Higher order weak approximations of stochastic differential equations with and without jumps

Higher order weak approximations of stochastic differential equations with and without jumps Hideyuki TANAKA Graduate School of Science and Engineering, Ritsumeikan University Rough Path Analysis and Related Topics, Nagoya University January 26, 212 1 / 37

Outline 1 Introduction 2 Approximations of SDEs without jumps Approximation of SDEs by random ODEs (Extended) Cubature on Wiener space Operator Splitting method as a Cubature 3 Approximations of SDEs with jumps I (finite jump intesity) Operator Splitting Jump-adapted approximation 4 Approximations of SDEs with jumps II (general Lévy process) Overview Small jump approximation How to deal with compound Poisson term 5 Further Research 2 / 37

Introduction Introduction 1 Introduction 2 Approximations of SDEs without jumps Approximation of SDEs by random ODEs (Extended) Cubature on Wiener space Operator Splitting method as a Cubature 3 Approximations of SDEs with jumps I (finite jump intesity) Operator Splitting Jump-adapted approximation 4 Approximations of SDEs with jumps II (general Lévy process) Overview Small jump approximation How to deal with compound Poisson term 5 Further Research 3 / 37

Introduction Topics 1 We want to know the value of the expectation E[f (X T )] where X t is a solution of a stochastic differential equation (SDE) 2 Applications: finance, control, filtering, physics, etc 3 However, we do not know the exact distribution of X t in general Therefore, we can not simulate X t by MonteCarlo 4 Goal: find a higher order approximation scheme X (n) t E[f (X T )] E[f (X (n) T )] = O(n k ) st (especially, k 2) 4 / 37

Introduction What are merits of higher order methods? Reducing computational cost (eg k = 1, n = 1 k = 2, n =several tens) Quasi Monte Carlo: It (sometimes) holds that E[f (X (n) T )] = g(y)p(dy) R α(n) g : R α(n) R p : measure on R α(n) If α(n) is not so large ( 1, 1?), QMC works well 5 / 37

Approximations of SDEs without jumps Approximations of SDEs without jumps 1 Introduction 2 Approximations of SDEs without jumps Approximation of SDEs by random ODEs (Extended) Cubature on Wiener space Operator Splitting method as a Cubature 3 Approximations of SDEs with jumps I (finite jump intesity) Operator Splitting Jump-adapted approximation 4 Approximations of SDEs with jumps II (general Lévy process) Overview Small jump approximation How to deal with compound Poisson term 5 Further Research 6 / 37

Approximations of SDEs without jumps Approximation of SDEs by random ODEs Setting Consider a Stratonovich SDE (R N -valued) X x t = x + d i= t V i (X x s ) dw i (s) (1) (W i ) 1 i d is a d-dimensional Brownian motion W (s) = s V i C b (RN ; R N ) Our purpose: Compute P t f (x) := E[f (X x t )] at time t = T, through time discretization methods 7 / 37

Approximations of SDEs without jumps Approximation of SDEs by random ODEs Random Ordinary Differential Equation Let t > be a fixed time-scaling parameter We want to construct an approximation process for r t X x r = x + d i= r V i (X x s)dω t i (s) (2) a driving random path ω has bounded variation paths ω W in some sense ω depends on fixed (small) time t; 8 / 37

Approximations of SDEs without jumps Approximation of SDEs by random ODEs Approximation Operator Approximation operator: For a scaling parameter t, Q t f (x) := E[f (X x t )] Time partition: π := { = t < t 1 < < t n = T } Operator Q t1 Q t2 t 1 Q tn t n 1 Markov chain whose transition follows from the random ODE (2) with scaling t = t j+1 t j Random ODE, defined in [, T ], driven by (ωπ) i Here ωπ(s)( i s T ) is defined for each time interval [t j, t j+1 ] independently: X t j,x r = x + d i= r ((ω tj+1 t j ) j=,,n 1 are independent) t j V i (X tj,x s )dω i t j+1 t j (s), r [t j, t j+1 ] 9 / 37

