Translated Title

Translated Abstract

Since　1970s,　the　nonlinear　science　has　been　a　hot　field　in　scientific　research.　Nonlinear　stochastic　differential　equations,　as　a　very　activemathematical　branch　in　nonlinear　science,　has　drawn　much　interests　since　it　appeared.　Its　applications　involve　almost　all　areas　in　natural　and　social　sciences.　Now　it　is　a　powerful　theoretical　formwork　to　explore　and　solve　many　complicated　problems　in　natural　science　and　engineering.　It　is　still　a　difficult　task　to　use　this　formwork　to　solve　the　problems　of　stability　and　numerical　solution　for　stochastic　differential　equations　with　jump.　In　this　thesis,　we　　make　a　systematic　and　deep　investigation　on　the　methods　of　stability,　attraction　and　numerical　solution　for　stochastic　differential　equations　with　Markov　and　jump-diffusion　processes.　The　main　contents　are　organized　as　follows:　In　Chapter　1,　we　outline　the　development　histories　and　the　state　of　the　art　of　some　current　research　on　　　stochastic　differential　equations　with　Markov　and　jump-diffusion　processe,　and　briefly　explain　the　main　results　obtained　in　this　thesis　　and　their　significance.　　　　　In　Chapter　2,　we　study　the　　exponential　stability　of　stochastic　functional　differential　　equations　with　jumping　parameters.　Firstly,　by　using　nonsingular　M-matrices　equivalent　statement,　a　sufficient　　condition　is　obtained　to　ensure　the　stability　of　　　stochastic　differential　equations　with　Markov　jumping　parameters　in　Hilbert　space.　This　result　are　improvement　and　extension　of　the　papers　[100-101],　since　the　new　coercivity　condition　introduced　in　this　work　permits　the　state　independent　term　to　be　time　dependent　and　nonnegative,　while　in　the　papers[100-101]　this　paramrter　is　required　to　be　constant　and　nonpositive.　Secondly,　　　exponential　stability　of　stochastic　functional　differential　delay　equations　with　Markov　　　jumping　parameters　in　Hilbert　space　is　investigated.　The　analyses　consist　in　using　exponential　martingale　formula,　Lyapunov　　　functional　and　some　special　inequalities　derived　for　our　stability　purposes.　Finally,　the　aim　of　this　work　is　to　obtain　sufficient　conditions　for　stability　　of　multidimensional　jump-diffusion　processes　　exponential　stability　　in　n-dimensional　Euclidean　space.　We　obtain　that　the　trivial　solution　of　　Eq.(2.1)　is　pth　moment　exponentially　stable　and　super　bound　of　the　pth　Lyapunov　exponent　is　obtained.　In　particular,this　result　are　improvement　and　extension　of　existing　results.In　mathematical　finance　theory,　one　of　the　principal　interest　is　focused　on　option　pricing.　In　Chapter　3,　the　aim　of　this　paper　is　to　investigate　stability　in　distribution　for　stability　　of　multidimensionaljump-diffusion　processes.　Most　of　these　papers　are　concerned　with　asymptotic　stability　in　probability　or　in　mean　square　(i.e.　the　solution　will　tend　to　zero　in　probability　or　in　mean　square).　However,　there　are　not　many　results　on　the　stability　in　distribution.　Under　equation　(2.1)　satisfies　the　local　Lipschitz　condition　and　the　linear　growth　condition,　this　work　is　to　obtain　sufficient　conditions　for　stability　　of　multidimensional　jump-diffusion　processes　in　the　sense　of　stability　in　distribution,　and　there　exists　the　unique　invariant　probability　measureStochastic　differential　equations　provide　a　useful　means　of　introducing　stochasticity　into　models　across　a　broad　range　of　systems　from　chemistry　to　population　biology　and　engineering　ect.　However,　in　many　applications　the　resulting　equations　have　so　far　proved　intractable　　to　direct　analytical　solution.　The　discussion　about　solution　of　equations,　under　equations　satisfy　the　global　Lipschitz　　condition　and　the　linear　growth　condition,　a　large　number　of　papers　give　numerical　solution　of　equations.　Unfortunately　these　conditions　are　often　not　met　by　systems　of　interest[103-104]　In　Chapter　4,　we　shall　first　introduce　Euler　approximation　and　the　equation　hypotheses　conditions.　　We　discuss　numerical　solutions　of　multidimensionaljump-diffusion　processes　on　local　Lipschitz　condition.　In　particular,　we　show　that　　the　Euler　scheme　applied　to　system,　converges　to　the　analytic　solution　x(t),　　and　in　doing　so　bound　the　order　of　this　approximation.　　We　demonstrate　that　the　Euler　approximate　solution　will　converge　to　the　true　solution　x(t)　if　both　remain　within　a　compact　set.　　Both　of　x(t)　and　the　approximate　solution,　are　bounded　in　probability.　Several　examples　are　studied　to　illustrate　　　the　results.　　　So　far　there　are　not　many　results　on　theattractor　for　the　solutions　of　stochastic　differential　delay　equations.　The　main　aim　of　Chapter　5　is　to　establish　new　results　on　the　attractor　for　stochastic　functional　differential　equations　with　Markov　switching.　The　semimartingale　convergence　theorem　and　the　generalized　Ito　formula　will　play　their　important　roles　in　this　paper.　We　give　conditions　which　the　equation　exists　weak　attractor　and　strong　attractor.Modern　portfolio　selection　theorycombines　probability　and　optimization　techniques　to　model　the　behavior　of　investment　under　uncertainty.　Usually,　investors　have　an　aversion　to　risk　and　pursue　maximum　utility,　the　optimal　portfolio　should　be　an　efficient　portfolio　withmaximum　utility.　Clearly,　the　efficient　ortfolio　selection　problem　is　a　very　important　topic　for　modern　portfolio　theory.　In　Chapter　6,　an　optimal　investment　and　consumption　model　is　established　when　r(t),　b(t)　are　continuously　varying.　It　is　given　that　analytic　expression　of　optimal　investment　and　optimal　consumption　policy　to　maximize　total　xpected　discounted　utility　of　consumption.

Translated Keyword

[]

Research Interests

Classification

Corresponding authors email