| Summary: | A stochastic difference game is considered in which a player wants to minimize the time spent by a controlled one-dimensional symmetric random walk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>X</mi><mi>n</mi></msub><mo>,</mo><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>}</mo></mrow></semantics></math></inline-formula> in the continuation region <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>:</mo><mo>=</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>}</mo></mrow></semantics></math></inline-formula>, and the second player seeks to maximize the survival time in <i>C</i>. The process starts at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><mi>x</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and the game ends the first time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>n</mi></msub><mo>≤</mo><mn>0</mn></mrow></semantics></math></inline-formula>. An exact expression is derived for the value function, from which the optimal solution is obtained, and particular problems are solved explicitly.
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