STRATHMORE UNIVERSITY
SCHOOL OF FINANCE AND APPLIED ECONOMICS
Bachelor of Business Science – Actuarial Science, Finance & Financial Economics
BSM 2215: INTRODUCTION TO STOCHASTIC MODELLING
Hitting and Exit-Times of a Stochastic Process
Syllabus objectives:
The student should be able to:
1. Define a Stopping Time
2. Explain and apply Doob’s Optional Stopping theorem
3. Use Doob’s Optional Stopping theorem to calculate the probability of Gambler’s Ruin.
4. Use Doob’s Optional Stopping theorem to calculate the expected duration of Gambler’s
Ruin.
5. Use Doob’s Optional Stopping theorem to calculate the probability of Exit from a
standard Brownian from an interval.
6. Use Doob’s Optional Stopping theorem to calculate the expected duration of Exit from a
standard Brownian from an interval.
Stopping Time
When you are playing a gambling game, you have to decide when to stop, when you are
speculating on the price of a share you have to decide when to sell or buy, or when you are
holding an American option you must decide when to exercise it. Normally, this decision is
based on the occurrence of an event happening a random time 𝜏 before or on time 𝑡 therefore
(𝜏 ≤ 𝑡) hence (𝜏 ≤ 𝑡) ∈ 𝐹( .
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Doob’s Optional stopping theorem
Let T be a stopping time and M a Martingale, then:
𝐸 𝑀+ = 𝐸 𝑀As long as any of the following conditions is satisfied:
1. T is bounded
2. |𝑀(/0 − 𝑀( | ≤ 𝐾
3. 𝐸[𝑇] < ∞
Applications of the Doob’s Optional Stopping Theorem
Gambler’s Ruin problem
Let us consider two players A and B. They toss a fair coin. If the coin is a head, A wins 1 dollar
from B, otherwise A losses 1 dollar to B. Given that A and B start with 𝑎 and 𝑏 initially. We are
interested in the following:
1. The wealth process of player A
2. The probability that A wins all the money from B
3. Probability that A loses all the money to B
4. The duration of the game
The Wealth process of player A
The wealth process of player A, the wealth of player A after the 𝑛 − 𝑡ℎ 𝑔𝑎𝑚𝑒 can be modelled
as follows:
𝑈A = 𝑎 +
Where: 𝑋D =
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A
DE0
𝑋D
1 𝐴 𝑤𝑖𝑛𝑠 1
−1 𝐴 𝑙𝑜𝑠𝑒𝑠 1
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The wealth process of player A is a Martingale
Proof:
𝑈A/0 = 𝑎 +
A/0
DE0
𝑋D = 𝑈A + 𝑋A/0
𝐸 𝑈A/0 𝐹A = 𝐸 𝑈A + 𝑋A/0 𝐹A
Applying measurability
= 𝑈A + 𝐸 𝑋A/0 𝐹A
Applying independence
= 𝑈A + 𝐸[𝑋A/0 ]
𝐸 𝑋A/0 = 1×0.5 + −1×0.5 = 0
= 𝑈A + 0 = 𝑈A
Hence the Wealth process of player A is a Martingale
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Another Martingale associated with Player A’s wealth process is the following:
𝑀A = (𝑈A − 𝑎)Q − 𝑛
Proof:
𝑀A/0 = (𝑈A/0 − 𝑎)Q − (𝑛 + 1)
= (𝑈A + 𝑋A/0 − 𝑎)Q − 𝑛 − 1