Approximations of SDEs without jumps Approximation of SDEs by random ODEs From local error to global error (Rough sketch): t j t j 1 T /n 1 Assume (P t Q t )g = O(t k+1 ) for smooth g 2 Then (P T QT n /n )f = (Pn T /n Qn T /n )f n 1 = Q j t/n (P T /n Q T /n )P (1 (j+1)/n)t f j= = n O(n (k+1) ) = O(n k ) 3 f : smooth & polynomial growth; see eg Talay & Tubaro(199), Tanaka & Kohatsu-Higa (29) 4 f : Lipschitz or more general, under Hörmander type condition; see Bally & Talay, Kohatsu-Higa, Kusuoka, etc 1 / 37

Approximations of SDEs without jumps Approximation of SDEs by random ODEs Stochastic Taylor expansion I By using Itô formula, (V i act as vector fields) f (X x t ) = f (x) + = f (x) + + i,j d d i= t d (V i f )(x) i= t s (V i f )(X x s ) dw i (s) t Notation for the order of convergence: For index (i 1,, i l ), i j =, 1,, d, dw i (s) (V j V i f )(X x r ) dw j (r) dw i (s) (i 1,, i l ) := l + #{j : i j = } 11 / 37

Approximations of SDEs without jumps Approximation of SDEs by random ODEs Stochastic Taylor expansion I (continued) Stochastic Taylor expansion: f (Xt x ) = (V i1 V il f )(x) (i 1,,i l ) m t s2 + R X m(t, f ) dw i 1 (s 1 ) dw i l (s l ) with E[ R X m(t, f ) ] C(T, f ) t (m+1)/2 A similar expansion holds for X x t ( dw is replaced by dω) 12 / 37

Higher order weak approximations of stochastic differential equations with and without jumps Approximations of SDEs without jumps (Extended) Cubature on Wiener space Cubature formulas (extended) 1 Definition: ω C,bv ([, 1]; R d+1 ) defines a cubature formula with degree m if for any index (i 1,, i l ) with (i 1,, i l ) m, [ 1 E s2 [ 1 = E s2 and for (i 1,, i l ) = m + 1 or m + 2, 1 ] dw i 1 (s 1 ) dw i l (s l ) ] dω i 1 (s 1 ) dω i l (s l ), s2 d ω i 1 (s 1 ) d ω i l (s l ) L 2 (Ω) 2 Lyons & Victoir (24) assume in addition, (We do not assume here) ω (s) = s ω has a discrete probability distribution on C,bv ([, 1]; R d+1 ) 13 / 37

Approximations of SDEs without jumps (Extended) Cubature on Wiener space Time Scaling 1 Note that t (i 1,,i l ) 1 t law = s2 2 Scaling for ω: For s t, s2 dw i 1 (s 1 ) dw i l (s l ) dw i 1 (s 1 ) dw i l (s l ) ω t (s) := tω (s/t) ω i t(s) := tω i t(s/t), 1 i d 14 / 37

Approximations of SDEs without jumps (Extended) Cubature on Wiener space Error estimates (Sketch) Assume that ω defines a cubature formula with degree m f C b Then E[f (Xt x )] E[f (X x t )] = E[Rm(t, X f )] E[Rm(t, X f )] = O(t (m+1)/2 ) Order k = (m 1)/2 scheme! For second order schemes, we need m = 5 15 / 37

Approximations of SDEs without jumps Operator Splitting method as a Cubature Stochastic Taylor expansion II Formal Taylor series of t e tl By taking expectations of stochastic Taylor expansions, E[f (X x t )] = f (x) + tlf (x) + t 2 2 L2 f (x) + t 3 3! L3 f (x) + where L := d i= L i ( the generator of P t ), L := V L i = 1 V 2 2 i, 1 i d Approximation of P t Approximation of exponential map 16 / 37

