Itô’s theory of stochastic integration was originally motivated as a direct method to construct diffusion processes (as subclass of Markov processes) as solution of stochastic differential equations [35] . As in [27] Itô integral is defined as a sum

∫ 0 T   X ( t ) d B ( t ) = ∑ i = 0 n − 1     C i ( B ( t i + 1 ) − B ( t i ) ) . (2)

Theorem 3.1. (Properties of stochastic integrals, [27] ) Let X ( t ) be a regular

adapted such that with probability one ∫ 0 T X 2 ( t ) d t < ∞ . Then Itô integral ∫ 0 T   X ( t ) d B ( t ) is defined and has the following properties.

1) Linearity. If Itô integrals of X ( t ) and Y ( t ) are defined and α and β are some constants then

∫ 0 T ( α X ( t ) + β Y ( t ) ) d B ( t ) = α ∫ 0 T   X ( t ) d B ( t ) + β ∫ 0 T   Y ( t ) d B (t)

2) ∫ 0 T X ( t ) I ( a , b ] ( t ) d B ( t ) = ∫ a b   X ( t ) d B ( t ) . The following two properties hold when the process satisfies an additional assumption

∫ 0 T E ( X 2 ( t ) ) d t < ∞ . (3)

3) Zero mean property. If condition 3 holds then E ( ∫ 0 T X ( t ) d B ( t ) ) = 0 .

4) Isometry property. If condition 3 holds. Then

E ( ∫ 0 T X ( t ) d B ( t ) ) 2 = ∫ 0 T E ( X 2 ( t ) ) d B (t)

5) Generalized Itô Isometry [31] . For f , g ∈ L a 2 d ( Ω , L 2 ( [ 0, T ] ) ) , we have

E [ ∫ 0 t     f ( s ) d W ( s ) ∫ 0 t     g ( s ) d W ( s ) ] = ∫ 0 t     E [ f ( s ) g ( s ) ] d t

Corollary 3.1.1. If X is a continuous adapted process then the Itô integral

∫ 0 T X ( t ) d B ( t ) exists. In particular, ∫ 0 T f ( B ( t ) ) d B ( t ) where f is a continuous function on R is well-defined.

A consequence of the isometry property is the expectation of the product of two Itô integrals.

Theorem 3.2. Let X ( t ) and Y ( t ) be regular adapted processes, such that ∫ 0 T   X ( t ) 2 d t < ∞ and ∫ 0 T   Y ( t ) 2 < ∞ . Then

E ( ∫ 0 T   X ( t ) d B ( t ) ∫ 0 T   Y ( t ) d B ( t ) ) = ∫ 0 T   E ( X ( t ) Y ( t ) ) d t

We denote by ℝ m n all real-valued m × n matrices and by

W ( t ) = ( W 1 ( t ) , ⋯ , W n ( t ) ) ′ , t ≥ 0.

Let [ a , b ] ∈ [ 0, ∞ [ and we put

C W ( [ a , b ] ) = { f : [ a , b ] × Ω → ℝ m n | ∀ 1 ≤ i ≤ m , ∀ 1 ≤ j ≤ n : f i j ∈ C W j ( [ a , b ] ) } ,

C I W ( [ a , b ] ) = { f : [ a , b ] × Ω → ℝ m n | ∀ 1 ≤ i ≤ m , ∀ 1 ≤ j ≤ n : f i j ∈ C I W j ( [ a , b ] ) }

and C I ( [ a , b ] ) respectively.

Definition 3.2.1. [37] If f : [ a , b ] × Ω → ℝ m n belongs to C I W ( [ a , b ] ) , then the stochastic integral with respect to W is the m-dimensional vector defined by

∫ a b   f ( t ) d W ( t ) = ( ∑ j = 1 n ∫ a b   f i j ( t ) d W j ( t ) ) ′ 1 ≤ i ≤ m (4)

where each of the integrals on the right-hand side is defined in the sense of Itô.

As in [38] the Itô formula for multidimensional Itô processes is defined in following way. If

X t i = X 0 i + ∫ 0 t     K s i d s + ∑ l = 1 k ∫ 0 t   H s i l d B s l ,

t ∈ [ 0, T ] , i = 1 , ⋯ , m , are Itô processes and F ∈ C 2 ( [ 0, T ] × ℝ m ) , then

F ( T , X T ) − F ( 0 , X 0 ) = ∑ i = 1 m ∫ 0 T ∂ F ∂ x i ( t , X t ) d X t i + ∑ 0 T ∂ F ∂ t ( t , X t ) + 1 2 ∑ i , j = 1 m ∫ 0 T ∂ 2 F ∂ x i ∂ x j ( t , X t ) d 〈 X i , X j 〉 t (5)

