Numerical Solution of Stochastic Differential Equations with Jumps in Finance

Transcription

1 Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen School of Finance and Economics and School of Mathematical Sciences University of Technology, Sydney Kloeden, P.E.&Pl, E.: Numerical Solution of Stochastic Differential Equations Springer, Applications of Mathematics 23 (1992,1995,1999). Pl, E.&Heath, D.: A Benchmark Approach to Quantitative Finance, Springer Finance (2010). Pl, E.&Bruti-Niberati, N.: Numerical Solution of SDEs with Jumps in Finance, Springer, Stochastic Modelling and Applied Probability 64 (2010).

2 Jump-Diffusion Multi-Factor Models Björk, Kabanov & Runggaldier (1997) Øksendal & Sulem (2005) Markovian explicit transition densities in special cases benchmark framework discrete time approximations suitable for simulation Markov chain approximations c Copyright E. Platen SDE Jump MC 1

3 Pathwise Approximations: scenario simulation of entire markets testing statistical techniques on simulated trajectories filtering hidden state variables Pl. & Runggaldier (2005, 2007) hedge simulation dynamic financial analysis extreme value simulation stress testing = higher order strong schemes predictor-corrector methods c Copyright E. Platen SDE Jump MC 2

4 Strong Convergence Applications: scenario analysis, filtering and hedge simulation strong order γ if ε s ( ) = E( XT Y N 2) K γ c Copyright E. Platen SDE Jump MC 3

5 Probability Approximations: derivative prices sensitivities expected utilities portfolio selection risk measures long term risk management = Monte Carlo simulation, higher order weak schemes, predictor-corrector, variance reduction, Quasi Monte Carlo, or Markov chain approximations, lattice methods c Copyright E. Platen SDE Jump MC 4

6 Weak Convergence Applications: derivative pricing, utilities, risk measures weak order β if ε w ( ) = E(g(X T )) E(g(Y N )) K β c Copyright E. Platen SDE Jump MC 5

7 Essential Requirements: parsimonious models long time horizons respect no-arbitrage in discrete time approximation numerically stable methods efficient methods for high-dimensional models higher order schemes c Copyright E. Platen SDE Jump MC 6

8 Continuous and Event Driven Risk Wiener processes W k, k {1,2,..., m} counting processes p k intensity h k jump martingaleq k dw m+k t = dq k t = ( dp k t hk t dt)( h k t ) 1 2 k {1,2,...,d m} W t = (W 1 t,...,wm t,q 1 t,...,qd m t ) c Copyright E. Platen SDE Jump MC 7

9 Primary Security Accounts Assumption 1 ds j t = S j t ( b j,k t a j t dt + d k=1 h k m t b j,k t dw k t ) k {m + 1,...,d}. Assumption 2 Generalized volatility matrixb t = [b j,k t ] d j,k=1 invertible. c Copyright E. Platen SDE Jump MC 8

10 market price of risk θ t = (θ 1 t,...,θd t ) = b 1 t [a t r t 1] primary security account ds j t = S j t ( r t dt + d k=1 b j,k t (θ k t dt + dwk t ) ) portfolio ds δ t = d j=0 δ j t ds j t c Copyright E. Platen SDE Jump MC 9

11 fraction π j δ,t = δj t S j t S δ t portfolio ds δ t = Sδ t { } r t dt + π δ,t b t(θ t dt + dw t ) c Copyright E. Platen SDE Jump MC 10

12 Assumption 3 h k m t > θ k t = numeraire portfolio (NP) exists generalized NP volatility c k t = θ k t for k {1,2,...,m} θ k t 1 θ k t (hk m t ) 1 2 for k {m + 1,...,d} NP fractions π δ,t = (π 1 δ,t,...,πd δ,t ) = ( c t b 1 t ) c Copyright E. Platen SDE Jump MC 11

13 Numeraire Portfolio - Benchmark ( ) ds δ t = S δ t r t dt + c t (θ tdt + dw t ) optimal growth rate m g δ t = r t d k=1 k=m+1 (θ k t )2 h k m t ln 1 + θ k t h k m t θ k t + θk t h k m t c Copyright E. Platen SDE Jump MC 12

