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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].ApplyingItô’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].ApplyingItô’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].ApplyingItô’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].ApplyingItô’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].ApplyingItô’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].ApplyingItô’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].ApplyingItô’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

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Joint works with P. E. Kloeden and M. Röckner. Faculty of Mathematics. Bielefeld University. 12th August 2010. A. Jentzen. Taylor Expansions for SPDEs

Taylor Expansions and Numerical Approximations for Stochastic Partial Differential Equations PDF

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Joint works with P. E. Kloeden and M. Röckner. Faculty of Mathematics. Bielefeld University. 12th August 2010. A. Jentzen. Taylor Expansions for SPDEs

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