= ∑ i = 1 m ∑ l = 1 k ∫ 0 T ∂ F ∂ x i ( t , X t ) H t i l d B t l + ∑ i = 1 m ∫ 0 T ∂ F ∂ x i ( t , X t ) K t i l d t       + ∫ 0 T ∂ F ∂ t ( t , X t ) d t + 1 2 ∑ i = 1 m ∑ l = 1 k ∫ 0 T ∂ 2 F ∂ x i ∂ x j ( t , X t ) H t i l H t j l d t . (6)

Morever, if F ∈ C 2,3 ( [ 0, T ] × ℝ m ) , then this formula can be written in term of the Stratonovich integral [38]

F ( T , X T ) − F ( 0 , X 0 ) = ∑ i = 1 m ∫ 0 T ∂ F ∂ x i ( t , X t ) ∘ d X t i + ∫ 0 T ∂ F ∂ t ( t , X t ) d t (7)

= ∑ i = 1 m ∑ l = 1 k ∫ 0 T ∂ F ∂ x i ( t , X t ) H t i l ∘ d B t + ∑ i = 1 m ∫ 0 T ∂ F ∂ x i ( t , X t ) K t i l d t + ∫ 0 T ∂ F ∂ t ( t , X t ) d t (8)

Definition 3.2.2. [35] Let X t and Y t be Itô processes. The Stratonovich integral of X t with respect to Y t is defined by

∫ a b   X t ∘ d Y t = ∫ a b X t d Y t + 1 2 ∫ a b ( d X t ) ( d Y t ) , (9)

or equivalently in the stochastic differential form

X t ∘ d Y t = X t d Y t + 1 2 ( d X t ) ( d Y t ) . (10)

Theorem 3.3. [35] Let f ( t , x ) be a continuous function with continuous partial derivatives ∂ F ∂ t , ∂ f ∂ t , and ∂ f ∂ x . Then

∫ a b f ( t , B ( t ) ) ∘ d B t = F ( t , B ( t ) ) | a b − ∫ a b ∂ B ∂ t ( t , B ( t ) ) ∘ d t . (11)

In particular, when the function f does not depend on t, we have

∫ a b   f ( B ( t ) ) ∘ d B t = F ( B ( t ) ) | a b . (12)

Theorem 3.4. [35] Let f ( t , x ) be a continuous function with continuous partial derivatives ∂ f ∂ t , ∂ f ∂ x , and ∂ 2 f ∂ x 2 . Then

∫ a b   X t ∘ d Y t = lim ‖ Δ ‖ → 0 ∑ i = 1 n   f ( t i * , 1 2 ( B ( t i − 1 ) + B ( t i ) ) ) ( B ( t i ) − B ( t i − 1 ) ) (13)

= lim ‖ Δ ‖ → 0 ∑ i = 1 n   f ( t i * , B ( t i − 1 + t i 2 ) ) ( B ( t i ) − B ( t i − 1 ) ) , (14)

in probability, where t i − 1 ≤ t i * ≤ t i , Δ = { t 0 , t 1 , ⋯ , t n − 1 , t n } is a partition of the finite interval [ a , b ] and ‖ Δ ‖ = max 1 ≤ i ≤ n ( t i − t i − n ) .

In [39] , the multidimensional Stratonovich integrals S m ( f ) can be expressed by the following formula using Itô integrals

S m ( f ) = ∑ 2 k ≤ m ! 2 k k ! ( m − 2 k ) I m − 2 k ( T r k f ) . (15)

where T r denoted the iterated traces that are defined formally starting with

T r f ( s 1 , ⋯ , s m − 2 ) = ∫ f ( s 1 , ⋯ , s m − 2 , s ) d s .

Another approach to formula (15) using Hida’s theory of white noise. Working on ℝ m instead of ℝ + m and assuming that f is a test-function, the integral S m ( f ) may indead be rewritten as

∫ f ( s 1 , ⋯ , s m ) X ˙ s 1 ( w ) ⋯ X ˙ s m ( w ) d s 1 ⋯ d s n = 〈 f , X ˙ ⊗ n 〉

where the derivative of Brownian motion is understood in the distribution sense. In the sense of Hu and Meyer [39] , a Stratonovich integral is given in rigorous form as

S ( f ) = ∑ m 1 m ! ∫ [ S )     f m ( s 1 , ⋯ , s m ) d X s 1 ( w ) ⋯ d X s m ( w ) (16)

where f is a finite sequence of coefficients f m ∈ L s 2 (ℝm)

and n ! = n × ( n − 1 ) × ⋯ × 1 .

