Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift. (22nd October 2014)

• Record Type: Journal Article
• Authors: Étoré, Pierre; Martinez, Miguel
• Is Part Of: ESAIM. Volume 18(2014)
• Page Start: 686
• Page End: 702
• Publication Date: 2014-10-22
• Subjects: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; 519.2
• DOI: 10.1051/ps/2013053
• Languages: English
• ISSNs: 1292-8100

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>In this note we propose an exact simulation algorithm for the solution of <disp-formula id="FD1"><label>(1)</label><alternatives><tex-math id="tex_eq1"><![CDATA[\begin{equation} \label{eds-intro} {\rm d}X_t={\rm d}W_t+\bar{b}(X_t){\rm d}t, \quad X_0=x, \end{equation}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxs5j" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq1" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mi mathvariant="italic">X</mml:mi><mml:mi mathvariant="italic">t</mml:mi></mml:msub><mml:mo mathvariant="normal">=</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mi mathvariant="italic">W</mml:mi><mml:mi mathvariant="italic">t</mml:mi></mml:msub><mml:mo mathvariant="normal">+</mml:mo><mml:mi mathvariant="italic">bÌ…</mml:mi><mml:mo mathvariant="normal">(</mml:mo><mml:msub><mml:mi mathvariant="italic">X</mml:mi><mml:mi mathvariant="italic">t</mml:mi></mml:msub><mml:mo mathvariant="normal">)</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="italic">t, </mml:mi><mml:mo> </mml:mo><mml:msub><mml:mi mathvariant="italic">X</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo mathvariant="normal">=</mml:mo><mml:mi mathvariant="italic">x, </mml:mi></mml:mrow></mml:math></alternatives></disp-formula>where <inline-formula><alternatives><tex-math id="tex_eq2"><![CDATA[\hbox{$\bar{b}$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxsb4" xmlns:xlink="http://www.w3.org/1999/xlink" /><textual-form><italic>bÌ…</italic></textual-form></alternatives></inline-formula> is a smooth real function except at point 0 where <inline-formula><alternatives><tex-math id="tex_eq3"><![CDATA[\hbox{$\bar{b}(0+)\neq \bar{b}(0-)$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxshq" xmlns:xlink="http://www.w3.org/1999/xlink" /><textual-form><italic>bÌ…</italic>(0 + ) ≠<italic>bÌ…</italic>(0 âˆ')</textual-form></alternatives></inline-formula>. The main idea is to sample an exact skeleton of <italic>X</italic> using an algorithm deduced from the convergence of the solutions of the skew perturbed equation <disp-formula id="FD2"><label>(2)</label><alternatives><tex-math id="tex_eq4"><![CDATA[\begin{equation} \label{edsbeta} {\rm d}X^\beta_t={\rm d}W_t+\bar{b}(X^\beta_t){\rm d}t + \beta {\rm d}L^0_t(X^\beta), \quad X_0=x \end{equation}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgkxs83" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq4" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mi mathvariant="italic">X</mml:mi><mml:mi mathvariant="italic">t</mml:mi><mml:mi mathvariant="italic">β</mml:mi></mml:msubsup><mml:mo mathvariant="normal">=</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mi mathvariant="italic">W</mml:mi><mml:mi mathvariant="italic">t</mml:mi></mml:msub><mml:mo mathvariant="normal">+</mml:mo><mml:mi mathvariant="italic">bÌ…</mml:mi><mml:mo mathvariant="normal">(</mml:mo><mml:msubsup><mml:mi mathvariant="italic">X</mml:mi><mml:mi mathvariant="italic">t</mml:mi><mml:mi mathvariant="italic">β</mml:mi></mml:msubsup><mml:mo mathvariant="normal">)</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="italic">t</mml:mi><mml:mo mathvariant="normal">+</mml:mo><mml:mi mathvariant="italic">β</mml:mi><mml:mi mathvariant="normal">d</mml:mi><mml:msubsup><mml:mi mathvariant="italic">L</mml:mi><mml:mi mathvariant="italic">t</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msubsup><mml:mo mathvariant="normal">(</mml:mo><mml:msup><mml:mi mathvariant="italic">X</mml:mi><mml:mi mathvariant="italic">β</mml:mi></mml:msup><mml:mo mathvariant="normal">)</mml:mo><mml:mi mathvariant="italic">, </mml:mi><mml:mo> </mml:mo><mml:msub><mml:mi mathvariant="italic">X</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo mathvariant="normal">=</mml:mo><mml:mi mathvariant="italic">x</mml:mi></mml:mrow></mml:math></alternatives></disp-formula>towards <italic>X</italic> solution of (1) as <italic>β</italic> ≠0 tends to 0. In this note, we show that this convergence induces the convergence of exact simulation algorithms proposed by the authors in [Pierre Étoré and Miguel Martinez. <italic>Monte Carlo Methods Appl. </italic><bold>19 </bold>(2013) 41â€"71] for the solutions of (2) towards a limit algorithm. Thanks to stability properties of the rejection procedures involved as <italic>β</italic> tends to 0, we prove that this limit algorithm is an exact simulation algorithm for the solution of the limit equation (<xref ref-type="disp-formula" rid="FD1">1</xref>). Numerical examples are shown to illustrate the performance of this exact simulation algorithm. </p> </abstract>