14 benchmarked portfolio Ŝ δ t = Sδ t S δ t Theorem 4 Any nonnegative benchmarked portfolio Ŝ δ is an (A,P)-supermartingale, that is, 0 t T < = no strong arbitrage Ŝ δ t E t(ŝ δ T ) Pl. & Heath (2010) c Copyright E. Platen SDE Jump MC 13

15 Multi-Factor Models model under benchmark approach mainly: benchmarked primary security accounts j {0,1,...,d} Ŝ j t = Sj t S δ t supermartingales, often SDE driftless, (local martingales, sometimes martingales) c Copyright E. Platen SDE Jump MC 14

16 model savings account S 0 t = exp { t 0 } r s ds = NP S δ t = S0 t Ŝ 0 t = stock S j t = Ŝ j t S δ t model additionally dividend rates and foreign interest rates c Copyright E. Platen SDE Jump MC 15

17 Example benchmarked security: ( ) dŝ t = Ŝ t Vt dw S t + dq t squared volatility: Bates model dv t = ξ(η V t )dt + q V t dw V t 3 2 model d 1 V t = ξ ) (η 1Vt 1 dt + q dw V t V t c Copyright E. Platen SDE Jump MC 16

18 time Figure 1: Simulated benchmarked primary security accounts. c Copyright E. Platen SDE Jump MC 17

19 time Figure 2: Simulated primary security accounts. c Copyright E. Platen SDE Jump MC 18

20 4.5 GOP EWI time Figure 3: Naive Diversification: NP and EWI ford = 50. c Copyright E. Platen SDE Jump MC 19

21 Figure 4: Benchmarked primary security accounts MMM. c Copyright E. Platen SDE Jump MC 20

22 Figure 5: Primary security accounts under the MMM. c Copyright E. Platen SDE Jump MC 21

23 EWI GOP Figure 6: NP and EWI. c Copyright E. Platen SDE Jump MC 22

24 fair security benchmarked security martingale fair minimal replicating portfolio fair nonnegative portfolio S δ with S δ τ = H τ = minimal price real world pricing formula V Hτ (t) = S δ t E t ( Hτ S δ τ ) No need for equivalent risk neutral probability measure! c Copyright E. Platen SDE Jump MC 23

25 Fair Hedging benchmarked fair portfolio Ŝ δ t = E t = minimal price martingale representation ( Hτ S δ τ ) H τ S δ τ = E t ( Hτ S δ τ ) + d τ k=1 t x k H τ (s)dw k s + M H τ (t) M Hτ - martingale (when pooled vanishing) M Hτ and W k orthogonal Föllmer & Schweizer (1991), Du&Pl. (2011) c Copyright E. Platen SDE Jump MC 24

26 Numerical Solution of SDEs Kloeden & Pl. (1999) Milstein (1995) Kloeden, Pl. & Schurz (2003) Jäckel (2002) Glasserman (2004) Pl. & Bruti-Liberati (2010) major problem: propagation of errors during simulation c Copyright E. Platen SDE Jump MC 25

27 Simulation of SDEs with Jumps strong schemes (paths) exact simulation Taylor (Wagner-Pl. expansion) explicit derivative-free predictor-corrector implicit, balanced implicit weak schemes (probabilities) Taylor (Wagner-Pl. expansion) simplified explicit derivative-free predictor-corrector, implicit c Copyright E. Platen SDE Jump MC 26

28 Jump-Adapted Time Discretization t 0 t 1 t 2 t 3 = T regular τ 1 τ 2 jump times jump-adapted t 0 t 1 t 2 t 3 t 4 t 5 = T c Copyright E. Platen SDE Jump MC 27

29 intensity of jump process regular schemes = high intensity jump-adapted schemes = low intensity c Copyright E. Platen SDE Jump MC 28