The Skorohod integral was introduced for the first time by A. Skorohod in 1975 as an extension of the Itô integral to non-adapted processes and is the adjoint of the Malliavin derivative which is fundamentals to the stochastic calculus of variations [40] [41] .

Definition 3.4.1. [40] Let u ( t ) , t ∈ [ 0, T ] , be a measurable stochastic process such that for all t ∈ [ 0, T ] the random variable u ( t ) is F T -measurable and E [ u 2 ( t ) ] < ∞ . Let its Wiener-Itô chaos expansion be

u ( t ) = ∑ n = 0 ∞     I n ( f n , t ) = ∑ n = 0 ∞     I n ( f n ( . , t ) ) . (17)

Then we define the Skorohod integral of u ( t ) by

δ ( u ) : = ∫ 0 T   u ( t ) δ W ( t ) : = ∑ n = 0 ∞   I n + 1 ( f ˜ n ) , (18)

where convergent in L 2 ( P ) . Here f ˜ n , n = 1 , 2 , ⋯ are the symmetric functions derived from f n ( ., t ) , n = 1,2, ⋯ . We say that u is Skorohod integrable, and we write u ∈ D o m ( δ ) if the series in (18) converges in L 2 ( P ) .

The Itô integral and others are based in a fundamental hypothesis of causal relationship. Shigeyoshi Ogawa [42] defined this following noncausal integral that is so-called Ogawa integral

∫ 0 T f ( t ) ∗ d W ( t ) = ∑ ∫ 0 t     f ( s ) m i ( s ) d s ∫ 0 t     m i ( s ) d W ( s ) ,     t ∈ [ 0, T ] (19)

where { m i ( t ) } is the complete orthnormal system on L 2 ( [ 0, T ] ) . Nualart and Zakai [43] proved that the Ogawa integral is equivalent to the Stratonovich integral of the Ogawa integral exists with the Stratonovich integral defined [31] as

∫ 0 t     f ( t ) ∘ d W ( t ) = lim | Π n | → 0 ∑ i = 1 n 1 t i + 1 − t i ∫ t i t i + 1 f ( s ) d s ( W ( t i + 1 ) − W ( t i ) ) . (20)

Here Π n = { 0 = t 1 < t 2 < ⋯ < t n − 1 < t n = t } is an arbitrary partition of the interval [ 0, t ] . The Ogawa integral coincides with the Stratonovich integral defined at the midpoints [31]

∫ 0 t   f ( t ) ∘ d W ( t ) = lim | Π n | → 0 ∑ i = 1 n     f ( t i + t i + 1 2 ) ( W ( t i + 1 ) − W ( t i ) ) . (21)

Here Π n = { 0 = t 0 < t 1 < t 2 < ⋯ < t n − 1 < t n = t } is an arbitrary partition of the interval [ 0, t ] .

Let X ( t ) be a diffusion in n dimensions described by the multi-dimensional stochastic differential equation

d X ( t ) = Φ ( X ( t ) , t ) d t + Ψ ( X ( t ) , t ) d B ( t ) , (22)

where Ψ is n × d matrix valued function, B is d-dimensional Brownian motion and and X and Φ are vector n-dimensional vector valued functions. The vector Φ ( X , t ) and the matrix Ψ ( X , t ) are the coefficients of the stochastic differential equation.

Theorem 4.1. (Unique and Existence of Solution). If the coefficients are locally Lipschitz in X with a constant independent of t, that is, for every N, there is a constant K depending only on T and N such that for all | x | , | y | ≤ N and all 0 ≤ t ≤ T ,

| Φ ( x , t ) − Φ ( y , t ) | + | Ψ ( x , t ) − Ψ ( y , t ) | ≤ K | x − y | , (23)

for any given X ( 0 ) the strong solution to stochastic differentional equation 26 is unique. If in addition to condition 23 the linear growth condition holds

| Φ ( x , t ) | + | Ψ ( x , t ) | ≤ K τ ( 1 + | x | ) , (24)

X ( 0 ) is independent of B, and E | X ( 0 ) | 2 < ∞ , then the strong solution exists and is unique on [ 0, T ] , moreover,

E ( sup | X ( t ) | 2 ) < C ( 1 + E | X ( 0 ) | 2 ) ,

where constant C depends only on K and T.

The following theorem gives the solution of stochastic differential equations as Markov processes.

Theorem 4.2. [1] (The solution of SDEs as Markov processes) If Equation (26) satisfies the conditions of the existence and uniqueness theorem 4.1, the solution X t of the equation for arbitrary initial values is a Markov process on the interval [ t 0 , T ] whose initial probability distribution at the instant to is the distribution of C and whose transition probabilities are given by

P ( s , x t , B ) = P ( X t ∈ B | X s = x ) = P ( X t ( s , x ) ∈ B ) (25)

where X t ( s , x ) is the solution of equation.

