Impact of Financial Crisis on Economic Growth: A Stochastic Model

Proof of Proposition 3.1

(i) Let t ≥ 0 . From the second equation of (3.2), we have

From the first equation of (3.2) and the positivity of y ⁢ ( t ) , we get

From the third equation of (3.2), we have

(ii) Since x ⁢ ( t ) and y ⁢ ( t ) are positive and x ⁢ ( t ) + y ⁢ ( t ) = 1 , then 0 ≤ x ⁢ ( t ) ≤ 1 and 0 ≤ y ⁢ ( t ) ≤ 1 . From the positivity of k ⁢ ( t ) and the third equation of (3.2), we have

That is,

Since 0 ≤ y ⁢ ( t ) ≤ 1 , (A.1) becomes

Thus, using the standard comparison theorem [19], we obtain

and if k ⁢ ( 0 ) ≤ ( λ 0 δ 0 + p ) 1 1 - α ,

From (A.2), we deduce that k ⁢ ( t ) ≤ ( λ 0 δ 0 + p ) 1 1 - α . ∎

Proof of Proposition 3.2

Consider the function Φ defined from R 3 to R 3 by

The function Φ is a continuously differentiable on R 3 ; hence it is locally Lipschitz. From the Cauchy–Lipschitz theorem [22], problem (3.2) admits a unique local solution. From Proposition 3.1, the solution is contained in a compact subset of R 3 and therefore is globally defined. ∎

Proof of Proposition 3.3

Since x ⁢ ( t ) + y ⁢ ( t ) = 1 , x ⁢ ( t ) = 1 - y ⁢ ( t ) and system (3.2) is equivalent to the following system:

( y , k ) is an equilibrium of system (A.3) if and only if ( 1 - y , y , k ) is an equilibrium of system (3.2).

A point ( y , k ) is an equilibrium of system (A.3) if it is a solution of the following system of algebraic equations:

From the first equation of (A.3), we have

If y = 0 , then x = 1 , and from the second equation of (A.4), we obtain k = ( λ 0 δ 0 + p ) 1 1 - α . If θ > ξ + μ + b , then y = θ - μ - ξ - b θ - μ is a valid solution of (A.4), i.e. 0 < θ - μ - ξ - b θ - μ < 1 . Hence

and from the second equation of (A.4), we obtain

Proof of Proposition 3.4

The first line of (A.3) is an equation with separable variables. After few transformations, it becomes

Solving (A.5) gives

where 𝐴 is a constant positive real number.

(i) If θ ≤ ξ + μ + b , taking the limit as 𝑡 tends to ∞ on both sides of (A.6) yields

That is, whatever the initial condition y ⁢ ( 0 ) is, the solution y ⁢ ( t ) converges to zero as 𝑡 tends to ∞. Therefore, as x ⁢ ( t ) = 1 - y ⁢ ( t ) , we have lim t → ∞ ⁡ x ⁢ ( t ) = 1 .

Since y ⁢ ( t ) converges to 0 as 𝑡 tends to ∞, for all ϵ > 0 sufficiently small, there exists a real number M > 0 such that, for all t > M ,

From the second equation of (A.3) and the second inequality of (A.7), we obtain

Using the nonnegativity of k ⁢ ( t ) and (A.8), we get

By the comparison theorem [19], (A.9) yields

Thus,

From the arbitrariness of 𝜖, we deduce from (A.10) that

Using once more the second equation of (A.3) and the first inequality of (A.7), we obtain

Using the positivity of k ⁢ ( t ) and the comparison theorem [19], we deduce that

Thus, from the arbitrariness of 𝜖, we deduce from (A.12) that

It then follows from (A.11) and (A.13) that

This lets us infer that

That is, lim t → ∞ ⁡ k ⁢ ( t ) = ( λ 0 δ 0 + p ) 1 1 - α .

(ii) If θ > ξ + μ + b , taking the limit as 𝑡 tends to ∞ on both sides of (A.6) yields

That is, whatever the initial condition y ⁢ ( 0 ) is, the solution y ⁢ ( t ) converges to θ - μ - ξ - b θ - μ as 𝑡 tends to ∞. Therefore, as x ⁢ ( t ) = 1 - y ⁢ ( t ) , we have lim t → ∞ ⁡ x ⁢ ( t ) = ξ + b θ - μ .

