<p>Considering the fractional Brownian motion <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textbf{B}_t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">B</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> as a generalized stochastic process in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textbf{S}'(\mathbb {R})\otimes (\textbf{S})_{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="bold">S</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊗</mo> <msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">S</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>, the fractional white noise <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{W}^H_t$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="bold">W</mi> <mi>t</mi> <mi>H</mi> </msubsup> </math></EquationSource> </InlineEquation> is defined as the distributional derivative of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{dB}^H_t$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="bold">dB</mi> <mi>t</mi> <mi>H</mi> </msubsup> </math></EquationSource> </InlineEquation>. Using the framework of white noise analysis and the integral representation of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textbf{B}^H_t$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="bold">B</mi> <mi>t</mi> <mi>H</mi> </msubsup> </math></EquationSource> </InlineEquation>, we explicitly calculate the coefficients of their chaos expansion and provide a recurrence formula for their effective calculation. As a novel stochastic model, we introduce the notion of fractional Brownian motion and fractional white noise with a distributed-order Hurst parameter <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textbf{H}\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">H</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Building on this construction, we derive numerical simulations of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textbf{B}^H_t$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="bold">B</mi> <mi>t</mi> <mi>H</mi> </msubsup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textbf{W}^H_t$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="bold">W</mi> <mi>t</mi> <mi>H</mi> </msubsup> </math></EquationSource> </InlineEquation> by truncating the obtained chaos expansions, which allows us to approximate their sample paths and estimate the truncation error. We illustrate the results with two examples from finance and life insurance: a stock price model and a mortality model, both driven by a fractional white noise process with distributed-order Hurst parameter.</p>

The chaos expansion of fractional Brownian motion and fractional white noise with distributed-order Hurst parameter

期刊论文
发表日期：

2026年4月24日

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Dora Seleši,
Stefan Tošić

摘要

Considering the fractional Brownian motion $\textbf{B}_t$ $B_{t}$ as a generalized stochastic process in $\textbf{S}'(\mathbb {R})\otimes (\textbf{S})_{-1}$ $S^{'} (R) \otimes {(S)}_{- 1}$ , the fractional white noise $\textbf{W}^H_t$ $W_{t}^{H}$ is defined as the distributional derivative of $\textbf{dB}^H_t$ ${dB}_{t}^{H}$ . Using the framework of white noise analysis and the integral representation of $\textbf{B}^H_t$ $B_{t}^{H}$ , we explicitly calculate the coefficients of their chaos expansion and provide a recurrence formula for their effective calculation. As a novel stochastic model, we introduce the notion of fractional Brownian motion and fractional white noise with a distributed-order Hurst parameter $\textbf{H}\in (0,1)$ $H \in (0, 1)$ . Building on this construction, we derive numerical simulations of $\textbf{B}^H_t$ $B_{t}^{H}$ and $\textbf{W}^H_t$ $W_{t}^{H}$ by truncating the obtained chaos expansions, which allows us to approximate their sample paths and estimate the truncation error. We illustrate the results with two examples from finance and life insurance: a stock price model and a mortality model, both driven by a fractional white noise process with distributed-order Hurst parameter.