Theorem 4.3. [1] (The solution of SDEs as Diffusion processes). The condition of the existence and uniqueness Theorem 4.1 are satisfied for the SDE

d X ( t ) = Φ ( X ( t ) , t ) d t + Ψ ( X ( t ) , t ) d B ( t ) , X t 0 = C , t 0 ≤ t ≤ T , (26)

where X t ∈ R d , Φ ( t , x ) ∈ R d , B ∈ R m and Ψ ( t , x ) is a d × m matrix. If in addition, the functions Φ and Ψ are continuous with respect to t, the solution X t is a d-dimensional diffusion process on [ t 0 , T ] with drift vector and diffusion matrix Π ( t , x ) = Ψ ( t , x ) Ψ ( t , x ) ′ . In particular, the solution of an autonomous SDE is always a homogeneous diffusion process on [ t 0 , ∞ ) .

Consider the Itô Stochastic Partial Differential Equation of the form as mentioned in [52]

d X t = [ A X t + F ( X t ) ] d t + B ( X t ) d W t ,     X t | ∂ D = 0, X = ξ (27)

for t ≥ 0 , where W t , is an infinite dimensional Wiener process of the

W t ( x , w ) = ∑ j = 1 ∞     C j W t j ( w ) ϕ j ( x ) ,   t ≥ 0,   x ∈ D (28)

with independent scalar Wiener processes W t j , j ∈ ℕ . Here the family ϕ j , j ∈ ℕ , is an orthonormal basis in, e.g., L 2 ( D , ℝ ) .

Assumptions: For uniqueness and existence of solution of this SPDE the following assumptions hold. A1) Linear operator A. Let L be a finite or countable set. In addition, let ( λ i ) i ∈ L be a family of real numbers with i n f i ∈ L > − ∞ and let ( ϑ i ) i ∈ L be an orthonormal basis of H. The linear operator A : D ( A ) → H is given by A υ = ∑ i = L − λ 〈 ϑ i , υ 〉 ϑ i for all υ ∈ D ( A ) with

D ( A ) = { ∑ i = L | λ | 2 | 〈 ϑ i , υ 〉 | 2 } ∈ H .

A2) Drift term F. Let α , δ ∈ ℝ be real numbers with δ − α < 1 and let F : H δ → H α be a globally Lipschitz continuous mapping.

A3) Diffusion term B.Let α , δ ∈ ℝ be real numbers with δ − β < 1 2 and let F : H δ → H S ( υ 0 , H β ) be a globally Lipschitz continuous mapping.

A4) Initial value ξ : Let γ ∈ [ δ , min ( α + 1 , β + 1 / 2 ) ] and p ∈ [ 2, ∞ ) be real numbers and let ξ : Ω → H γ be an F 0 / B ( H γ ) -measurable mapping with

E [ ‖ ξ ‖ H γ p ] < ∞ .

The literature contains many existence and uniqueness theorems for mild solutions of SPDEs. Theorem below provides an existence, uniqueness, and regularity result for solutions of SPDEs with globally Lipschitz continuous coefficients in the Equation (27).

Theorem 4.4. [52] Assume that the Assumptions A1)-A4) are fulfilled. Then there exists a unique predictable stochastic process X : [ 0, T ] × Ω → H γ

satisfying sup t ∈ [ 0 , T ] E [ ‖ ξ ‖ H γ p ] < ∞ and

X t = e A t ξ + ∫ 0 t     e A ( t − s ) F ( X s ) d s + ∫ 0 t     e A ( t − s ) B ( X s ) d W s (29)

ℙ − a . s . for all t ∈ [ 0, T ] . In addition,

X ∈ ∩ r ∈ ( − ∞ , γ ] C min ( γ − r , 1 / 2 ) ( [ 0 , T ] , L p ( Ω , H r ) ) .

Here we assume that the Assumptions that X : [ 0, T ] × Ω → H is a predictable stochastic process, which satisfies 27. Let t 0 ∈ [ 0, T ) . Then the solution process X also satisfies

X t = e A ( t − t 0 ) X t 0 + ∫ t 0 t     e A ( t − s ) F ( X s ) d s + ∫ t 0 t     e A ( t − s ) B ( X s ) d W s , ℙ − a . s . (30)

for every t ∈ [ t 0 ∈ T ] .

