A new proof for the conditions of Novikov and Kazamaki

Convergence of local supermartingales and Novikov-Kazamaki type conditions for processes with jumps

We characterize the event of convergence of a local supermartingale. Conditions are given in terms of its predictable characteristics and quadratic variation. The notion of extended local integrability plays a key role. We then apply these characterizations to provide a novel proof for the sufficiency and necessity of Novikov-Kazamaki type conditions for the martingale property of nonnegative local martingales with jumps.

When a Stochastic Exponential is a True Martingale. Extension of a Method of Bene^s

Let $\mathfrak{z}$ be a stochastic exponential, i.e., $\mathfrak{z}_t=1+\int_0^t\mathfrak{z}_{s-}dM_s$, of a local martingale $M$ with jumps $\triangle M_t>-1$. Then $\mathfrak{z}$ is a nonnegative local martingale with $\E\mathfrak{z}_t\le 1$. If $\E\mathfrak{z}_T= 1$, then $\mathfrak{z}$ is a martingale on the time interval $[0,T]$. Martingale property plays an important role in many applications. It is therefore of interest to give natural and easy verifiable conditions for the martingale property. In this paper, the property $\E\mathfrak{z}_{_T}=1$ is verified with the so-called linear gro…

A simple proof of a result of A. Novikov

We give simple proofs that for a continuous local martingale M_t: 1) \liminf_{ε->0} ε\log Ee^{(1-ε) <M>_\infty /2} < \infty ==> E\exp(M_\infty - <M>_\infty /2) = 1, 2) \liminf_{ε->0} ε\log\sup_{t>=0} Ee^{(1-ε)M_t/2} < \infty ==> E\exp(M_\infty - <M>_\infty /2) = 1 .

A few comments on a result of A. Novikov and Girsanov's theorem

We give a simple proof that for a continuous local martingale $M_{t}$ $$ \liminf_{\varepsilon\downarrow0}\varepsilon \log Ee^{(1-\varepsilon) \langle M\rangle_{\infty}/2}<\infty \Longrightarrow E\exp(M_{\infty}-\langle M\rangle_{\infty}/2)=1. $$

从CKLS过程到CIR型和OU型过程：利用二次可导映射和广义Girsanov定理

From CKLS Process to CIR-type Process: Using a Twice-differentiable Mapping and Generalized Girsanov's Theorem

本文通过二次可导映射和广义Girsanov定理，将CKLS模型转换为CIR和OU过程，修正了原有模型中的错误，并建立了金融应用中的有效性条件。

A Weak Convergence Criterion Constructing Changes of Measure

Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability conditions (such as the well-known Novikov condition). The weak convergence approach that we propose allows to replace integrability conditions by a suitable tightness condition. We then provide several applications of this approach ranging from simplified proofs of classical results to characterizations of processes conditioned on first passage time events and…