Existence and Uniqueness of Solutions for a Class of Nonlinear Stochastic Differential Equations

Definition 1. An F_t-dimensional process x(t) is said to be the solution of (1) if x(t₀) = x  and

holds for every t ≥ t₀ with probability 1.

A solution x(t) is said to be unique if for any continuous solutions x(t),   y(t) such that x(t₀) = y(t₀) = x one has $$ for all  t ≥ t₀.

In this work we use the method of successive approximations to establish existence and uniqueness (pathwise) of the solution of (1). We study its probability properties. We prove that x(t) is diffusion process and find coefficients of diffusion.

Theorem 2. Assume that  G(t, x) is a continuous function; a(t, x),   σ(t, x) are measurable functions for x ∈ R^d,   t ≥ t₀ and satisfy the following conditions:

• (1)

  there exists a constant   C > 0, such that |G(t, x)| + |a(t, x)| + ∥σ(t, x)∥ ≤ C(1 + |x|) for x ∈ R^d, t ≥ t₀;

• (2)

  there exist constants L₁ > 0 and L > 0 such that |G(t, x) − G(t, x^′)| ≤ L₁|x − x^′| and

  $$ ()

•  

  for x, x^′ ∈ R^d, t ≥ t₀.

Proof. To find a solution of the integral equation (2), we use the method of successive approximations. We start by choosing an initial approximation x₀(t) = x.

At the next step,

Consequently, Consider an interval [t₀, T] such that $$. Note that it can be always achieved by choosing a sufficiently small T − t₀ and by the conditions on L₁.

We prove that a solution exists on this interval.

From (4) it follows that

where C₁ is a constant independent of t₀, T, and x.

Next, estimate

Therefore, The estimation of the stochastic integral follows from its properties. Similar to [11, p.20], we have Thus, we have Since B < 1, it then follows that the series converges.

This implies the uniform convergence with probability 1 of the series

on the interval [t₀, T]. Its sum is x(t). So x_n(t) converges to some random process. Every x_n(t) is continuous with probability 1. Whence it follows that the limit x(t) is continuous with probability 1, too.

Then prove uniqueness of this continuous solution. Assume that there exists a second continuous solution y(t) of (1). Denote  by ϰ_N(t) the random variable which equals 1 if |x(s)| ≤ N, |y(s)| ≤ N  and it equals 0 otherwise. Then

Therefore, The last inequality implies the estimation By the Gronwall-Bellman inequality and the above inequality together, we have So, Probabilities at the right-hand side of the above inequality approach to zero as N → ∞, because x(t)  and  y(t) are continuous with probability 1. Therefore x(t) and  y(t) are stochastic equivalent.

We conclude that

Hence, the existence and uniqueness of the solution are proved on [t₀, T].

Next, we show boundedness of a second moment of the solution. Denote again by ϰ_N(t) the indicator of the set

Then Similarly, we have So, Use the Gronwall-Bellman inequality to find  the estimation where D is a constant independent of N. Applying the Fatou lemma to the last inequality and assuming that N → ∞, we have Since E|x|² ≤ ∞, then it follows from [8] that E|x|² < ∞  for every t ∈ [t₀, T].

Thus existence, uniqueness, and boundedness of the second moment of the solution x(t) are proved on [t₀, T]. Since the constant B is dependent only on T − t₀ and E|x|² < ∞, and x(T) is independent of  W(s) − W(T) for s⩾T, then by similar manner we can prove the existence and uniqueness of the solution of the IVP with initial conditions (T, x(T)) on the interval [T, T₁], where T₁ is chosen such that the inequality $$.

This procedure can be repeated in order to extend the solution of (1) to the entire semiaxis t⩾t₀. The theorem is proved.

Theorem 3. Under conditions (1)-(2) of Theorem 2 the solution x(t) of (1) is a Markov process with a transition probability defined by

where x_t,x(s) is a solution of that equation where s⩾t⩾t₀, x ∈ R^d.