30 SDE with Jumps dx t = a(t,x t )dt + b(t,x t )dw t + c(t,x t )dp t X 0 R d p t = N t : Poisson process, intensity λ < p t = N t i=1 (ξ i 1): compound Poisson, ξ i i.i.d r.v. Poisson random measure c Copyright E. Platen SDE Jump MC 29

31 time discretization t n = n discrete time approximation Y n+1 = Y n + a(y n ) + b(y n ) W n + c(y n ) p n c Copyright E. Platen SDE Jump MC 30

32 Literature on Strong Schemes with Jumps Pl (1982), Mikulevicius&Pl (1988) = γ {0.5, 1,...} Taylor schemes and jump-adapted Maghsoodi (1996, 1998) = strong schemes γ 1.5 Jacod & Protter (1998) = Euler scheme for semimartingales Gardoǹ (2004) = γ {0.5, 1,...} strong schemes Higham & Kloeden (2005) = implicit Euler scheme Bruti-Liberati & Pl (2007) = γ {0.5,1,...} explicit, implicit, derivative-free, predictor-corrector c Copyright E. Platen SDE Jump MC 31

33 Euler Scheme Euler scheme where Y n+1 = Y n + a(y n ) + b(y n ) W n + c(y n ) p n W n N(0, ) and p n = N tn+1 N tn Poiss(λ ) strong orderγ = 0.5 c Copyright E. Platen SDE Jump MC 32

34 Strong Taylor Scheme Wagner-Platen expansion (strong orderγ = 1.0) = Y n+1 = Y n + a(y n ) + b(y n ) W n + c(y n ) p n + b(y n )b (Y n )I (1,1) +b(y n )c (Y n )I (1, 1) + {b(y n + c(y n )) b(y n )}I ( 1,1) +{c(y n + c(y n )) c(y n )}I ( 1, 1) with I (1,1) = 1 {( W 2 n) 2 }, I ( 1, 1) = 1 2 {( p n) 2 p n } I (1, 1) = N(t n+1 ) i=n(t n )+1 W τ i p n W tn, I ( 1,1) = p n W n I (1, 1) simulation jump times τ i : W τi = I (1, 1) and I ( 1,1) Computational effort heavily dependent on intensity λ c Copyright E. Platen SDE Jump MC 33

35 Derivative-Free Strong Schemes avoid computation of derivatives order1.0 derivative-free strong scheme c Copyright E. Platen SDE Jump MC 34

36 implicit methods Talay (1982) Klauder & Petersen (1985) Milstein (1988) Hernandez & Spigler(1992, 1993) Saito & Mitsui(1993a, 1993b) Kloeden & Pl. (1992, 1999) Milstein, Pl. & Schurz (1998) Higham (2000) Alcock & Burrage (2006) solve algebraic equation c Copyright E. Platen SDE Jump MC 35

37 ad hoc attempts lead to explosions balanced implicit methods Milstein, Pl. & Schurz (1998) Alcock & Burrage (2006) c Copyright E. Platen SDE Jump MC 36

38 Implicit Strong Schemes wide stability regions implicit Euler scheme order1.0 implicit strong Taylor scheme c Copyright E. Platen SDE Jump MC 37

39 Predictor-Corrector Euler Scheme corrector Y n+1 = Y n + ( ) θā η (Ȳ n+1 ) + (1 θ)ā η (Y n ) n + ā η = a ηbb ( ) ηb(ȳ n+1 ) + (1 η)b(y n ) W n + p(t n+1 ) i=p(t n )+1 c(ξ i ) predictor Ȳ n+1 = Y n + a(y n ) n + b(y n ) W n + p(t n+1 ) i=p(t n )+1 c(ξ i ) θ, η [0, 1] degree of implicitness c Copyright E. Platen SDE Jump MC 38

40 Jump-Adapted Strong Approximations jump-adapted time discretisation jump times included in time discretisation jump-adapted Euler scheme and Y tn+1 = Y tn + a(y tn ) tn + b(y tn ) W tn Y tn+1 = Y tn+1 + c(y tn+1 ) p n strong orderγ = 0.5 c Copyright E. Platen SDE Jump MC 39

