The　supply　and　demand　of　financial　assets　for　investors,　who　can　place　and　cancel　limit　orders　at　any　time,　often　changes　continuously　with　market　fluctuations.　And　it　causes　the　randomness　of　market　liquidity.　This　paper　studies　the　optimal　portfolio　execution　problem　with　the　stochastic　market　depth　in　the　limit　order　book　(LOB)　market.　We　develop　successfully　the　analytical　optimal　execution　strategy　and　minimum　execution　cost　for　such　a　problem　by　using　the　dynamic　programming　method.　The　revealed　results　show　that　the　obtained　optimal　execution　strategy　is　a　linear　function　with　respect　to　the　displacement　and　remaining　position.　Moreover,　the　optimal　execution　cost　is　a　non-linear　function　of　the　initial　asset　quantity.　The　numerical　examples　show　that　the　investor　would　pay　higher　transaction　costs　and　reduce　the　level　of　investment　welfare　when　they　ignore　the　random　fluctuation　characteristics　of　market　depth　and　the　cross-effect　between　asset　prices.　©　2022　Systems　Engineering　Society　of　China.　All　rights　reserved.