Since y ⁢ ( t ) converges to θ - μ - ξ - b θ - μ as 𝑡 tends to ∞, for all ϵ > 0 sufficiently small, there exists a real number N > 0 such that, for all t > N ,

From the second equation of (A.3) and the second inequality of (A.14), we obtain

Using the nonnegativity of k ⁢ ( t ) and (A.15), we get

By the comparison theorem [19], (A.16) yields

Thus,

From the arbitrariness of 𝜖, we deduce from (A.17) that

Using once more the second equation of (A.3) and the first inequality of (A.14), we obtain

Using the positivity of k ⁢ ( t ) and the comparison theorem [19], we deduce that

Thus, from the arbitrariness of 𝜖, we deduce from (A.19) that

It then follows from (A.18) and (A.20) that

This lets us infer that

That is, lim t → ∞ ⁡ k ⁢ ( t ) = ( λ 0 δ 0 + p ) 1 1 - α ⁢ ( ξ + b θ - μ ) τ 1 - α . ∎

Proof of Proposition 3.5

Let ( x ⁢ ( 0 ) , y ⁢ ( 0 ) , k ⁢ ( 0 ) ) ∈ Γ . Since the coefficients of system (3.1) are C 1 functions, they are locally Lipschitz continuous. Therefore, for any given initial value ( x ⁢ ( 0 ) , y ⁢ ( 0 ) , k ⁢ ( 0 ) ) , there is a unique local solution ( x ⁢ ( t ) , y ⁢ ( t ) , k ⁢ ( t ) ) on t ∈ [ 0 , τ e ) , where τ e is the explosion time. Let q 0 > 1 be sufficiently large so that x ⁢ ( 0 ) and y ⁢ ( 0 ) lie within the interval [ 1 q 0 , 1 ] and k ⁢ ( 0 ) is in [ 1 q 0 , ( λ 0 δ 0 + p ) 1 1 - α ] . For each integer q ≥ q 0 , define the stopping time τ q by

where, throughout this paper, we set inf ⁡ ϕ = ∞ (as usual, 𝜙 denotes the empty set).

According to the definition, τ q is increasing as k → ∞ . Denoting τ * = lim q → ∞ ⁡ τ q , we have τ * ≤ τ e almost surely (a.s.). If we can show that τ * = ∞ , then τ e = ∞ , and the solution ( x ⁢ ( t ) , y ⁢ ( t ) , k ⁢ ( t ) ) is such that x ⁢ ( t ) > 0 , y ⁢ ( t ) > 0 and k ⁢ ( t ) > 0 a.s. for all t ≥ 0 . In fact, if, for all t ≥ 0 , x ⁢ ( t ) and y ⁢ ( t ) are positive a.s., then, by definition of these variables, the equality x ⁢ ( t ) + y ⁢ ( t ) = 1 leads to 0 ≤ x ⁢ ( t ) ≤ 1 and 0 ≤ y ⁢ ( t ) ≤ 1 a.s.

Furthermore, using the almost sure positivity of y ⁢ ( t ) and k ⁢ ( t ) , we can proceed as in Proposition 3.1 and prove that

In other words, to complete the proof, all we need to show is τ * = ∞ a.s. If this assertion is false, then there exists a pair of constants T > 0 and ϵ ∈ ( 0 , 1 ) such that P ⁢ ( { τ * ≤ T } ) > ϵ . Hence there is an integer q 1 ≥ q 0 such that P ⁢ ( { τ q ≤ T } ) ≥ ϵ for all q ≥ q 1 . Now define a C 2 function V : R + 3 → R by

where R + 3 = { ( x 1 , x 2 , x 3 ) ∈ R , x i > 0 , i = 1 , 2 , 3 } .

Applying Itô’s formula [17] to system (3.1) for ω ∈ { τ * ≤ T } and t ∈ [ 0 , τ * ) , we obtain

where

Setting

we have

By virtue of x + y = 1 , k ≤ ( λ 0 δ 0 + p ) 1 1 - α and the continuity of 𝐺 on R + 3 , there exists a constant M 1 < 0 such that

Hence, for any q ≥ q 1 , integrating both sides of (A.21) from 0 to T ∧ τ q = min ⁡ ( T , τ q ) , we have

It is straightforward that ∫ 0 T ∧ τ q σ ⁢ y ⁢ ( u ) ⁢ d W u and ∫ 0 T ∧ τ q σ ⁢ x ⁢ ( u ) ⁢ d W u are martingales, and this lets us infer that

where 𝔼 is the expectation with respect to probability ℙ.