Proposition 2 Let assumption A1)-A4) be satisfied and let γ ∈ ( 0,1 ) be given by Assumption A3. Then there is an up-to-modification unique predictable stochastic process X : [ 0, T ] × Ω → D ( ( κ − A ) γ ) with

sup 0 ≤ t ≤ T E ‖ ( κ − A ) γ A t ‖ H p < ∞

for p ∈ [ 0, ∞ ) and with

P [ X t = e A ( t − t 0 ) X 0 + ∫ 0 t     e A ( t − s ) F ( X s ) d s + ∫ 0 t     e A ( t − s ) B ( X s ) d W s ] = 1 , (31)

for all t ∈ [ 0, T ) . Moreover, X is the unique mild solution of the SPDE 27 in the sense of Equation (31).

In real world, some phenomena or economic policy decisions are governed under uncertainty with jumps. Therefore, stochastic differential equation with jumps modeling can be considered as a useful econometric approach [53] . Consider a one-dimensional SDE, d = 1, in the form

d X t = a ( t , X t ) d t + b ( t , X t ) d W t + ∫ ε     c ( t , X t − , v ) p φ ( d v , d t ) (32)

for t ∈ [ 0, T ] , with X 0 ∈ ℝ , and W = { W t , t ∈ [ 0, T ] } an F t -adapted one-dimensional Wiener process. We assume an an F t -adapted Poisson measure p φ ( d v , d t ) with mark space ε ⊆ ℝ \ { 0 } and with intensity measure φ ( d v ) d t = λ F ( d v ) d t , where F ( . ) is a given probability distribution function for the realizations of the marks.

Consider a one-dimensional SDE with Jumps (32) in integral form, is of the form

X t = X 0 + ∫ 0 t     a ( s , X s ) d s + ∫ 0 t     b ( s , X s ) d W s + ∑ i = 1 p φ ( t ) c ( τ i , X τ i ) (33)

Consider the following Stochastic Delay Differential Equations with constant delay in Stratonovich form [54]

d X ( t ) = f ( X ( t ) , X ( t − τ ) ) d t + ∑ l = 1 r     g l ( X ( t ) , X ( t − τ ) ) ∘ d W l ( t ) , (34)

X ( t ) = ϕ ( t ) (35)

where τ > 0 is a constant ( W ( t ) , F t ) = ( { W l ( t ) ,1 ≤ l ≤ r } , F ) is a system of one dimensional independent standard Wiener process, the function f : ℝ d × ℝ d → ℝ d , g l : ℝ d × ℝ d → ℝ d , ϕ ( t ) : [ − τ ,0 ] → ℝ d are continuous with E ‖ ϕ ‖ L ∞ 2 < ∞ . and ϕ is F -measurable. For mean-square stability of (35), we assume that f , g l , ∂ x g l g q and ∂ x τ g l g q ( ∂ x and ∂ x τ denote the derivatives with respect to the first and second variables respectively), l , q = 1 , 2 , ⋯ , r , in (35) satisfy the Lipschitz and linear growth conditions.

The Euler-Maruyama Scheme. The Euler-Maruyama method is a method for the approximate numerical solution of a stochastic differential equation. It is a simple generalization of the Euler method for ordinary differential equations to stochastic differential equations. It is named after a Swiss mathematician, physicist, geograph, astronomer, engineer, and logician Leonhard Euler (1707-1783) and a Japanese mathematician Gisiro Maruyama (1916-1986). Consider a scalar Itô stochastic ordinary differential equation [52]

d X t = f ( t , X t ) d t + g ( t , X t ) d W t (36)

with a standard scalar Wiener process W t , t ≥ 0 . This Equation (36) is in fact a symbolic representation for the stochastic integral equation

X t = X t 0 + ∫ t 0 t     f ( t , X t ) d t + ∫ t 0 t     g ( t , X t ) d W t (37)

The simplest numerical scheme for the stochastic ordinary differential Eqaution (36) is the Euler-Maruyama Scheme given by

Y n + 1 = Y n + f ( t n , Y n ) ∫ t n t n + 1   d s + g ( t n , Y n ) ∫ t n t n + 1   d W s (38)

where one usually writes

Δ n = ∫ t n t n + 1   d s ,     Δ W n = ∫ t n t n + 1   d W s ,

for n = 0 , 1 , ⋯ , M T − 1 and where t 0 < t 1 < ⋯ < t M = T with M T ∈ ℕ is an arbitrary partition of [ t 0 , T ] . The Euler-Maruyama approximation of an m-dimensional stochastic differential equation X h = ( X 1 h , X 2 h , ⋯ , X m h ) is defined by [38]

X t p + 1 h = X t p h + μ ( t p , X t p h ) h + σ ( t p , X t p h ) Δ B p (39)

X 0 h = x , Δ B p : = B t p + 1 − B t p , t p = p h .

As a strong approximation, it is of order 1/2, while as a weak approximation it s of order 1. In other words, sup t ≤ T E | X t h − X t | = O ( h 1 / 2 ) and

E f ( X t h ) − E f ( X t ) = O ( h ) , h → 0 , for all f ∈ C μ 4 ( ℝ m ) .

