Rough descriptions of ruin for a general class of surplus processes

1998 ◽  
Vol 30 (04) ◽  
pp. 1008-1026 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Stochastic Process ◽  
General Class ◽  
Positive Real Number ◽  
Ruin Probabilities ◽  
Positive Real ◽  
Exponential Estimates ◽  
Large Deviations Theory ◽  
Time Of Ruin ◽  
The Time Of Ruin ◽  
Time Point

Let {Y n | n = 1, 2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Y n > M} (T = +∞ if Y n ≤ M for n = 1, 2,…). Using the techniques of large deviations theory we obtain rough exponential estimates for ruin probabilities for a general class of processes. Special attention is given to the probability that ruin occurs up to a certain time point. We also generalize the concept of the safety loading and consider its importance to ruin probabilities.

Rough descriptions of ruin for a general class of surplus processes

1998 ◽  
Vol 30 (4) ◽  
pp. 1008-1026 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Stochastic Process ◽  
General Class ◽  
Positive Real Number ◽  
Ruin Probabilities ◽  
Positive Real ◽  
Exponential Estimates ◽  
Large Deviations Theory ◽  
Time Of Ruin ◽  
The Time Of Ruin ◽  
Time Point

Let {Yn | n = 1, 2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Yn > M} (T = +∞ if Yn ≤ M for n = 1, 2,…). Using the techniques of large deviations theory we obtain rough exponential estimates for ruin probabilities for a general class of processes. Special attention is given to the probability that ruin occurs up to a certain time point. We also generalize the concept of the safety loading and consider its importance to ruin probabilities.

Large deviations for the time of ruin

1999 ◽  
Vol 36 (3) ◽  
pp. 733-746 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Large Deviations ◽  
Positive Real Number ◽  
Rate Function ◽  
Positive Real ◽  
Large Deviations Principle ◽  
Borel Sets ◽  
Time Of Ruin ◽  
The Family ◽  
The Time Of Ruin ◽  
Simulation Problem

Let {Yn | n=1,2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Yn > M} (T = +∞ if Yn ≤ M for n=1,2,…). We are interested in the ruin probabilities for large M. Define the family of measures {PM | M > 0} by PM(B) = P(T/M ∊ B) for B ∊ ℬ (ℬ = Borel sets of ℝ). We prove that for a wide class of processes {Yn}, the family {PM} satisfies a large deviations principle. The rate function will correspond to the approximation P(T/M ≈ x) ≈ P(Y⌈xM⌉/M ≈ 1) for x > 0. We apply the result to a simulation problem.

Rough limit results for level-crossing probabilities

1994 ◽  
Vol 31 (2) ◽  
pp. 373-382 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Real Number ◽  
Rate Of Convergence ◽  
Generating Functions ◽  
Positive Real Number ◽  
Level Crossing ◽  
Positive Real ◽  
Large Deviations Theory ◽  
Crossing Probabilities

Let Y1, Y2, · ·· be a stochastic process and M a positive real number. Define TM = inf{n | Yn > M} (TM = + ∞ if for n = 1, 2, ···)· We are interested in the probabilities P(TM <∞) and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn).

Large deviations for the time of ruin

1999 ◽  
Vol 36 (03) ◽  
pp. 733-746 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Large Deviations ◽  
Positive Real Number ◽  
Rate Function ◽  
Positive Real ◽  
Large Deviations Principle ◽  
Borel Sets ◽  
Time Of Ruin ◽  
The Family ◽  
The Time Of Ruin ◽  
Simulation Problem

Let {Y n | n=1,2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Y n &gt; M} (T = +∞ if Y n ≤ M for n=1,2,…). We are interested in the ruin probabilities for large M. Define the family of measures {P M | M &gt; 0} by P M (B) = P(T/M ∊ B) for B ∊ ℬ (ℬ = Borel sets of ℝ). We prove that for a wide class of processes {Y n }, the family {P M } satisfies a large deviations principle. The rate function will correspond to the approximation P(T/M ≈ x) ≈ P(Y ⌈xM⌉/M ≈ 1) for x &gt; 0. We apply the result to a simulation problem.

Rough limit results for level-crossing probabilities

1994 ◽  
Vol 31 (02) ◽  
pp. 373-382 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Real Number ◽  
Rate Of Convergence ◽  
Generating Functions ◽  
Positive Real Number ◽  
Level Crossing ◽  
Positive Real ◽  
Large Deviations Theory ◽  
Crossing Probabilities

Let Y 1, Y 2, · ·· be a stochastic process and M a positive real number. Define TM = inf{n | Yn &gt; M} (TM = + ∞ if for n = 1, 2, ···)· We are interested in the probabilities P(TM &lt;∞) and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn ).

Stochastic differential equations for ruin probabilities

1995 ◽  
Vol 32 (01) ◽  
pp. 74-89 ◽  
Author(s):  
Christian Max Møller
Keyword(s):  
Differential Equations ◽  
Point Processes ◽  
Diffusion Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Ruin Probabilities ◽  
Probability Of Ruin ◽  
Time Of Ruin ◽  
Markov Structure ◽  
The Time Of Ruin

The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin. Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon.

scholarly journals Power estimates for ruin probabilities

2005 ◽  
Vol 37 (03) ◽  
pp. 726-742 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Time Horizon ◽  
Heavy Tails ◽  
Random Variables ◽  
Partial Sums ◽  
Ruin Probabilities ◽  
Infinite Time ◽  
Infinite Time Horizon ◽  
Time Of Ruin ◽  
The Time Of Ruin ◽  
Crude Estimate

Let X 1, X 2,… be real-valued random variables. For u&gt;0, define the time of ruin T = T(u) by T = inf{n: X 1+⋯+X n &gt;u} or T=∞ if X 1+⋯+X n ≤u for every n = 1,2,…. We are interested in the ruin probabilities of general processes {X n } for large u. In the presence of heavy tails, one often finds power estimates. Our objective is to specify the associated powers and provide the crude estimate P(T≤xu)≈u −R(x) for large u, for a given x∈ℝ. The rate R(x) will be described by means of tails of partial sums and maxima of {X n }. We also extend our results to the case of the infinite time horizon.

Stochastic differential equations for ruin probabilities

1995 ◽  
Vol 32 (1) ◽  
pp. 74-89 ◽  
Author(s):  
Christian Max Møller
Keyword(s):  
Differential Equations ◽  
Point Processes ◽  
Diffusion Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Ruin Probabilities ◽  
Probability Of Ruin ◽  
Time Of Ruin ◽  
Markov Structure ◽  
The Time Of Ruin

The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin.Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon.

scholarly journals Power estimates for ruin probabilities

2005 ◽  
Vol 37 (3) ◽  
pp. 726-742 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Time Horizon ◽  
Heavy Tails ◽  
Random Variables ◽  
Partial Sums ◽  
Ruin Probabilities ◽  
Infinite Time ◽  
Infinite Time Horizon ◽  
Time Of Ruin ◽  
The Time Of Ruin ◽  
Crude Estimate

Let X1, X2,… be real-valued random variables. For u>0, define the time of ruin T = T(u) by T = inf{n: X1+⋯+Xn>u} or T=∞ if X1+⋯+Xn≤u for every n = 1,2,…. We are interested in the ruin probabilities of general processes {Xn} for large u. In the presence of heavy tails, one often finds power estimates. Our objective is to specify the associated powers and provide the crude estimate P(T≤xu)≈u−R(x) for large u, for a given x∈ℝ. The rate R(x) will be described by means of tails of partial sums and maxima of {Xn}. We also extend our results to the case of the infinite time horizon.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

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