Chekhov is not the kind of writer who would depict such a character change in a middle-aged man like Gurov. He knew that in real life people rarely if ever change that radically. What happens is that the cynical and corrupt Gurov meets and falls in love with the much younger Anna Sergeyevna who is naive and innocent and, as often happens in love affairs, the two exchange character traits, so that each becomes a different person by a sort of spiritual infection. He becomes somewhat more like her and she becomes somewhat more like him. An example of how Anna changes is that she is willing to deceive her husband regularly in the same way that Gurov has been cheating on his wife. But neither Gurov nor Anna is willing to "risk disgrace and divorce for his [or her] grand passion." Chekhov was a physician, and he often sounds more like a diagnostician than a poet. He had years of experience treating patients whom he was unable to cure and had to watch die. He sounds this way in "The Lady with the Pet Dog." He can't help these two people. Their case is terminal. There is no solution, radical, romantic, or otherwise. At the end of the long story the author characteristically leaves them with their problem unresolved and unresolvable.
And it seemed as though in a little while the solution would be found, and then a new and splendid life would begin; and it was clear to both of them that they had still a long, long road before them, and that the most complicated and difficult part of it was only just beginning.
This is an example of what is so splendid about Anton Chekhov and why so many contemporary writers have idolized and imitated him. We have gone beyond the Romantic Age. Readers won't accept romantic solutions to problems by ordinary people. Gurov knows he can't get a divorce and run off with a woman who is about the same age as his daughter. They would have nothing to live on. They are imprisoned in their environments and in their marriages. They both realize that she would feel remorse for deserting her children, if not for deserting her husband. In time her naive infatuation would dissipate and she would see the aging Gurov for what he is--and what he knows himself to be. As bad as their situation is at the end of the story, it is probably better than any such radical solution as divorce. They are just two people having an affair. It is complicated by the fact that they live far away from each other, but it is still just one of the countless extramarital affairs beginning, continuing, or terminating in many parts of the world. In time theirs will probably come to a natural end and only remain with them as a bittersweet memory. Gurov will have changed, no doubt. He won't be such a womanizer, and he may be more understanding of people in general and of women in particular. Anna will have changed at least in so far as she has lost some of her naivete and idealism and will probably not feel guilty about her past sinfulness but rather will treasure the memory.
Raymond Carver was one of the many American writers influenced by Chekhov. He had a picture of Chekhov pinned to the wall above his writing desk. Many of Carver's stories end without a resolution to the focal character's problem. The problem itself seems to be the conclusion. A good example is Carver's "Where I'm Calling From." But there are many others included in the collection which takes its title from that marvelous short story.