Therefore, taking the expectation in both sides of (A.24) leads to

From (A.22) and (A.23), (A.25) leads to

where A ⁢ ( x ⁢ ( T ∧ τ q ) , y ⁢ ( T ∧ τ q ) , k ⁢ ( T ∧ τ q ) ) = E ⁢ ( V ⁢ ( x ⁢ ( T ∧ τ q ) , y ⁢ ( T ∧ τ q ) , k ⁢ ( T ∧ τ q ) ) ) - V ⁢ ( x ⁢ ( 0 ) , y ⁢ ( 0 ) , k ⁢ ( 0 ) ) .

Set Ω q = { ω ∈ Ω , τ q ⁢ ( ω ) ≤ T } for q ≥ q 1 . Letting I Ω q be the indicator function of Ω q , we have P ⁢ ( Ω q ) ≥ ϵ .

On the other hand, noting that x + y = 1 and k ≤ ( λ 0 δ 0 + p ) 1 1 - α , we have

Letting q → ∞ , (A.26) and (A.27) yield the contradiction - ∞ ≥ M 1 ⁢ T > - ∞ . Therefore, we obtain τ * = ∞ a.s. This completes the proof. ∎

Proof of Proposition 3.6

Consider the functions

We have ϕ 1 ⁢ ( 0 ) = ϕ 2 ⁢ ( 1 ) = θ - ( μ + ξ + b + σ 2 2 ) .

(i) Let us assume that θ < ξ + μ + b + σ 2 2 . That is, ϕ 1 ⁢ ( 0 ) < 0 . Since ϕ 1 is a continuous function, one can choose sufficiently small ρ > 0 such that

Consider the Lyapunov function Ψ ⁢ ( x , y ) = y β , where β ∈ ( 0 , 1 ) is a constant to be determined. Applying Itô’s formula, we have

where L ⁢ Ψ ⁢ ( x , y ) = β ⁢ y β ⁢ ϕ 1 ⁢ ( y ) + 1 2 ⁢ β 2 ⁢ y β ⁢ σ 2 ⁢ x 2 . For a constant 𝑤 in ( 0 , 1 ) , denote

By continuity of ϕ 1 , we can choose w ∈ ( 0 , 1 ) sufficiently small such that, for any y ∈ [ 0 , w ) ,

In addition, we let β ∈ ( 0 , 1 ) be sufficiently small such that, for any w ∈ Γ w 1 , we have

Thus, when 𝛽 and 𝑤 are small enough, for any ( x , y ) ∈ Γ w 1 , we have

By [23, Theorem 3.3, Chapter 4] and (A.28), for any γ > 0 , there is w 1 ∈ ( 0 , w ) such that

From the arbitrariness of 𝛾, we deduce that

Moreover, we further prove that any solution ( x ⁢ ( t ) , y ⁢ ( t ) , k ⁢ ( t ) ) starting in Γ is such that ( x ⁢ ( t ) , y ⁢ ( t ) ) will eventually enter Γ w 1 1 . Define T w 1 = inf ⁡ { t > 0 : y ⁢ ( t ) ≤ w 1 } . Consider the function Ψ ¯ ⁢ ( x , y ) = - ( y + 1 ) ν , where 𝜈 is a sufficiently large number such that

In fact, applying Itô’s formula to Ψ ¯ ⁢ ( x , y ) , we have

where

Using (A.30), we get

By Dynkin’s formula and the boundedness of ( 1 + y ) ν - 1 ⁢ x ⁢ y , we have

Since Ψ ¯ ⁢ ( x , y ) is bounded on [ 0 , 1 ] × [ 0 , 1 ] , we have E ⁢ [ T w 1 ] < ∞ . Thanks to the strong Markov property, E ⁢ [ T w 1 ] < ∞ and (A.29) imply that, for any γ > 0 ,

From the arbitrariness of 𝛾, we deduce that

From (A.31), we deduce that, for ϵ > 0 sufficiently small, there exists M 2 > 0 such that, for t ≥ M 2 ,

Proceeding as in item (i) of the proof for Proposition 3.4, we obtain

Using (A.31) and (A.32), we deduce that if θ < ξ + μ + b + σ 2 2 , the financial crisis dies out and ( x ⁢ ( t ) , y ⁢ ( t ) , k ⁢ ( t ) ) tends to ( 1 , 0 , ( λ 0 δ 0 + p ) 1 1 - α ) almost surely as 𝑡 tends to ∞.