Approximations of SDEs without jumps Operator Splitting method as a Cubature Splitting of exponential maps on noncommutative algebra 1 Now we can not solve directly e tl = e t(l +L 1 + +L d ) 2 However, each e tl i may be solvable Approximation of e tl by {e tl,, e tl d } {L i } : noncommutative 17 / 37

Approximations of SDEs without jumps Operator Splitting method as a Cubature Second order method via splitting e t(l i +L j ) = I + t(l i + L j ) + t 2 2 (L i + L j ) 2 + e tl i e tl j = I + t(l i + L j ) + t2 2 (L2 i + L 2 j + 2L i L j ) + Socond-order methods: e t(l i +L j ) e t(l i +L j ) = 1 2 etl i e tl j + 1 2 etl j e tl i + O(t 3 ) = e t/2l i e tl j e t/2l i + O(t 3 ) The idea can be found in the works of Strang (196s) 18 / 37

Approximations of SDEs without jumps Operator Splitting method as a Cubature ODE and coordinate SDE What s e tl i? Notation: exp(v )x is the solution of at time t = 1 dz t (x) dt = V (z t (x)), z (x) = x exp(w i t V i)x is the solution of the SDE dx x i,t = x + t V i (X x i,s) dw i (s) R N So, e tl i is solvable in the sense of e tl i f (x) = E[f (X x i,t)] = E[f (exp(w i t V i )x)] e tl i e tl j two step flow exp(w j t V j) exp(w i t V i)x 19 / 37

Approximations of SDEs without jumps Operator Splitting method as a Cubature Ninomiya-Victoir scheme Ninomiya-Victoir scheme exp(t/2v ) exp( tz 1 V 1 ) exp( tz d V d ) exp(t/2v )x, X x if Λ = 1, t = exp(t/2v ) exp( tz d V d ) exp( tz 1 V 1 ) exp(t/2v )x, if Λ = 1 Z = (Z i ) 1 i d and Λ are independent Z N(, I d ) P(Λ = ±1) = 1/2 If the ODE has no closed-form solution, then we can use Runge-Kutta methods 2 / 37

Approximations of SDEs without jumps Operator Splitting method as a Cubature Ninomiya-Victoir scheme as an extended cubature on Wiener space Ninomiya-Victoir as cubature ω i (r) = r ηi (s)ds, r 1: η i (s) = (d + 1), if i =, s [, 1 2(d+1) ) [ 2d+1 2(d+1), 1), (d + 1)Z i, if i 1, Λ = 1, s [ 2d 2i+1 2(d+1), 2d 2i+3 2(d+1) ), (d + 1)Z i, if i 1, Λ = 1, s [ 2i 1 2(d+1), 2i+1 2(d+1) ),, otherwise ω : degree 5 formula (ω (s) s) Theorem Let f C 6 pol (RN ), and ω is defined by the above Then P T f (x) QT n const /nf (x) n 2 21 / 37

Approximations of SDEs without jumps Operator Splitting method as a Cubature Remark : Cubature formula for Gaussian measure Cubature for Gaussian measure with degree m: A discrete-valued random variable Z such that E[Z α ] = x α 1 R d ( 2π) d e x for α m 2 /2 dx N-V + Gaussian cubature with degree 5 Cubature on Wiener space with degree 5, and with finite number of paths (but ω (s) s) The number of paths is "2 (the number of points of Gaussian cubature)" 22 / 37

Approximations of SDEs with jumps I (finite jump intesity) Approximations of SDEs with jumps I (finite jump intesity) 1 Introduction 2 Approximations of SDEs without jumps Approximation of SDEs by random ODEs (Extended) Cubature on Wiener space Operator Splitting method as a Cubature 3 Approximations of SDEs with jumps I (finite jump intesity) Operator Splitting Jump-adapted approximation 4 Approximations of SDEs with jumps II (general Lévy process) Overview Small jump approximation How to deal with compound Poisson term 5 Further Research 23 / 37

