TaylorexpansionsforSODEs TaylorexpansionsforSPDEs AnewnumericalmethodforSPDEswithnon-additivenoise AnewnumericalmethodforSPDEswithadditivenoise Taylor Expansionsand Numerical Approximationsfor Stochastic Partial Differential Equations A.Jentzen JointworkswithP.E.KloedenandM.Röckner FacultyofMathematics BielefeldUniversity 12thAugust2010 A.Jentzen TaylorExpansionsforSPDEs TaylorexpansionsforSODEs TaylorexpansionsforSPDEs AnewnumericalmethodforSPDEswithnon-additivenoise AnewnumericalmethodforSPDEswithadditivenoise Content 1 TaylorexpansionsforSODEs 2 TaylorexpansionsforSPDEs 3 AnewnumericalmethodforSPDEswithnon-additivenoise 4 AnewnumericalmethodforSPDEswithadditivenoise A.Jentzen TaylorExpansionsforSPDEs TaylorexpansionsforSODEs TaylorexpansionsforSPDEs AnewnumericalmethodforSPDEswithnon-additivenoise AnewnumericalmethodforSPDEswithadditivenoise Content 1 TaylorexpansionsforSODEs 2 TaylorexpansionsforSPDEs 3 AnewnumericalmethodforSPDEswithnon-additivenoise 4 AnewnumericalmethodforSPDEswithadditivenoise A.Jentzen TaylorExpansionsforSPDEs TaylorexpansionsforSODEs TaylorexpansionsforSPDEs AnewnumericalmethodforSPDEswithnon-additivenoise AnewnumericalmethodforSPDEswithadditivenoise LetT > 0andlet(Ω,F,P)beaprobabilityspace.Letf,g : R→ Rbe smoothfunctionsandlet(W) beascalarBrownianmotion. t t∈[0,T] ConsidertheSODE: dX = f(X )dt +g(X )dW, t t t t whichisunderstoodas t t X = X + f(X )ds+ g(X )dW t 0 s s s Z0 Z0 P-a.s.forallt ∈ [0,T].ApplyingItˆo’sformulatotheintegrandsaboveyields t t s X ≈ X +f(X )·t +g(X )· dW +g′(X )g(X )· dW dW t 0 0 0 s 0 0 u s Z0 Z0 Z0 1 = X +f(X )·t +g(X )·W + ·g′(X )g(X )· (W)2−t P-a.s. 0 0 0 t 0 0 t 2 (cid:0) (cid:1) Milstein’sapproximation(1974). A.Jentzen TaylorExpansionsforSPDEs TaylorexpansionsforSODEs TaylorexpansionsforSPDEs AnewnumericalmethodforSPDEswithnon-additivenoise AnewnumericalmethodforSPDEswithadditivenoise LetT > 0andlet(Ω,F,P)beaprobabilityspace.Letf,g : R→ Rbe smoothfunctionsandlet(W) beascalarBrownianmotion. t t∈[0,T] ConsidertheSODE: dX = f(X )dt +g(X )dW, t t t t whichisunderstoodas t t X = X + f(X )ds+ g(X )dW t 0 s s s Z0 Z0 P-a.s.forallt ∈ [0,T].ApplyingItˆo’sformulatotheintegrandsaboveyields t t s X ≈ X +f(X )·t +g(X )· dW +g′(X )g(X )· dW dW t 0 0 0 s 0 0 u s Z0 Z0 Z0 1 = X +f(X )·t +g(X )·W + ·g′(X )g(X )· (W)2−t P-a.s. 0 0 0 t 0 0 t 2 (cid:0) (cid:1) Milstein’sapproximation(1974). A.Jentzen TaylorExpansionsforSPDEs TaylorexpansionsforSODEs TaylorexpansionsforSPDEs AnewnumericalmethodforSPDEswithnon-additivenoise AnewnumericalmethodforSPDEswithadditivenoise LetT > 0andlet(Ω,F,P)beaprobabilityspace.Letf,g : R→ Rbe smoothfunctionsandlet(W) beascalarBrownianmotion. t t∈[0,T] ConsidertheSODE: dX = f(X )dt +g(X )dW, t t t t whichisunderstoodas t t X = X + f(X )ds+ g(X )dW t 0 s s s Z0 Z0 P-a.s.forallt ∈ [0,T].ApplyingItˆo’sformulatotheintegrandsaboveyields t t s X ≈ X +f(X )·t +g(X )· dW +g′(X )g(X )· dW dW t 0 0 0 s 0 0 u s Z0 Z0 Z0 1 = X +f(X )·t +g(X )·W + ·g′(X )g(X )· (W)2−t P-a.s. 0 0 0 t 0 0 t 2 (cid:0) (cid:1) Milstein’sapproximation(1974). A.Jentzen TaylorExpansionsforSPDEs TaylorexpansionsforSODEs TaylorexpansionsforSPDEs AnewnumericalmethodforSPDEswithnon-additivenoise AnewnumericalmethodforSPDEswithadditivenoise LetT > 0andlet(Ω,F,P)beaprobabilityspace.Letf,g : R→ Rbe smoothfunctionsandlet(W) beascalarBrownianmotion. t t∈[0,T] ConsidertheSODE: dX = f(X )dt +g(X )dW, t t t t whichisunderstoodas t t X = X + f(X )ds+ g(X )dW t 0 s s s Z0 Z0 P-a.s.forallt ∈ [0,T].ApplyingItˆo’sformulatotheintegrandsaboveyields t t s X ≈ X +f(X )·t +g(X )· dW +g′(X )g(X )· dW dW t 0 0 0 s 0 0 u s Z0 Z0 Z0 1 = X +f(X )·t +g(X )·W + ·g′(X )g(X )· (W)2−t P-a.s. 0 0 0 t 0 0 t 2 (cid:0) (cid:1) Milstein’sapproximation(1974). A.Jentzen TaylorExpansionsforSPDEs TaylorexpansionsforSODEs TaylorexpansionsforSPDEs AnewnumericalmethodforSPDEswithnon-additivenoise AnewnumericalmethodforSPDEswithadditivenoise LetT > 0andlet(Ω,F,P)beaprobabilityspace.Letf,g : R→ Rbe smoothfunctionsandlet(W) beascalarBrownianmotion. t t∈[0,T] ConsidertheSODE: dX = f(X )dt +g(X )dW, t t t t whichisunderstoodas t t X = X + f(X )ds+ g(X )dW t 0 s s s Z0 Z0 P-a.s.forallt ∈ [0,T].ApplyingItˆo’sformulatotheintegrandsaboveyields t t s X ≈ X +f(X )·t +g(X )· dW +g′(X )g(X )· dW dW t 0 0 0 s 0 0 u s Z0 Z0 Z0 1 = X +f(X )·t +g(X )·W + ·g′(X )g(X )· (W)2−t P-a.s. 0 0 0 t 0 0 t 2 (cid:0) (cid:1) Milstein’sapproximation(1974). A.Jentzen TaylorExpansionsforSPDEs TaylorexpansionsforSODEs TaylorexpansionsforSPDEs AnewnumericalmethodforSPDEswithnon-additivenoise AnewnumericalmethodforSPDEswithadditivenoise LetT > 0andlet(Ω,F,P)beaprobabilityspace.Letf,g : R→ Rbe smoothfunctionsandlet(W) beascalarBrownianmotion. t t∈[0,T] ConsidertheSODE: dX = f(X )dt +g(X )dW, t t t t whichisunderstoodas t t X = X + f(X )ds+ g(X )dW t 0 s s s Z0 Z0 P-a.s.forallt ∈ [0,T].ApplyingItˆo’sformulatotheintegrandsaboveyields t t s X ≈ X +f(X )·t +g(X )· dW +g′(X )g(X )· dW dW t 0 0 0 s 0 0 u s Z0 Z0 Z0 1 = X +f(X )·t +g(X )·W + ·g′(X )g(X )· (W)2−t P-a.s. 0 0 0 t 0 0 t 2 (cid:0) (cid:1) Milstein’sapproximation(1974). A.Jentzen TaylorExpansionsforSPDEs TaylorexpansionsforSODEs TaylorexpansionsforSPDEs AnewnumericalmethodforSPDEswithnon-additivenoise AnewnumericalmethodforSPDEswithadditivenoise LetT > 0andlet(Ω,F,P)beaprobabilityspace.Letf,g : R→ Rbe smoothfunctionsandlet(W) beascalarBrownianmotion. t t∈[0,T] ConsidertheSODE: dX = f(X )dt +g(X )dW, t t t t whichisunderstoodas t t X = X + f(X )ds+ g(X )dW t 0 s s s Z0 Z0 P-a.s.forallt ∈ [0,T].ApplyingItˆo’sformulatotheintegrandsaboveyields t t s X ≈ X +f(X )·t +g(X )· dW +g′(X )g(X )· dW dW t 0 0 0 s 0 0 u s Z0 Z0 Z0 1 = X +f(X )·t +g(X )·W + ·g′(X )g(X )· (W)2−t P-a.s. 0 0 0 t 0 0 t 2 (cid:0) (cid:1) Milstein’sapproximation(1974). A.Jentzen TaylorExpansionsforSPDEs