Nonstandard construction of the stochastic integral and applications to stochastic differential equations. II

Abstract

H. J. Keisler has recently used a nonstandard theory of Itô integration (due to R. M. Anderson) to construct solutions of Itô integral equations by solving an associated internal difference equation. In this paper we use the same general approach to find solutions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">y(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of semimartingale integral equations of the form <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y left-parenthesis t comma omega right-parenthesis equals h left-parenthesis t comma omega right-parenthesis plus integral Subscript 0 Superscript t Baseline f left-parenthesis s comma omega comma y left-parenthesis dot comma omega right-parenthesis right-parenthesis d z left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi> ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi> ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mo> ∫ </mml:mo> <mml:mn>0</mml:mn> <mml:mi>t</mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi> ω </mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo> ⋅ </mml:mo> <mml:mo>,</mml:mo> <mml:mi> ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>d</mml:mi> <mml:mi>z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">y(t,\omega ) = h(t,\omega ) + \int _0^t {f(s,\omega ,y(\cdot ,\omega ))\,dz(s)}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z"> <mml:semantics> <mml:mi>z</mml:mi> <mml:annotation encoding="application/x-tex">z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a given semimartingale, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a right-continuous process and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis s comma omega comma dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi> ω </mml:mi> <mml:mo>,</mml:mo> <mml:mo> ⋅ </mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(s,\omega , \cdot )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is continuous on the space of right-continuous functions with left limits, with the topology of uniform convergence on compacts. In addition, we generalize Keisler’s continuity theorem and give necessary and sufficient conditions for an internal martingale to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -continuous.