Approximations of SDEs with jumps I (finite jump intesity) Setting: jump-type SDE Let J(t) be a compound Poisson process Consider the following SDE with jumps X x t = x + d i= t t V i (Xs x ) dw i (s) + h(xs )dj(s) x Question: How can we construct a higher order scheme? Point 1 : W and J are independent Point 2 : Number of jumps = finite 24 / 37

Approximations of SDEs with jumps I (finite jump intesity) Operator Splitting Approach (OS): Operator Splitting 1 Consider the equation t dxd+1,t x = x + h(xd+1,s )dj(s) x easy to simulate (if we can easily simulate jump size) 2 Generator: L d+1 f (x) := (f (x + h(x)y) f (x))ν(dy) R d where ν is a finite Lévy measure 25 / 37

Higher order weak approximations of stochastic differential equations with and without jumps Approximations of SDEs with jumps I (finite jump intesity) Operator Splitting Approach (OS): Operator Splitting (continued) 3 As in the continuous diffusion case, we can consider approximations of e t d+1 i= L i by {e tl,, e tl d, e tl d+1 } 4 We can construct a second order scheme via 1 e tl 2 etl d+1 + 1 2 etl d+1 e tl = e t d+1 i= L i + O(t 3 ) (This is first considered by Fujiwara (26, Master thesis)) 26 / 37

Approximations of SDEs with jumps I (finite jump intesity) Jump-adapted approximation Approach (JSAS): Jump (Size) Adapted Simulation Ref : Mordecki & Szepessy & Tempone & Zouraris (28) 1 First, simulate J(t): Jump time: < τ 1 < < τ k < T 2 Since τ j+1 τ j may be large, we use another partition { t j } so that {t i } := { t j } {τ j } satisfies t j+1 t j 3 (Jump adapted simulation) Continuous term approximation: X tj = Euler or cubature scheme in time interval [t j 1, t j ) for continuous part X t j 1,x t j Jump simulation: = x + d tj i= t j 1 V i (X t j 1,x s ) dw i (s) { X tj + h(x tj ) (jump size), if t j : jump time, X tj = X tj, otherwise 27 / 37

Approximations of SDEs with jumps I (finite jump intesity) Jump-adapted approximation Practical Problems Which is better, (OS) or (JSAS)? The best (adapted) choice of partition t 1 < < t n How to simulate by QMC (We need a restriction of the number of jumps) 28 / 37

Approximations of SDEs with jumps II (general Lévy process) Approximations of SDEs with jumps II (general Lévy process) 1 Introduction 2 Approximations of SDEs without jumps Approximation of SDEs by random ODEs (Extended) Cubature on Wiener space Operator Splitting method as a Cubature 3 Approximations of SDEs with jumps I (finite jump intesity) Operator Splitting Jump-adapted approximation 4 Approximations of SDEs with jumps II (general Lévy process) Overview Small jump approximation How to deal with compound Poisson term 5 Further Research 29 / 37

Approximations of SDEs with jumps II (general Lévy process) Overview Setting : Infinite Activity Lévy-driven SDE Genaral Lévy-driven SDE: X x t = x + d i= t t V i (Xs x ) dw i (s) + h(xs )dz x (s) (3) where Z (t) is a Lévy process without Brownian term, and Z t p>1 L p Generator: L = L + + L d + L d+1, L d+1 f (x) := f (x)h(x)b + (f (x + h(x)y) f (x) f (x)h(x)1 y 1 )ν(dy), R d where b R d, ν is a Lévy measure (Note: y 1 y 2 ν(dy) < ) 3 / 37

Approximations of SDEs with jumps II (general Lévy process) Overview Basic Strategy Extend the schemes (OS) & (JSAS) to the general case (3) 1 (OS): Tanaka & Kohatsu(29) Kohatsu & Ngo(211, submitted) 2 (JSAS): Kohatsu & Tankov(29) without Brownian term ie V i, 1 i d Approach (OS): Decompose L d+1 = L (1) d+1 + L(2) d+1 + L(3) d+1 where L (1) d+1 f (x) = b ϵ(hf )(x), L (2) d+1 f (x) = (f (x + h(x)y) f (x) f (x)h(x)y)ν(dy), L (3) d+1 f (x) = (f (x + h(x)y) f (x))ν(dy), y ϵ y >ϵ where ϵ < 1, b ϵ := b ϵ< y 1 yν(dy) 31 / 37