The Milstein Scheme The Mistein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Russian mathematician Grigori N. Milstein (who first published the method in 1974. The another useful numerical scheme for the SODE (36) is the Milstein Scheme given in [52] by

Y n + 1 = Y n + f ( t n , Y n ) ∫ t n t n + 1   d s + g ( t n , Y n ) ∫ t n t n + 1   d W s     + g ( t n , Y n ) ∂ g ∂ x ( t n , Y n ) ∫ t n t n + 1 ∫ t n s   d W u d W s (40)

The Milstein approximation of an m-dimensional stochastic differential equation X h = ( X 1 h , X 2 h , ⋯ , X m h ) is defined by [38]

X i t p + 1 h = X i t p h + μ i ( t p , X i t p h ) h + ∑     σ i j ( t p , X i t p h ) Δ B p j + ∑ j , l , q ∂ σ i q ∂ x j σ j l Δ C p l q ,

Δ C p l q : = ∫ t p t p + 1 ( B s l − B t p ) d B s q , X i t p h = x i , t p = p h .

The Runge-Kutta Scheme. The Runge-Kutta methods are a family of implicit and explicit iterative methods, which include the well-known routine called the Euler Method, used in temporal discretization for the approximate solutions of ordinary differential equations. These methods were developed around 1900 by the German mathematicians Carl Runge (1856-1927) and Wilhelm Kutta (1867-1944). Consider an m-dimensional Stratonovich differential equation of the form [62] [63]

d X = f ( t , X ) d t + g ( t , X ) ∘ d W ( t ) ,     X ( t 0 ) = X 0 , (41)

where f is an m-vector-valued function, g is an m × p matrix-valued function and W ( t ) is a p-dimensional process having independent scalar Wiener process components and the solution X ( t ) is an m-vector process. A general class of stochastic Runge-Kutta method in which [62] [63]

X i = x n + h ∑ j = 1 s     a i j f ( X j ) + ∑ j = 1 s     Z i j g ( X j ) ,   i = 1 , ⋯ , s (42)

x n + 1 = x n + h ∑ j = 1 s     α i f ( X j ) + ∑ j = 1 s     Z j g ( X j ) , (43)

where Z and z are respectively, an s × s matrix and s × 1 vector whose elements are themselves arbitrary random variables. By letting

Z i j = b i j ( 1 ) J 1 + b i j J 10 / h ,   i , j = 1 , ⋯ , s ,

Z j = γ j ( 1 ) J 1 + γ j ( 2 ) / h ,   j = 1 , ⋯ , s .

Here J 10 represents the Stratonovich multiple integral of order two given by

J 10 = ∫ t 0 t ∫ t 0 s ∘ d W ( s t ) d s .

This method is defined to be of order p if the local truncation error is O ( h p ) .

The Euler scheme for SDE with jumps (32), is given by the algorithm, Platen [53] [64] [65]

Y n + 1 = Y n + a Δ n + b Δ W n + ∫ t n t n + 1 ∫ ε     c ( v ) p φ ( d v , d z )

Y n + 1 = Y n + a Δ n + b Δ W n + ∑ i = p φ ( t ) + 1 p φ ( t n + 1 )     c ( ξ i ) (44)

for n ∈ { 0,1, ⋯ , N − 1 } with initial value Y 0 = X 0 . Here Δ n = t n + 1 − t n is the length of the time interval [ t n , t n + 1 ] and Δ W n = W t n + 1 − W t n is the n^th Gaussian N ( 0, Δ n ) distributed increment of the Wiener process W, n ∈ { 0,1, ⋯ , N − 1 } , p φ ( t ) = p φ ( ε , [ 0 , t ] ) represents the total number of jumps of Poisson random measure up to time t, which is Poisson distributed with mean λ t .

In the multidimensional case with mark-indepedent jump size we obtain the k^th component of the Euler scheme

Y n + 1 k = Y n k + a k Δ n + ∑ i = p φ ( t ) + 1 p φ ( t n + 1 )     b k , j Δ W n + c k Δ p n . (45)

This material is from [66]

X ( t j + 1 n ) = S ( t j + 1 n − t j n ) X ( t j n ) + ∫ t j n t j + 1 n     S ( t j + 1 n − s ) B ( S ( s − t j n ) X ( t j n )     + ∫ t j n s     B X ( r ) d r + ∫ t j n s     S ( s − r ) G ( X ( r ) ) d M ( r ) ) d s     + ∫ t j n t j + 1 n     S ( t j + 1 n − s ) B ( S ( s − t j n ) X ( t j n )     + ∫ t j n s     B X ( r ) d r + ∫ t j n s     S ( s − r ) G ( X ( r ) ) d M ( r ) ) d M ( s ) (46)

For SPDE with multiplicative noise, (27), there are two stochastic numerical methods that are used in the literature the linear-mplicit Euler and the linear-implicit Crank-Nicolson schemes [52] .

