Goran Peskir

A Change-of-Variable Formula with Local Time on Surfaces

Let \(X = (X_t)_{t \geq 0}\) be a continuous semimartingale and let \(b: \mathbb{R}_+ \rightarrow \mathbb{R}\) be a continuous function of bounded variation. Setting \(C = \{(t, x) \in \mathbb{R} + \times \mathbb{R} | x b(t)\}\) suppose that a continuous function \(F: \mathbb{R}_+ \times \mathbb{R} \rightarrow \mathbb{R}\) is given such that F is C1,2 on \(\bar{C}\) and F is \(C^{1,2}\) on \(\bar{D}\). Then the following change-of-variable formula holds: \(\eqalign{ F(t,X_t) = F(0,X_0)+\int_0^{t} {1 \over 2} (F_t(s, X_s+) + F_t(s,X_s-)) ds\cr + \int_0^t {1 \over 2} (F_x(s,X_s+) + F_x(s,X_s-))dX_s\cr + {1 \over 2} \int_0^t F_{xx} (s,X_s)I (X_s \neq b(s)) d \langle X, X \rangle_s\cr + {1 \over 2} \int_0^t (F_x(s,X_s+)-F_x(s,X_s-)) I(X_s = b(s)) d\ell_{s}^{b} (X),\cr} \) where \(\ell_{s}^{b}(X)\) is the local time of X at the curve b given by \(\ell_{s}^{b}(X) = \mathbb{P} - \lim_{\varepsilon \downarrow 0} {1 \over 2 \varepsilon} \int_0^s I(b(r)- \varepsilon < X_r < b(r) + \varepsilon) d \langle X, X \rangle_{r} \) and \(d\ell_{s}^{b}(X)\) refers to the integration with respect to \(s \mapsto \ell_{s}^{b}(X)\). A version of the same formula derived for an Ito diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping.

The Russian option: Finite horizon

Abstract.We show that the optimal stopping boundary for the Russian option with finite horizon can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation (an explicit formula for the arbitrage-free price in terms of the optimal stopping boundary having a clear economic interpretation). The results obtained stand in a complete parallel with the best known results on the American put option with finite horizon. The key argument in the proof relies upon a local time-space formula.

Solving the Poisson Disorder Problem

The Poisson disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ‘disorder’ when the intensity of an observed Poisson process changes from one value to another. Partial answers to this question are known to date only in some special cases, and the main purpose of the present paper is to describe the structure of the solution in the general case. The method of proof consists of reducing the initial (optimal stopping) problem to a free-boundary differential-difference problem. The key point in the solution is reached by specifying when the principle of smooth fit breaks down and gets superseded by the principle of continuous fit. This can be done in probabilistic terms (by describing the sample path behaviour of the a posteriori probability process) and in analytic terms (via the existence of a singularity point of the free-boundary equation).

Stopping Brownian Motion Without Anticipation as Close as Possible to Its Ultimate Maximum

Let

The law of the supremum of a stable Lévy process with no negative jumps

B=(B_t)_{0 \le t \le 1}

Optimal Stopping Games for Markov Processes

be the standard Brownian motion started at 0, and let

The trap of complacency in predicting the maximum

S_t=

Maximal inequalities for the Ornstein-Uhlenbeck process

\max_{ 0 \le r \le t} B_r

The Wiener Sequential Testing Problem with Finite Horizon

for