Approximations of SDEs with jumps II (general Lévy process) Small jump approximation Asmussen-Rosinski approximation for L (2) d+1 L (1) d+1 and L(3) d+1 respectively correspond to "drift" & "compound Poisson" term ( ) What is L (2) d+1? : Define Σ ϵ := y ϵ y iy j ν(dy) 1 i,j d ( ) L (2) d+1 = h(x)σ ϵh (x)d 2 f (x) + O y 3 ν(dy) y ϵ So we replace L (2) d+1 by a new "small diffusion"(itô form) term L (2) d+1 := h(x)σ ϵh (x)d 2 f (x) as an approximation If necessary for simulation, we can modify (L (1) (2) (2) d+1, L d+1 ) so that L d+1 becomes the generator of the SDE of Stratonovich form Control (ϵ, n) as y ϵ y 3 ν(dy) O(n k ) 32 / 37

Higher order weak approximations of stochastic differential equations with and without jumps Approximations of SDEs with jumps II (general Lévy process) How to deal with compound Poisson term The case y ν(dy) < + y 1 1 The number of jumps wrt L (3) d+1 goes to +, as ϵ To avoid this, we need some restriction for jumps 2 If we assume that y ν(dy) < +, then y 1 Note: b ϵ < + So we can use (OS) approximations for (L + L (1) d+1 ) : (drift) (2) L 1,, L d, L d+1 : (2d-diffusion) L (3) d+1 : (Jumps) 3 For L (3) d+1, we can construct an approximation process which has single or double jumps in (fixed) small time interval See Tanaka & Kohatsu, or Kohatsu & Ngo 33 / 37

Higher order weak approximations of stochastic differential equations with and without jumps Approximations of SDEs with jumps II (general Lévy process) How to deal with compound Poisson term The case y ν(dy) = + y 1 1 Kohatsu & Ngo discuss a case study where Z is a subordinated Brownian motion They construct an algorithm under the case y ν(dy) = +, but yν(dy) < + y 1 y 1 2 If we assume that y ν(dy) = +, then in general, y 1 Note: b ϵ = + We should not decompose (L (1) d+1 + L(3) d+1 ), since L(1) d+1 includes the truncation function for L (3) d+1 Consider (OS) approximations for (2) L 1,, L d, L d+1 : (2d-diffusion) L + (L (1) d+1 + L(3) d+1 ) : (ODE with Jumps) What can we do for (L (1) d+1 + L(3) d+1 )? 34 / 37

Further Research Further Research 1 Introduction 2 Approximations of SDEs without jumps Approximation of SDEs by random ODEs (Extended) Cubature on Wiener space Operator Splitting method as a Cubature 3 Approximations of SDEs with jumps I (finite jump intesity) Operator Splitting Jump-adapted approximation 4 Approximations of SDEs with jumps II (general Lévy process) Overview Small jump approximation How to deal with compound Poisson term 5 Further Research 35 / 37

Further Research Problems in time discretization 1 Find higher-order methods with degree more than 7 (d 2) (We have known only "existence"!) 2 High order approximation of Lévy driven SDEs How to deal with small jumps (which occurs infinitely many times) How to simulate Lévy processes whose Lévy measure has high singularity ( y ν(dy) = + ) y 1 3 Problems in computing conditional expectations E[ F t ]: Pricing American/Bermudan options Simulating forward-backward SDEs (FBSDEs) Recombination techniques: eg Chevance(1997), Lyons-Litterer(forthcoming, AAP), Tanaka(211, submitted) 36 / 37

Further Research Thank you for your attention 37 / 37