41 Merton SDE :µ = 0.05, σ = 0.2, ψ = 0.2, λ = 10, X 0 = 1, T = X T Figure 7: Plot of a jump-diffusion path. c Copyright E. Platen SDE Jump MC 40

42 Error T Figure 8: Plot of the strong error for Euler(red) and 1.0 Taylor(blue) scheme. c Copyright E. Platen SDE Jump MC 41

43 Merton SDE :µ = 0.05, σ = 0.1, λ = 1, X 0 = 1, T = Log 2 Error Euler EulerJA 1Taylor 1TaylorJA 15TaylorJA Log 2 dt Figure 9: Log-log plot of strong error versus time step size. c Copyright E. Platen SDE Jump MC 42

44 Literature on Weak Schemes with Jumps Mikulevicius & Pl (1991) = jump-adapted order β {1, 2...} weak schemes Liu & Li (2000) = order β {1,2...} weak Taylor, extrapolation and simplified schemes Kubilius&Pl (2002) and Glasserman & Merener (2003) = jump-adapted Euler with weaker assumptions on coefficients Bruti-Liberati&Pl (2006) = jump-adapted orderβ {1,2...} derivative-free, implicit and predictor-corrector schemes c Copyright E. Platen SDE Jump MC 43

45 Simplified Euler Scheme Euler scheme = weak order β = 1 simplified Euler scheme Y n+1 = Y n + a(y n ) + b(y n ) Ŵ n + c(y n )(ˆξ n 1) ˆp n if Ŵ n and ˆp n match the first 3 moments of W n and p n up to an O( 2 ) error = weak orderβ = 1 P( W n = ± ) = 1 2 c Copyright E. Platen SDE Jump MC 44

46 Jump-Adapted Taylor Approximations jump-adapted Euler scheme = weak orderβ = 1 jump-adapted order 2 weak Taylor scheme Y tn+1 = Y tn + a tn + b W tn + bb ) (( W tn ) 2 tn + a b Z tn (aa + 12 ) 2 a b 2 2 t n + (ab + 12 ) b b 2 { W tn tn Z tn } and Y tn+1 = Y tn+1 + c(y tn+1 ) p n weak orderβ = 2 (can be simplified and made derivative free) c Copyright E. Platen SDE Jump MC 45

47 Predictor-Corrector Schemes predictor-corrector = numerical stability and efficiency jump-adapted predictor-corrector Euler scheme Y tn+1 = Y tn { } a(ȳ tn+1 ) + a tn + b W tn with predictor Ȳ tn+1 = Y tn + a tn + b W tn weak orderβ = 1 c Copyright E. Platen SDE Jump MC 46

48 3 EulerJA 2 ImplEulerJA PredCorrJA Log 2 Error Log 2 dt Figure 10: Log-log plot of weak error versus time step size. c Copyright E. Platen SDE Jump MC 47

49 Regular Approximations higher order schemes : time, Wiener and Poisson multiple integrals random jump size difficult to handle higher order schemes: computational effort dependent on intensity c Copyright E. Platen SDE Jump MC 48

50 Numerical Stability roundoff and truncation errors propagation of errors numerical stability priority over higher order c Copyright E. Platen SDE Jump MC 49

51 specially designed test equations Hernandez & Spigler (1992, 1993) Milstein (1995) Kloeden & Pl. (1999) Saito & Mitsui(1993a, 1993b, 1996) Hofmann & Pl. (1994, 1996) Higham (2000) c Copyright E. Platen SDE Jump MC 50

52 linear test dynamics X t = X 0 exp {(1 α)λt + } α λ W t α,λ R = P ( ) lim X t = 0 t = 1 (1 α)λ < 0 c Copyright E. Platen SDE Jump MC 51

53 linear Itô SDE dx t = (1 32 ) α λx t dt + α λ X t dw t corresponding Stratonovich SDE dx t = (1 α)λx t dt + α λ X t dw t α = 0 no randomness α = 2 3 Itô SDE no drift = martingale α = 1 Stratonovich SDE no drift c Copyright E. Platen SDE Jump MC 52