= 𝑈A + 𝑋A/0
Q
− 2𝑎 𝑈A + 𝑋A/0 + 𝑎Q − 𝑛 − 1
= 𝑈A Q + 2𝑈A 𝑋A/0 + 𝑋A/0 Q − 2𝑎 𝑈A + 𝑋A/0 + 𝑎Q − 𝑛 − 1
𝐸[𝑈A Q + 2𝑈A 𝑋A/0 + 𝑋A/0 Q − 2𝑎 𝑈A + 𝑋A/0 + 𝑎Q − 𝑛 − 1|𝐹A ]
Applying measurability, linearity and constant property of conditional expectation
𝐸[𝑈A Q + 2𝑈A 𝑋A/0 + 𝑋A/0 Q − 2𝑎 𝑈A + 𝑋A/0 + 𝑎Q − 𝑛 − 1|𝐹A ]
𝑈A Q + 2𝑈A 𝐸 𝑋A/0 + 𝐸 𝑋A/0 Q 𝐹A − 2𝑎𝑈A − 2𝑎𝐸 𝑋A/0 𝐹A + 𝑎Q − 𝑛 − 1
Applying independence
𝑈A Q + 2𝑈A 𝐸[𝑋A/0 ] + 𝐸[𝑋A/0 Q ] − 2𝑎𝑈A − 2𝑎𝐸[𝑋A/0 ] + 𝑎Q − 𝑛 − 1
𝐸 𝑋A/0 = 1×0.5 + −1×0.5 = 0
𝐸 [𝑋A/0 Q = 1Q ×0.5 + (−1)Q ×0.5 = 1
𝑈A Q + 1 − 2𝑎𝑈A + 𝑎Q − 𝑛 − 1
𝑈A Q − 2𝑎𝑈A + 𝑎Q − 𝑛 = 𝑈A − 𝑎
Q
− 𝑛 = 𝑈A
Hence it is a Martingale
Gambler Ruin Probability:
Since 𝐸[|𝑈A/0 − 𝑈A |] < 1 we can apply the Doob’s optional stopping theorem
Define 𝜏 = inf {𝑡 > 0|𝑈+ = 𝑎 + 𝑏 𝑜𝑟 𝑈+ = 0}
𝐸 𝑈+ = 𝐸 𝑈- = 𝑎
𝐸 𝑈+ = 𝐸(𝑈+ |𝐴 𝑤𝑖𝑛𝑠 𝑎𝑙𝑙 𝑓𝑟𝑜𝑚 𝐵)𝑃(𝑈+ = 𝑎 + 𝑏) + 𝐸(𝑈+ |𝐴 𝑖𝑠 𝑟𝑢𝑖𝑛𝑒𝑑)𝑃(𝑈+ = 0)
𝑎 = 𝑎 + 𝑏 ×𝑃 𝑈+ = 𝑎 + 𝑏 + (0)×𝑃(𝑈+ = 0)
𝑎 + 𝑏 ×𝑃 𝑈+ = 𝑎 + 𝑏 = 𝑎
Probability that A wins
𝑃 𝑈+ = 𝑎 + 𝑏 =
𝑎
𝑎+𝑏
The probability of Ruin of player A
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But 𝑃 𝑈+ = 𝑎 + 𝑏 + 𝑃 𝑈+ = −𝑎 = 1
𝑃 𝑈+ = 0 = 1 −
𝑎
𝑏
=
𝑎+𝑏 𝑎+𝑏
Duration of a fair game
Define 𝜏 = inf {𝑡 > 0|𝑈+ = 𝑎 + 𝑏 𝑜𝑟 𝑈+ = 0}
𝑀+ = (𝑈+ − 𝑎)Q − 𝜏
𝐸 𝜏 = 𝐸[ 𝑈+ − 𝑎
Q
− 𝑀+ ]
𝐸 𝜏 ≤ 𝑏Q
We can therefore apply the Doob’s optional stopping theorem
𝐸 𝑀+ = 𝐸 𝑀- = 0
𝐸 𝜏 = 𝐸 𝑈+ − 𝑎
𝐸 𝜏 = 𝑎Q ×
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Q
= 𝑎Q 𝑃 𝐴 𝑟𝑢𝑖𝑛𝑒𝑑 + 𝑏 Q 𝑃 𝐴 𝑤𝑖𝑛𝑠 𝑎𝑙𝑙
𝑏
𝑎
(𝑎 + 𝑏)
+ 𝑏Q ×
= 𝑎𝑏
= 𝑎𝑏
𝑎+𝑏
𝑎+𝑏
(𝑎 + 𝑏)
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Hitting and Exit Time of a Standard Brownian Motion from an interval
Let 𝑊( be a standard Brownian Motion with 𝑊- = 0, and 𝑎 > 0 𝑎𝑛𝑑 𝑏 > 0. We want to
calculate two things:
1. The exit probability of the Standard Brownian Motion from an interval
Ø The 𝑃(𝐵+ ≤ −𝑎)
Ø The 𝑃(𝐵+ ≥ 𝑏)
2. The expected duration of the exit time from an interval
The Exit probability of a standard Brownian Motion from an interval
Define 𝜏 = inf {𝑡 > 0 𝐵( ∉ (−𝑎, 𝑏)
𝐵( is a Martingale and is bounded and is bounded, so applying Optional Stopping theorem
𝐸 𝐵- = 0 = 𝐸 𝐵+ = −𝑎×𝑃 𝐵+ = −𝑎 + 𝑏×𝑃 𝐵+ = 𝑏
But 𝑃 𝐵+ = −𝑎 + 𝑃 𝐵+ = 𝑏 = 1
−𝑎[1 − 𝑃 𝐵+ = 𝑏 ] + 𝑏×𝑃 𝐵+ = 𝑏 = 0
−𝑎 + 𝑎𝑃 𝐵+ = 𝑏 + 𝑏×𝑃 𝐵+ = 𝑏
𝑎𝑃 𝐵+ = 𝑏 + 𝑏×𝑃 𝐵+ = 𝑏 = 𝑎
𝑃 𝐵+ = 𝑏 𝑏 + 𝑎 = 𝑎
𝑎
𝑃 𝐵+ = 𝑏 =
𝑎+𝑏
𝑎
𝑎+𝑏−𝑎
𝑏
𝐵+ = 𝑎 = 1 −
=
=
𝑎+𝑏
𝑎+𝑏
𝑎+𝑏
Expected duration of standard Brownian Motion from an interval
Consider a Martingale 𝑁( = 𝐵( Q − 𝑡
𝑁+ = 𝐵+ Q − 𝜏
𝐸 𝜏 = 𝐸 𝐵+ Q − 𝐸[𝑁+ ]
Since 𝐵+ Q is bounded 𝐸 𝜏 < ∞ has upper bound and hence the Optional stopping Theorem
applies
𝐸 𝑁- = 𝐸 𝑁+ = 0
𝑎Q ×
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𝑏
𝑎
+ 𝑏Q ×
=0
𝑎+𝑏
𝑎+𝑏
𝑎𝑏(𝑎 + 𝑏)
= 𝑎𝑏
𝑎+𝑏
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