(ii) Let us assume that θ > ξ + μ + b + σ 2 2 . We prove that the equation ϕ 2 ⁢ ( x ) = 0 admits a unique solution x * in ( 0 , 1 ) . Since θ > ξ + μ + b + σ 2 2 , the discriminant Δ of the polynomial ϕ 2 ⁢ ( x ) satisfies

Therefore, the equation ϕ 2 ⁢ ( x ) = 0 admits two solutions x 1 and x 2 defined by

It is straightforward that

Thus, the solutions x 1 and x 2 are positive. It is also easy to verify that x 1 < x 2 .

Since ϕ 2 ⁢ ( 0 ) = - ( ξ + b ) < 0 and ϕ 2 ⁢ ( 1 ) = θ - ( ξ + μ + b + σ 2 2 ) > 0 , by the Bolzano–Cauchy intermediate-value theorem [34], at least one of these solutions, x 1 , belongs to the interval ( 0 , 1 ) . If both of them belong to ( 0 , 1 ) , using the equality ϕ 2 ⁢ ( x ) = - σ 2 2 ⁢ ( x - x 1 ) ⁢ ( x - x 2 ) leads to ϕ 2 ⁢ ( 1 ) = - σ 2 2 ⁢ ( 1 - x 1 ) ⁢ ( 1 - x 2 ) < 0 , which is in contradiction with ϕ 2 ⁢ ( 1 ) > 0 . Therefore, the unique solution in ( 0 , 1 ) is x * = x 1 .

Next, we prove that the values of x ⁢ ( t ) for t ≥ 0 oscillate around x * almost surely. That is,

First, we prove that lim sup t → ∞ ⁡ x ⁢ ( t ) > x * . If it is not true, then there exists a sufficiently small ϵ ∈ ( 0 , 1 ) such that P ⁢ ( Ω 1 ) > ϵ , where Ω 1 = { ω ∈ Ω , lim sup t → ∞ ⁡ x ⁢ ( t , ω ) ≤ x * - 2 ⁢ ϵ } . Hence, for every ω ∈ Ω 1 , there is T 1 = T 1 ⁢ ( ω ) such that

Since ϕ 2 ′ ⁢ ( x * ) = ( θ - μ ) 2 - 2 ⁢ σ 2 ⁢ ( ξ + b ) > 0 , the function ϕ 2 is monotonically increasing at x * and (A.34) leads to

On the other hand, by the large number theorem for martingales, there is an Ω 2 ⊂ Ω with P ⁢ ( Ω 2 ) = 1 such that, for every ω ∈ Ω 2 ,

Now, fix any ω ∈ Ω 1 ∩ Ω 2 . From (A.35), for t ≥ T 1 ⁢ ( ω ) , we get

Using (A.36), (A.37) leads to lim sup t → ∞ ⁡ ln ⁡ ( y ⁢ ( t , ω ) ) t ≤ ϕ 2 ⁢ ( x * - ϵ ) < 0 . Thus,

From (A.38), we deduce that lim t → ∞ ⁡ x ⁢ ( t , ω ) = 1 , which contradicts (A.34). Therefore, we must have the desired assertion. That is,

Next, let us prove that lim inf t → ∞ ⁡ x ⁢ ( t ) < x * . If it is not true, then there is a sufficiently small ϵ 1 ∈ ( 0 , 1 ) such that P ⁢ ( Ω 3 ) > ϵ 1 , where Ω 3 = { ω ∈ Ω , lim inf t → ∞ ⁡ x ⁢ ( t , ω ) ≥ x * + 2 ⁢ ϵ 1 } . Hence, for every ω ∈ Ω 3 , there is a T 2 = T 2 ⁢ ( ω ) such that

Since the function ϕ 2 is monotonically increasing at x * , (A.39) leads to

Now, fixing ω ∈ Ω 2 ∩ Ω 3 , from (A.40), we get

Using (A.36), (A.41) leads to lim inf t → ∞ ⁡ ln ⁡ ( y ⁢ ( t , ω ) ) t ≥ ϕ 2 ⁢ ( x * + ϵ 1 ) > 0 . Hence, lim t → ∞ ⁡ y ⁢ ( t , ω ) = ∞ , which contradicts y ⁢ ( t , ω ) ≤ 1 . This completes the proof of the assertion lim inf t → ∞ ⁡ x ⁢ ( t ) < x * .

Relation (A.33) means that y ⁢ ( t ) is always oscillating between 0 and 1. That is, the financial crisis is persistent. The third equation of (3.1) can be rewritten as

Hence, using (A.33), we obtain the asymptotic behavior of the economic growth. It oscillates almost surely around the value ( λ 0 ⁢ x * τ δ 0 + p ) 1 1 - α . This completes the proof of Proposition 3.6. ∎