\gamma _{2}}$ and estimates (40), (41), we finally obtain The substitution of (30) and (44) into (28) leads to inequality That completes the proof of the lemma.  □

The item (1) is proved in the same way as item (i) in [16], using the following iteration process: ${u^{(0)}}(t,x)=0$, consequently, we give only a brief version of the proof. Denote Then for each $\omega \in \Omega $ the following estimates hold: and we can prove by induction that and the series ${\textstyle\sum _{n=0}^{\infty }}{g_{n}}(t)$ converges uniformly in $[0,T]$. Hence there exists a limit function $u(t,x)={\lim \nolimits_{n\to \infty }}{u^{(n)}}(t,x)$, which is the solution of (6). Prove that it is unigue. Let $w(t,x)$ be another solution of (6); then, using the same arguments as in the proof of (46), we obtain that for a function $g(t)={\sup _{x\in \bar{B}}}|u(t,x)-w(t,x)|$, and for each $n\ge 1$. Sending n to infinity, we obtain that $u=w$.

In order to prove item (2), we represent (45) as where We will prove that function ${u^{(n)}}$ is Hölder continuous in $[\delta ,T]\times \bar{{B^{\prime }}}$ for each fixed $\omega \in \Omega $ with the exponent ${\gamma _{1}}$ by induction on n; if $n=0$, the statement is obvious. The function ${u_{1}}(t,x)$ satisfies the equation $\mathcal{L}{u_{1}}=0$ in $(0,T]\times B$ (see, for example, the proof of Theorem 4.3 in [8]), and, consequently, in $[\delta ,T]\times \bar{{B^{\prime }}}$. On the other hand, [8, Theorem 4.3] implies that function ${u_{2}^{(n)}}$ is a solution of the problem The Hölder continuity of $f(s,y,{u^{(n-1)}}(s,y))$ by y follows from the inequalities where ${\beta _{1}}=\min \{\beta (f),{\gamma _{1}}\}$. Theorem 1 in [6] implies that for each $\epsilon \in (0,1)$ where constant ${C_{2}}$ depends only on ϵ and the operator $\mathcal{L}$. Applying Lemma 3, we obtain that there exist the versions ${\tilde{u}_{n}^{(x)}}$ of the functions ${u^{(n)}}$ such that where constant ${L_{{\tilde{u}^{(x)}}}}$ does not depend on n. Sending n to infinity, we obtain the statement of the item.

The beginning of the proof of the item (3) is similar to the proof of the item (2), we just use Lemma 4 instead of Lemma 3 and get that where constant ${L_{{\tilde{u}^{(t)}}}}$ does not depend on n. Therefore, there exists a version ${\tilde{u}^{(t)}}$ of a function u such that On the other hand, we have already built a version ${\tilde{u}^{(x)}}$, which satisfies (11). We exclude all $\omega \in \Omega $ such that ${\tilde{u}^{(x)}}(t,x)\ne {\tilde{u}^{(t)}}(t,x)$ for at least one pair of rational $(t,x)\in [\delta ,T]\times \bar{{B^{\prime }}}$. For other $\omega \in \Omega $ we take $\tilde{u}={\tilde{u}^{(t)}}={\tilde{u}^{(x)}}$ for rational $(t,x)$ and define $\tilde{u}$ for other pairs $(t,x)\in [\delta ,T]\times \bar{{B^{\prime }}}$ by continuity. The function $\tilde{u}$ which is built in such way is Hölder continuous on $[\delta ,T]\times \bar{{B^{\prime }}}$.  □