In this section, we study the first hitting time of the solution of an uncertain differential equation, and give the uncertainty distributions in different cases.
First hitting time of strictly increasing function of the solution
Theorem 13.
Let X
t
and be the solution and α-path of the uncertain differential equation
with an initial value X
0, respectively. Given a strictly increasing function J(x), and a level z > J(X
0), the first hitting time τ
z
that J(X
t
) reaches z has an uncertainty distribution
Proof.
Write
Since J(x) is a strictly increasing function, we have
By Theorem 4 and the monotonicity of uncertain measure, we have
It follows from the duality axiom of uncertain measure that
This completes the proof. □
For a strictly increasing function J(x), in order to calculate the uncertainty distribution Ψ(s) of the first hitting time τ
z
that J(X
t
) reaches z when J(X
0) < z, we design a numerical method as below.
Step 1: Fix ε as the accuracy, and fix h as the step length. Set N = s/h.
Step 2: Employ the recursion formula
for N times, and calculate If
then return 1 − ε and stop.
Step 3: Employ the recursion formula
for N times, and calculate If
then return ε and stop.
Step 4: Set α
1 = ε, α
2 = 1 − ε.
Step 5: Set α = (α
1 + α
2)/2.
Step 6: Employ the recursion formula
for N times, and calculate If
then set α
1 = α. Otherwise, set α
2 = α.
Step 7: If |α
2 − α
1| ≤ ε, then return 1 − α and stop. Otherwise, go to Step 5.
Theorem 14.
Let X
t
be the solution of an uncertain differential equation dX
t
= f(t,X
t
)dt + g(t,X
t
)dC
t
with an initial value X
0. Assume X
t
has an uncertainty distribution Φ
t
(x) at each time t. Then given a strictly increasing function J(x) and a level z > J(X
0), the first hitting time τ
z
that J(X
t
) reaches z has an uncertainty distribution
Proof.
Since the event {τ
z
≤ s} is equivalent to the event
provided z > J(X
0), it follows from Theorem 6 that
This completes the proof. □
Theorem 15.
Let X
t
and be the solution and α-path of the uncertain differential equation
with an initial value X
0, respectively. Given a strictly increasing function J(x) and a level z < J(X
0), the first hitting time τ
z
that J(X
t
) reaches z has an uncertainty distribution
Proof.
Write
Then
By Theorem 4 and the monotonicity of uncertain measure, we have
It follows from the duality axiom of uncertain measure that
This completes the proof. □
For a strictly increasing function J(x), in order to calculate the uncertainty distribution ϒ(s) of the first hitting time τ
z
that J(X
t
) reaches z when J(X
0) > z, we design a numerical method as below.
Step 1: Fix ε as the accuracy, and fix h as the step length. Set N = s/h.
Step 2: Employ the recursion formula
for N times, and calculate If
then return 1 − ε and stop.
Step 3: Employ the recursion formula
for N times, and calculate If
then return ε and stop.
Step 4: Set α
1 = ε, α
2 = 1 − ε.
Step 5: Set α = (α
1 + α
2)/2.
Step 6: Employ the recursion formula
for N times, and calculate If
then set α
1 = α. Otherwise, set α
2 = α.
Step 7: If |α
2 − α
1| ≤ ε, then return α and stop. Otherwise, go to Step 5.
Theorem 16.
Let X
t
be the solution of an uncertain differential equation dX
t
= f(t,X
t
)dt + g(t,X
t
)dC
t
with an initial value X
0. Assume X
t
has an uncertainty distribution Φ
t
(x) at each time t. Then given a strictly increasing function J(x) and a level z < J(X
0), the first hitting time τ
z
that J(X
t
) reaches z has an uncertainty distribution
Proof.
Since the event {τ
z
≤ s} is equivalent to the event
provided z < J(X
0), it follows from Theorem 10 that
This completes the proof. □
First hitting time of strictly decreasing function of the solution
Theorem 17.
Let X
t
and be the solution and α-path of the uncertain differential equation
with an initial value X
0, respectively. Given a strictly decreasing function J(x), and a level z > J(X
0), the first hitting time τ
z
vthat J(X
t
) reaches z has an uncertainty distribution
Proof.
Write
Since J(x) is a strictly decreasing function, we have
By Theorem 4 and the monotonicity of uncertain measure, we have
It follows from the duality axiom of uncertain measure that
This completes the proof. □
Theorem 18.
Let X
t
be the solution of an uncertain differential equation dX
t
= f(t,X
t
)dt + g(t,X
t
)dC
t
with an initial value X
0. Assume X
t
has an uncertainty distribution Φ
t
(x) at each time t. Then given a strictly decreasing function J(x) and a level z > J(X
0), the first hitting time τ
z
that J(X
t
) reaches z has an uncertainty distribution
Proof.
Since the event {τ
z
≤ s} is equivalent to the event
provided z > J(X
0), it follows from Theorem 10 that
This completes the proof. □
Theorem 19.
Let X
t
and be the solution and α-path of the uncertain differential equation
with an initial value X
0, respectively. Given a strictly decreasing function J(x) and a level z < J(X
0), the first hitting time τ
z
that J(X
t
) reaches z has an uncertainty distribution
Proof.
Write
Then
By Theorem 4 and the monotonicity of uncertain measure, we have
It follows from the duality axiom of uncertain measure that
This completes the proof. □
Theorem 20.
Let X
t
be the solution of an uncertain differential equation dX
t
= f(t,X
t
)dt + g(t,X
t
)dC
t
with an initial value X
0. Assume X
t
has an uncertainty distribution Φ
t
(x) at each time t. Then given a strictly decreasing function J(x) and a level z < J(X
0), the first hitting time τ
z
that J(X
t
) reaches z has an uncertainty distribution
Proof.
Since the event {τ
z
≤ s} is equivalent to the event
provided z < J(X
0), it follows from Theorem 6 that
This completes the proof. □