On a nonlinear stochastic integral equation of the Hammerstein type (1973)

A nonlinear stochastic integral equation of the Hammerstein type in the form <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t semicolon omega right-parenthesis equals h left-parenthesis t semicolon omega right-parenthesis plus integral Underscript s Endscripts k left-parenthesis t comma s semicolon omega right-parenthesis f left-parenthesis s comma x left-parenthesis s semicolon omega right-parenthesis right-parenthesis d mu left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>;</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>;</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>;</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>;</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mi>d</mml:mi> <mml:mi>μ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x(t;\omega ) = h(t;\omega ) + \int _s {k(t,s;\omega )f(s,x(s;\omega )} )d\mu (s)</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> is studied where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t element-of upper S comma a"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">t \in S,a</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <italic>a</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-finite measure space with certain properties, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega element-of normal upper Omega"> <mml:semantics> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\omega \in \Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the supporting set of a probability measure space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Omega comma upper A comma upper P right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\Omega ,A,P)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the integral is a Bochner integral. A random solution of the equation is defined to be a second order vector-valued stochastic process <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t semicolon omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>x</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:mrow> <mml:annotation encoding="application/x-tex">x(t;\omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <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> which satisfies the equation almost certainly. Using certain spaces of functions, which are spaces of second order vector-valued stochastic processes on <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>, and fixed point theory, several theorems are proved which give conditions such that a unique random solution exists.