54 Definition 5 Y = {Y t,t 0} is called asymptotically stable if P ( ) lim Y t = 0 t = 1. impact of perturbations declines asymptotically over time c Copyright E. Platen SDE Jump MC 53

55 stability region Γ those pairs(λ,α) (,0) [0,1) for which approximationy asymptotically stable c Copyright E. Platen SDE Jump MC 54

56 transfer function Y n+1 Y n = G n+1(λ,α) Y asymptotically stable E(ln(G n+1 (λ,α))) < 0 Higham (2000) c Copyright E. Platen SDE Jump MC 55

57 Euler scheme Y n+1 = Y n + a(y n ) + b(y n ) W n G n+1 (λ,α) = ( )λ 2 α + αλ W n W n N(0, ) c Copyright E. Platen SDE Jump MC 56

58 Figure 11: A-stability region for the Euler scheme c Copyright E. Platen SDE Jump MC 57

59 semi-drift-implicit predictor-corrector Euler method Y n+1 = Y n ( a(ȳn+1 ) + a(y n ) ) + b(y n ) W n Ȳ n+1 = Y n + a(y n ) + b(y n ) W n G n+1 (λ,α) = ( 1 + λ 1 3 ){ 2 α ( λ (1 32 ) α + αλ W n )} + αλ W n c Copyright E. Platen SDE Jump MC 58

60 Figure 12: -stability region for semi-drift-implicit predictor-corrector Euler method c Copyright E. Platen SDE Jump MC 59

61 Figure 13: A-stability region for the predictor-corrector Euler method with θ = 0 and η = 1 2 c Copyright E. Platen SDE Jump MC 60

62 Figure 14: A-stability region for the symmetric predictor-corrector Euler method c Copyright E. Platen SDE Jump MC 61

63 p-stability Pl. & Shi (2008) Definition 6 if For p > 0 a process Y = {Y t,t > 0} is called p-stable lim E( Y t p ) = 0. t Forα [0, 1 ) and λ < 0 test SDE isp-stable. 1+p/2 c Copyright E. Platen SDE Jump MC 62

64 Stability region those triplets (λ,α,p) for which Y isp-stable. c Copyright E. Platen SDE Jump MC 63

65 For λ < 0, α [0,1) and p > 0 Y p-stable E((G n+1 (λ,α)) p ) < 1 forp > 0 = E(ln(G n+1 (λ,α))) 1 p E((G n+1(λ,α)) p 1) < 0 = asymptotically stable c Copyright E. Platen SDE Jump MC 64

66 Λ p Α Figure 15: Stability region for the Euler scheme c Copyright E. Platen SDE Jump MC 65

67 Λ p Α Figure 16: Stability region for semi-drift-implicit predictor-corrector Euler method c Copyright E. Platen SDE Jump MC 66 1

68 Λ p Α Figure 17: Stability region for the predictor-corrector Euler method with θ = 0 and η = 1 2 c Copyright E. Platen SDE Jump MC 67 1

69 Stability of Some Implicit Methods semi-drift implicit Euler scheme Y n+1 = Y n (a(y n+1) + a(y n )) + b(y n ) W n full-drift implicit Euler scheme Y n+1 = Y n + a(y n+1 ) + b(y n ) W n solve algebraic equation c Copyright E. Platen SDE Jump MC 68

70 Λ p Α Figure 18: Stability region for semi-drift implicit Euler method c Copyright E. Platen SDE Jump MC 69

71 Λ p Α Figure 19: Stability region for full-drift implicit Euler method c Copyright E. Platen SDE Jump MC 70

72 balanced implicit Euler method Milstein, Pl. & Schurz (1998) Y n+1 = Y n + (1 32 ) α λy n + α λ Y n W n +c W n (Y n Y n+1 ) c Copyright E. Platen SDE Jump MC 71

73 Λ p Α Figure 20: Stability region for a balanced implicit Euler method c Copyright E. Platen SDE Jump MC 72