The Euler-Maruyama scheme

Y k + 1 N , M , L = ( I − h A N ) − 1 ( Y k + 1 N , M , L + h F N ( Y k + 1 N , M , L ) )                         + ( I − h A N ) − 1 B N , L ( Y k + 1 N , M , L ) Δ W k ,   ℙ − a . s . (47)

The Crank-Nicolson scheme. The Crank-Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is implicit in time and can be written as an implicit Runge-Kutta method, and it is numerically stable. The method was developed by a British mathematical physicist John Crank (1867-1944) and a British mathematician Phyllis Nicolson (1916-1968) in the mid 20th century [67] .

Y k + 1 N , M , L = ( I − h 2 A N ) − 1 ( ( I + h 2 A N ) Y k + 1 N , M , L + h F N ( Y k + 1 N , M , L ) )                         + ( I − h 2 A N ) − 1 B N , L ( Y k N , M , L ) Δ W k ,   ℙ − a . s . (48)

for k ∈ { 0,1, ⋯ , M − 1 } and N , M , L ∈ ℕ . Here it is necessary to assume that λ ≥ 0 for all i ∈ L in Assumption ( ) in order to ensure that ( I − h A ) is inversible for every h ≥ 0 .

Convergence of SPDE with multiplicative noise. The convergence of the exponential Euler scheme will proved under the following assumptions.

Assumption 5.0.1 (A5). (Linear operator A). There exist sequences of real eigenvalues 0 < λ 1 ≤ λ 2 ≤ ⋯ and orthonormal eigenfunctions ( e n ) n ≥ 1 of − A such that the linear operator A : D ( A ) ∈ H → H is given by

A v = ∑ n = 1 ∞ − λ n 〈 e n , v 〉 ,

for all v ∈ D ( A ) with D ( A ) = { v ∈ H : ∑ n = 1 ∞ | λ n | 2 〈 e n , v 〉 2 < ∞ } .

(A6) (nonlinearity F). The nonlinearity F : H → H is two times continuously Fréchet differentiable and its derivatives satisfy

‖ F ′ ( x ) − F ′ ( y ) ‖ ≤ L | x − y | H ,

| ( − A ) − ( − r ) F ′ ( x ) ( − A ) r v | H ≤ L | v | H ,

for all x , y ∈ H , v ∈ D ( ( − A ) r ) , and r = 0 , 1 / 2 , 1 , and

| A − 1 F ″ ( x ) ( v , w ) | ≤ L | ( − A ) − 1 / 2 v | H | ( − A ) − 1 / 2 w | H ,

for all x , y ∈ H , where L > 0 is a positive constant.

(A7) (Cylindrical Q-Wiener process W t ). There exist a sequence ( q n ) n ≥ 1 of positive real numbers and a real number γ ∈ ( 0,1 ) such that

∑ n = 1 ∞     λ n 2 γ − 1 q n < ∞

and pairwise independent scalar F t -adapted Wiener process ( W t ) t ≥ 0 for n ≥ 1 . The cylindrical Q-Wiener process W t is given formally by

W t = ∑ n = 1 ∞ q n W t n e n . (49)

(A8) (Initial value). The random variable x 0 : Ω → D ( ( − A ) γ ) satisfies E | ( − A ) γ x 0 | H 4 < ∞ , where γ > 0 is given in A7.

There are many numerical schemes for solving stochastic delay differential equations. As given in [54] , we give three schemes to solve (35). The first scheme is the Predictor-correction scheme given by

X n + 1 = X n + h 2 [ f ( X n , X n − m ) + f ( X ¯ n + 1 , X n − m + 1 ) ]     + 1 2 ∑ l = 1 r [ g l ( X n , X n − m ) + g l ( X ¯ n + 1 , X n − m + 1 ) ] Δ W l , n , (50)

X ¯ n + 1 = X n + h f ( X n , X n − m ) + ∑ l = 1 r     g l ( X n , X n − m ) Δ W l , n (51)

The second is the Midpoint scheme given by

X n + 1 = X n + h f ( X n + X n + 1 2 , X n − m + X n − m + 1 2 )     + ∑ l = 1 r [ g l ( X n + X n + 1 2 , X n − m + X n − m + 1 2 ) ] Δ W ¯ l , n (52)

where we have Δ W l , n with Δ W ¯ l , n .