Proof. As in Theorem 2, we solve (1) by the method of successive approximations. It can be shown that  x_t,x(s)  is completely defined by the nonrandom initial value x and the process W(s) − W(t) for s > t, which are independent of F_t. Since x(t) is a solution of (2), it is F_t measurable. Hence x_t,x(s) is independent of x(t) and events from F_t. Note that, from the uniqueness of the solution x(s) for s > t, it follows that it is a unique solution of the equation

Since the process x_t,x(t)(s) is also a solution of this equation, then x(s) = x_t,x(t)(s) with probability 1.

As for the rest, the proof is the same as the proof of the theorem for ordinary stochastic equations [11]. The theorem is proved.

Corollary 4. Suppose that conditions of Theorem 2 are satisfied. Then

• (1)

  if functions G,  a, and σ are independent of t, then the solution x(t) is a homogeneous Markov process;

• (2)

  if functions G,  a, and σ are periodic functions with period θ, then a transition probability is periodic function; that is, P(t + θ, x, s + θ, A) = P(t, x, s, A).

Lemma 5. Let x_t,x(s) be a solution of (2) such that x_t,x(t) = x, x ∈ R^d. If conditions of Theorem 2 are satisfied and G_t(t, x) is continuous for t ≥ t₀, x ∈ R^d, then an inequality

holds. Here H > 0 is a constant dependent only on C, L₁, L, and G_t(t, x).

Proof. Denote by ϰ_N(s) the indicator of the set

Similarly to finding estimation (24), we have where o_x points dependence on x.

The last estimate follows from Hölder’s inequality and properties of stochastic integral. From inequality (30) and condition for L₁, we have

where C₂ depends only on C.

Further, the right-hand side of (31) can be estimated by

where $$ and  R₁ = 6⁵L⁴,   t ≤ s ≤ t + 1.

Using the lemma from [11, p.38], we obtain

where H > 0 depends only on C₂, L, and G_t(t, x).

Now, assume that N → ∞  and obtain estimate (28). This completes the proof of the Lemma.

Theorem 6. If conditions (1)-(2) of Theorem 2 are satisfied and functions a(t, x), σ(t, x), G(t, x), G_t(t, x), $$, and $$, $$, are continuous for t⩾t₀, x ∈ R^d. Functions G_t(t, x), $$, and $$ satisfy Lipschitz condition with respect to x in neighborhood of every point (t, x). Then the solution of (1) is diffusion process with

and diffusion matrix with

Proof. Take any point (t, x) from a region t⩾t₀, x ∈ R^d. Consider a stochastic ITO equation in a closed neighborhood of this point

From the conditions of the theorem, it follows that there exists a closed neighborhood of the point  (t, x), such that coefficients of the equation are Lipschitz with respect to x. Fix this neighborhood. Extend these coefficients on the all region t ≥ t₀, x ∈ R^d such that they remain continuous by both variables, Lipschitz and linear with respect to x.  

Then the equation

with these extended coefficients has a unique solution of the IVP y(t) = x   for  s > t.

From ITO formula and coincidence of coefficients of (36) and (37) in this neighborhood of the point (t, x), one can show that processes x_t,x and y(s) coincide in this neighborhood.

It is known that under these assumptions the process y(s) is diffusion. Its diffusion coefficients in the point (t, x) are defined by coefficients of (36).

Next, show that x(s) has the same diffusion coefficients. To do this it is sufficient to estimate the following limits and use estimation (28)

where ɛ₀ is chosen such that the point (s, y) is laying in this neighborhood.

We get

Similarly, obtain the second limit in (38): So, the existence of limits in (38) is proved for every point (t, x) and fixed ɛ₀, chosen for this point. It is clear that from (28), these limits exist for every ɛ > 0 and they are independent of ɛ. From the above and the definition of diffusion process [12, p.67] we can finish the proof. The theorem is proved.