74 Λ p Α Figure 21: Stability region for the simplified symmetric Euler method c Copyright E. Platen SDE Jump MC 73

75 Λ p Α Figure 22: Stability region for the simplified symmetric implicit Euler Scheme c Copyright E. Platen SDE Jump MC 74 1

76 Λ p Α Figure 23: Stability region for the simplified fully implicit Euler Scheme c Copyright E. Platen SDE Jump MC 75

77 Variance Reduction via Integral Representations Heath & Platen (2002) The HP Variance Reduced Estimator SDE dx s,x t = a(t,x s,x t )dt + m j=1 b j (t,x s,x t )dw j t c Copyright E. Platen SDE Jump MC 76

78 L 0 f(t,x) = f(t,x) t + d i=1 a i (t,x) f(t,x) x i d m i,k=1 j=1 b i,j (t,x)b k,j (t,x) 2 f(t,x) x i x k c Copyright E. Platen SDE Jump MC 77

79 u(0,x) = E ( h(τ,x 0,x τ ) ) = E ( ū(τ,x 0,x τ ) ) = ū(0,x) + E = ū(0,x) + T 0 ( τ 0 E ) L 0 ū(t,x 0,x t )dt ( ) 1 {t<τ} L 0 ū(t,x 0,x t ) dt unbiased estimator for u(0,x) HP estimator Z τ = ū(0,x) + τ 0 L 0 ū(t,x 0,x t )dt c Copyright E. Platen SDE Jump MC 78

80 Figure 24: Simulated outcomes for the intrinsic value (S 0,x1 t K) +, t [0,T]. c Copyright E. Platen SDE Jump MC 79

81 Figure 25: Simulated outcomes for the estimator Z t, t [0,T]. c Copyright E. Platen SDE Jump MC 80

82 References Alcock, J. T. & K. Burrage (2006). A note on the balanced method. BIT Numerical Mathematics 46, Björk, T., Y. Kabanov, & W. J. Runggaldier (1997). Bond market structure in the presence of marked point processes. Math. Finance 7, Bruti-Liberati, N. & E. Platen (2006). On weak predictor-corrector schemes for jump-diffusion processes in finance. Technical report, University of Technology, Sydney. QFRC Research Paper 179. Bruti-Liberati, N. & E. Platen (2007). Strong approximations of stochastic differential equations with jumps. J. Comput. Appl. Math. 205(2), Föllmer, H. & M. Schweizer (1991). Hedging of contingent claims under incomplete information. In M. H. A. Davis and R. J. Elliott (Eds.), Applied Stochastic Analysis, Volume 5 of Stochastics Monogr., pp Gordon and Breach, London/New York. Gardoǹ, A. (2004). The order of approximations for solutions of Itô-type stochastic differential equations with jumps. Stochastic Anal. Appl. 22(3), Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering, Volume 53 of Appl. Math. Springer. Glasserman, P. & N. Merener (2003). Numerical solution of jump-diffusion LIBOR market models. Finance Stoch. 7(1), Heath, D. & E. Platen (2002). A variance reduction technique based on integral representations. Quant. Finance 2(5), c Copyright E. Platen SDE Jump MC 81

83 Hernandez, D. B. & R. Spigler (1992). A-stability of implicit Runge-Kutta methods for systems with additive noise. BIT 32, Hernandez, D. B. & R. Spigler (1993). Convergence and stability of implicit Runge-Kutta methods for systems with multiplicative noise. BIT 33, Higham, D. J. (2000). Mean-square and asymptotic stability of numerical methods for stochastic ordinary differential equations. SIAM J. Numer. Anal. 38, Higham, D. J. & P. E. Kloeden (2005). Numerical methods for nonlinear stochastic differential equations with jumps. Numer. Math. 110(1), Hofmann, N. & E. Platen (1994). Stability of weak numerical schemes for stochastic differential equations. Comput. Math. Appl. 28(10-12), Hofmann, N. & E. Platen (1996). Stability of superimplicit numerical methods for stochastic differential equations. Fields Inst. Commun. 9, Jäckel, P. (2002). Monte Carlo Methods in Finance. Wiley. Jacod, J. & P. Protter (1998). Asymptotic error distribution for the Euler method for stochastic differential equations. Ann. Probab. 26(1), Klauder, J. R. & W. P. Petersen (1985). Numerical integration of multiplicative-noise stochastic differential equations. SIAM J. Numer. Anal. 6, Kloeden, P. E. & E. Platen (1992). Higher order implicit strong numerical schemes for stochastic differential equations. J. Statist. Phys. 66(1/2), Kloeden, P. E. & E. Platen (1999). Numerical Solution of Stochastic Differential Equations, Volume 23 of Appl. Math. Springer. Third printing, (first edition (1992)). c Copyright E. Platen SDE Jump MC 82

