Construction of positivity preserving numerical schemes for multidimensional stochastic differential equations

In this note we work on the construction of positive preserving numerical schemes for systems of stochastic differential equations. We use the semi discrete idea that we have proposed before proposing now a numerical scheme that preserves positivity on multidimensional stochastic differential equations converging strongly in the mean square sense to the true solution.


Introduction
Throughout, let T > 0 and (Ω, F , {F t } 0≤t≤T , P) be a complete probability space, meaning that the filtration {F t } 0≤t≤T satisfies the usual conditions, i.e. is right continuous and F 0 includes all P−null sets, and let an m-dimensional Wiener process W (t) defined on this space.
Let the following multidimensional stochastic differential equation, where a(·) : R d → R d , b j (·) : R d → R d , j = 1, .., m, x(0) = x and x : Ω → R d is F 0 measurable. That is, a(x) = (a 1 (x), ..., a d (x)), b j (x) = (b 1 , ..., b d ) where j = 1, ..., m. Assumption A Suppose that there exists some functions f i (x, y) : R 2d → R and g ij (x, y) : R 2d → R such that f i (x, x) = a i (x) and g ij (x, x) = b ij (x) with x = (x 1 , ..., x d ), y = (y 1 , ..., y d ), i = 1, ..., d and j = 1, ..., m. Let E x p 2 < A for some p > 2. Suppose further that f, g satisfy the following condition for any R > 0, i = 1, ..., d, j = 1, ..., m and x 1 , x 2 , y 1 , y 2 such that x 1 2 ∨ x 2 2 ∨ y 1 2 ∨ y 2 2 ≤ R, where the constant C R depends on R and x ∨ y denotes the maximum of x, y. Here Let the equidistant partition 0 = t 0 < t 1 < ... < t N = T and ∆ = T /N. We propose the following numerical scheme, for i = 1, ..., d. Note that y(s) = (y 1 (s), ..., y d (s)) and y(ŝ) = (y 1 (ŝ), ..., y d (ŝ)), with y i (ŝ) = y i (t k ) when s ∈ [t k , t k+1 ] and y i (0) = x i . This is again, in general, a system of stochastic differential equations and we suppose that has a unique strong solution. However, in practice, we will choose f, g in a way that the resulting system has less than d dependent equations or/and having known explicit solution. An interesting choice that we will see in our example is that the resulting system is not in fact a system of SDEs but d independent equations. Each of these equations is linear with known explicit solution and we solve it independently of the others. Below we state a convergence, in the mean square sense, theorem of y t to the true solution as ∆ ↓ 0. The proof of this theorem is exactly the same as in [1] changing the absolute values by the Euclidean norm in R d .

Theorem 1 Suppose Assumption A holds and (2) has a unique strong solution. Let also
for some p > 2 and A > 0. Then the semi-discrete numerical scheme (2) converges to the true solution of (1) in the mean square sense, that is Proof. We use the same arguments as in [1], [4].
Using exactly the same arguments as in [4] we obtain, First, let us estimate the quantity E y(t ∧ θ) − y( t ∧ θ) 2 2 , beginning with, Taking expectations, using Ito's isometry and the fact that |f i (y s , yŝ)|, |g ij (y s , yŝ)| ≤ C R we have that, and from this it follows that We can write now, Taking expectations on both sides, using Doob's martingale inequality for the second term at the right hand side and Assumption A for g(·, ·) we arrive at, Therefore, we have that Using Gronwall's inequality and continuing as in [1] we arrive at the desired result.

Example
Consider the following multidimensional SDE.
Assumption B Assume that E| ln x i | + Ex 2p i < C for all i = 1, ..., d for some p > 2 and C > 0. Consider the following system of SDEs, Each of the above equation has only one unknown function and is linear with known explicit solution and thus preserves positivity.

Theorem 2 Under Assumption B we have
Proof. For the moment bound of the true solution we use Lemma 3.2 of [4] since the drift coefficient satisfies the monotonicity condition and the diffusion term the linear growth condition.
We will prove now that the approximate solution has bounded moments. Set the stopping time θ R = inf{t ∈ [0, T ] : y i (t) > R}. Using Ito's formula on y q i (t ∧ θ R ) (with q = 2p) we obtain Taking expectations we arrive at the following inequality, Ey q i (s)ds.
Using now Gronwall's inequality we get that Ey q i (t ∧ θ R ) < A and using Fatou's lemma we arrive at the bound Ey q i (t) < A for i = 1, ..., d. Therefore, we have that E( y t q q ) < A. Using again Ito's formula on y p i (t) we obtain, Taking expectations and using Doob's inequality on the diffusion term we arrive at Using the equivalence of the norms in R d we obtain the desired result.
Taking absolute values and then expectations, using Jensen inequality and then Ito's isometry on the diffusion term we arrive at We have used also the moment bound for the true solution (see Theorem 2). Therefore ln R P ({θ R ≤ R}) < C, and thus P ({θ R ≤ R}) → 0 as R → ∞. But In this example we have a(x) = x− x 2 2 x and b(x) = x with x = (x 1 , ..., x d ), that is m = 1. We choose f i (x, y) = x i − y 2 2 x i and g i (x, y) = x i where x = (x 1 , ..., x d ) and y = (y 1 , ..., y d ). It is easy to see that f i (x, x) = a(x) and g i (x, x) = b(x). Moreover, f i , g i satisfies Assumption A. Therefore, our proposed numerical scheme preserves positivity and converges strongly in the mean square sense to the true solution.
Conclusion Concerning super linear SDEs it is well known that the usual Euler scheme diverges and the tamed-Euler scheme [5] does not preserve positivity. For scalar SDEs there are some numerical schemes that preserves positivity (see for example [1], [2], [3], [6] and the references therein) but for the multidimensional case it is not clear how can be extended. So we extend here our semi discrete method ( [1]), that we have proposed before for scalar SDEs, to the multidimensional case. There is also the possibility in some multidimensional SDEs to combine the semi discrete method with another method designed for scalar SDEs in order to construct positivity preserving numerical schemes. Let us note that we can use the semi discrete method also in the case when the diffusion term is super linear. Our goal in the future is to apply the semi discrete method to more complicated systems of SDEs.