The last scheme is the Milstein-like scheme given by

X n + 1 = X n + h f ( X n , X n − m ) + ∑ l = 1 r     g l ( X n , X n − m ) I 0     + ∑ l = 1 r ∑ l = 1 r     ∂ x g l ( X n , X n − m ) g q ( X n , X n − m ) I q , l , t n , t n + 1 , 0     + ∑ l = 1 r ∑ l = 1 r     ∂ x τ g l ( X n , X n − m ) g q ( X n − m , X n − 2 m ) χ t n ≥ τ I q , l , t n , t n + 1 (53)

where I 0 = ∫ t n t n + 1   d W ˜ l ( t ) , I q , l , t n , t n + 1 , 0 = ∫ t n t n + 1 ∫ t n t     d W ˜ q ( s ) d W ˜ l ( t ) ,

I q , l , t n , t n + 1 , τ = ∫ t n t n + 1 ∫ t n − τ t − τ     d W ˜ q ( s ) d W ˜ l ( t ) , t n + 1 ≥ 0 ,

∫ t n t n + 1   d W ˜ l ( t ) = Δ W l , n = W l ( t n + 1 ) − W l ( t n ) .

Definition 6.0.1. (Strong Convergence) We say that a numerical scheme for solving the SDE (36) converges strongly on [ 0, T ] to the solution X of the SDE if for the final time T have

lim δ → 0 E ( | X ( T ) − Y δ ( T ) | ) = 0. (54)

A strongly convergent scheme is said o have convergence rate γ if for some constants C and δ 0 > 0 we have

lim δ → 0 E ( | X ( T ) − Y δ ( T ) | ) ≤ C δ γ , ∀ δ ∈ [ 0, δ 0 ] . (55)

Theorem 6.1. (Strong Convergence: Euler-Maruyama scheme) Under assumptions of Lipschitz and linear growth of coefficients and additionally

| a ( t , x ) − a ( s , x ) | + | σ ( t , x ) − σ ( s , x ) | ≤ K ( 1 + | x | ) | t − s | 1 / 2 (56)

for some suitable constant K, the Euler-Maruyama scheme converges strongly with a convergence rate of γ = 1 / 2 .

Theorem 6.2. (Strong convergence of Milstein scheme) In addition to the

assumption of Theorem , let σ ( t , x ) and ∂ σ ( t , x ) ∂ x satisfy the conditions on

the coefficients of the Theorem . If further we have a ∈ C 1,1 , σ ∈ C 1,2 , then the Milstein converges strongly with a convergence rate of γ = 1 .

The convergence of error for SPDE is given by the following theorem.

Theorem 6.3. (Convergence Theorem, [52] ) Suppose that assumptions (A1)-(A8) are satisfied. Then there is a constant C T > 0 such that

sup k = 0 , ⋯ , M ( E | X t k − Y k ( N , M ) | H 2 ) 1 2 ≤ C T ( λ N − γ + log ( M ) M ) , (57)

holds for all N , M ∈ ℕ , where X t is the solution of SPDE (27), Y k ( N , M ) is the

numerical solution given by (46), t k = T k M for k = 0 , 1 , ⋯ , M , and γ > 0 is

the constant given in Assumption (A8).

As with the Euler-Maruyama method, the Milstein method is very easy to implement which is a reason that it is also quite popular among practitioners in finance.

Algorithm 6.3.1. (The Euler-Maruyama Scheme). Let Δ t : = T / N for a given N. Then approximate the SDE via

1) Set Y N ( 0 ) = X ( 0 ) = x 0

2) For j = 0 to N − 1 do

a) Simulate a standard normally distributed random number Z j

b) Set Δ W ( j Δ t ) = Δ t Z j and

Y N ( ( j + 1 ) Δ t ) = Y N ( j Δ t ) + a ( j Δ t , Y N ( j Δ t ) ) Δ t + σ ( j Δ t , Y N ( j Δ t ) ) Δ W ( j Δ t ) .

Algorithm 6.3.2. (The Milstein Scheme) Let Δ t : = T / N for a given N. Then approximate the SDE via

1) Set Y N ( 0 ) = X ( 0 ) = x 0

2) For j = 0 to N − 1 do

a) Simulate a standard normally distributed random number Z j

b) Set Δ W ( j Δ t ) = Δ t Z j and

Y N ( ( j + 1 ) Δ t ) = Y N ( j Δ t ) + a ( j Δ t , Y N ( j Δ t ) ) Δ t + σ ( j Δ t , Y N ( j Δ t ) ) Δ W ( j Δ t )     + 1 2 σ ( j Δ t , Y N ( j Δ t ) ) σ ′ ( j Δ t , Y N ( j Δ t ) ) ( Δ W ( j Δ t ) 2 − Δ t ) .