84 Kloeden, P. E., E. Platen, & H. Schurz (2003). Numerical Solution of SDEs Through Computer Experiments. Universitext. Springer. Third corrected printing, (first edition (1994)). Kubilius, K. & E. Platen (2002). Rate of weak convergence of the Euler approximation for diffusion processes with jumps. Monte Carlo Methods Appl. 8(1), Liu, X. Q. & C. W. Li (2000). Weak approximation and extrapolations of stochastic differential equations with jumps. SIAM J. Numer. Anal. 37(6), Maghsoodi, Y. (1996). Mean-square efficient numerical solution of jump-diffusion stochastic differential equations. SANKHYA A 58(1), Maghsoodi, Y. (1998). Exact solutions and doubly efficient approximations of jump-diffusion Itô equations. Stochastic Anal. Appl. 16(6), Mikulevicius, R. & E. Platen (1988). Time discrete Taylor approximations for Ito processes with jump component. Math. Nachr. 138, Mikulevicius, R. & E. Platen (1991). Rate of convergence of the Euler approximation for diffusion processes. Math. Nachr. 151, Milstein, G. N. (1988). A theorem of the order of convergence of mean square approximations of systems of stochastic differential equations. Theory Probab. Appl. 32, Milstein, G. N. (1995). Numerical Integration of Stochastic Differential Equations. Mathematics and Its Applications. Kluwer. Milstein, G. N., E. Platen, & H. Schurz (1998). Balanced implicit methods for stiff stochastic systems. SIAM J. Numer. Anal. 35(3), Øksendal, B. & A. Sulem (2005). Applied stochastic control of jump-duffusions. Universitext. Springer. c Copyright E. Platen SDE Jump MC 83

85 Platen, E. (1982). An approximation method for a class of Itô processes with jump component. Liet. Mat. Rink. 22(2), Platen, E. & N. Bruti-Liberati (2010). Numerical Solution of SDEs with Jumps in Finance. Springer. in preparation. Platen, E. & K. Du (2011). Benchmarked risk minimization for jump diffusion markets. UTS Working Paper. Platen, E. & D. Heath (2010). A Benchmark Approach to Quantitative Finance. Springer Finance. Springer. Platen, E. & W. J. Runggaldier (2005). A benchmark approach to filtering in finance. Asia-Pacific Financial Markets 11(1), Platen, E. & W. J. Runggaldier (2007). A benchmark approach to portfolio optimization under partial information. Asia-Pacific Financial Markets 14(1-2), Platen, E. & L. Shi (2008). On the numerical stability of simulation methods for SDEs. Technical report, University of Technology, Sydney. QFRC Research Paper 234. Saito, Y. & T. Mitsui (1993a). Simulation of stochastic differential equations. Ann. Inst. Statist. Math. 45, Saito, Y. & T. Mitsui (1993b). T-stability of numerical schemes for stochastic differential equations. World Sci. Ser. Appl. Anal. 2, Saito, Y. & T. Mitsui (1996). Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33(6), Talay, D. (1982). Convergence for each trajectory of an approximation scheme of SDE. Computes Rendus c Copyright E. Platen SDE Jump MC 84

86 Acad. Sc. Paris, Séries I Math. 295(3), (in French). c Copyright E. Platen SDE Jump MC 85